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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1

Michael Winkler
Published Online: 2019-06-16 | DOI: https://doi.org/10.1515/anona-2020-0013

Abstract

In bounded n-dimensional domains Ω, the Neumann problem for the parabolic equation

ut=(A(x,t)u)+(b(x,t)u)f(x,t,u)+g(x,t)(*)

is considered for sufficiently regular matrix-valued A, vector-valued b and real valued g, and with f representing superlinear absorption in generalizing the prototypical choice given by f(⋅, ⋅, s) = sα with α > 1. Problems of this form arise in a natural manner as sub-problems in several applications such as cross-diffusion systems either of Keller-Segel or of Shigesada-Kawasaki-Teramoto type in mathematical biology, and accordingly a natural space for initial data appears to be L1(Ω).

The main objective thus consists in examining how far solutions can be constructed for initial data merely assumed to be integrable, with major challenges potentially resulting from the interplay between nonlinear degradation on the one hand, and the possibly destabilizing drift-type action on the other in such contexts. Especially, the applicability of well-established methods such as techniques relying on entropy-like structures available in some particular cases, for instance, seems quite limited in the present setting, as these typically rely on higher initial regularity properties.

The first of the main results shows that in the general framework of (*), nevertheless certain global very weak solutions can be constructed through a limit process involving smooth solutions to approximate variants thereof, provided that the ingredients of the latter satisfy appropriate assumptions with regard to their stabilization behavior.

The second and seemingly most substantial part of the paper develops a method by which it can be shown, under suitably stregthened hypotheses on the integrability of b and the degradation parameter α, that the solutions obtained above in fact form genuine weak solutions in a naturally defined sense. This is achieved by properly exploiting a weak integral inequality, as satisfied by the very weak solution at hand, through a testing procedure that appears to be novel and of potentially independent interest.

To underline the strength of this approach, both these general results are thereafter applied to two specific cross-diffusion systems. Inter alia, this leads to a statement on global solvability in a logistic Keller-Segel system under the assumption α > 2n+4n+4 on the respective degradation rate which seems substantially milder than any previously found condition in the literature. Apart from that, for a Shigesada-Kawasaki-Teramoto system some apparently first results on global solvability for L1 initial data are derived.

Keywords: rough initial data; generalized solutions; cross-diffusion

MSC 2010: 35D30 (primary); 35K55; 92C17; 35Q92 (secondary)

1 Introduction

A common feature of numerous evolution equations stemming from population models in mathematical biology is the appearance of superlinear degradation terms. In applications typically interpreted, depending on the respective modeling approach, as accounting for diminution due to competition, or as more generally representing abilities of systems to spontaneously prevent overcrowding, such expressions typically arise in the form of algebraic zero-order absorption terms. In the simplest case combined merely with diffusion and thus resulting in semilinear heat equations such as

ut=Δu+λuμ|u|α1u,α>1,λR,μ>0,(1.1)

degradation mechanisms of this type usually provide additional dissipation resulting in accordingly enhanced relaxation features. A favorable mathematical effect thereof is that despite their nonlinear character, such absorptive nonlinearities do not essentially counteract existence theories; in fact, sufficiently elaborate analysis shows that the superlinear damping in (1.1) can be used to even expand the well-known solution theory for the heat equation so as to construct solutions even for very singular initial data with regularity properties far below integrability (see [4, 28, 29, 43] and the detailed discussion in the latter, for instance).

That this situation may substantially change when such absorption interacts with further and possibly destabilizing mechanisms is indicated by findings on extensions of (1.1) to systems involving cross-diffusion, such as the logistic Keller-Segel system ([15])

ut=Δu(uv)+λuμuα,τvt=Δvv+u,(1.2)

or the Shigesada-Kawasaki-Teramoto system ([25])

ut=d1Δu+a11Δu2+a12Δ(uv)+μ1u(1ua1v),vt=d2Δv+a22Δv2+a21Δ(uv)+μ2v(1va2u).(1.3)

Indeed, the solution theories for both these systems are much less developed than that for e.g. (1.1), which may be viewed as partially reflecting a certain singularity-supporting potential of the respective transport processes therein; drastic caveats in this direction are provided by studies reporting the taxis-driven occurrence of large densities in several versions of (1.2) for α = 2 ([14, 16, 39, 41]), and even detecting finite-time blow-up of some solutions to (1.2) in n-dimensional balls with n ≥ 3, for τ = 0 and α ∈ (1, α0(n)) with some α(n) ∈ (1, 2), even for smooth initial data ([42], cf. also [38]).

Apart from accordingly implied natural limitations, the construction of global solutions to both (1.2) and (1.3) in the literature has been confronted with significant additional and possibly technical challenges, and thus in successful cases been strongly relying on the presence of particular global dissipative features expressed in corresponding energy or at least quasi-energy inequalities. For instance, the discovery of an appropriate Lyapunov-like functional has given rise to a breakthrough in the existence theory, within suitably weak solution concepts, for (1.3) with widely arbitrary parameters therein ([7]), thus complementing and extending results on global solvability in classes of smooth functions but under various types of more or less restrictive assumptions on the system ingredients ([8, 12, 19, 21, 22]). Similarly, the use of certain quasi-energy structures in (1.2) has formed an essential fundament for the construction of global bounded solutions in suitable parameter regimes and in presence of sufficiently regular initial data ([23, 30, 36]).

Beyond the evident circumstance that such structures are commonly quite sensitive with respect to changes in the system ingredients, an apparent application-relevant restraint stems from the observation that a corresponding analysis usually requires the initial data to be regular enough so as to have the associated energy be finite at the initial instant. In the context of (1.3), this leads to the requirement, apparently underrun nowhere in the literature, that u0 := u|t=0 at least be an element of an Llog L-type Zygmund class; as for (1.2), most works even assume continuity of the initial data. Up to one single exception addressing global existence of certain generalized solutions to (1.2) in the simple case τ = 0 with α > 2 − 1n, however, the literature does not provide any result on solvability in parabolic drift-diffusion systems of the form (1.2) or (1.3), to say nothing of providing a generalizing or even unifying point of view, in situations when initial data are merely assumed to be integrable, and thus to comply with essentially minimal requirements meaningful in the context of applications in which ∫ u0 usually plays the role of a total population size.

Main objective: Construction of generalized solutions with initial data in L1.

Methodologically, the main challenges going along with the treatment of less regular initial data seem to be linked to the derivation of appropriate compactness properties of the respective superlinear reaction terms, thereby allowing for suitable limit procedures in conveniently designed approximate problems. Here we especially emphasize that due to the presence of additional drift-type mechanisms therein, the accessibility of cross-diffusion systems like (1.2) and (1.3) to compactness-revealing techniques based on duality arguments, as recently developed to quite a comprehensive extent in frameworks of certain pure reaction-diffusion systems generalizing (1.1) to corresponding multi-component problems ([5, 24]), seems very limited.

Accordingly, a common characteristic feature of virtually all precedent solution constructions for (1.2) and (1.3) consists in asserting equi-integrability properties of the nonlinearities in question by tracking the time evolution of convex functionals of the crucial unknown u, with ∫ u ln u consituting the most frequently seen representative. Due to the absorptive character of degradation, namely, the associated testing procedures, essentially involving increasing functions of u as test functions in the respective first equations, yield favorably signed contributions that involve functionals of u with conveniently fast growth as u → ∞. Indeed, corresponding multiplication by ln u, e.g. in (1.2) resulting in space-time L1 estimates for uα ln u and hence implying suitable (equi-)integrability features of uα, has been at the core of various existence proofs in (1.2) as well as in several related taxis-type systems ([17, 27, 44]); through their mere nature, however, such techniques seem restricted to cases in which, again, not only u0 but even some superlinear functional of u0 is integrable.

The purpose of the present work is to develop an apparently alternative approach toward the construction of generalized solutions, firstly mild enough with regard to the initial data so as to be applicable to data merely belonging to L1, and secondly sufficiently robust in not relying on fragile structures like entropies. We shall accordingly be concerned with a rather general class of systems involving superlinear degradation, possibly furthermore perturbed by drift terms, by subsequently considering the no-flux type parabolic problem

ut=(A(x,t)u)+(b(x,t)u)f(x,t,u)+g(x,t),xΩ,t(0,T),(A(x,t)u)ν+b(x,t)uν=0,xΩ,t(0,T),u(x,0)=u0(x),xΩ,(1.4)

where T ∈ (0, ∞] and Ω ⊂ ℝn is a bounded domain with smooth boundary. Here we assume throughout that the diffusion operator generalizes the Laplacian in that with some positive constants kA and KA,

A=(Aij)i,j{1,...,n}L(Ω×(0,T);Rn×n)is such thatAij(x,t)=Aji(x,t)for all(x,t)Ω×(0,T)andi,j{1,...,n}with(A(x,t)ξ)ξkA|ξ|2for all (x,t,ξ)Ω×(0,T)×Rnand |Aij(x,t)|KAfor all (x,t)Ω×(0,T) and i,j{1,...,n},(1.5)

that the drift coefficient satisfies the crucial square integrability condition

bLloc2(Ω¯×[0,T);Rn),(1.6)

that the nonlinear part of the reaction term,

fC1(Ω¯×[0,T);×[0,)),(1.7)

essentially represents power-type superlinear absorption of the style in (1.2) and (1.3) in satisfying

kfsαf(x,t,s)Kfsαfor all(x,t)Ω×(0,T)and each ss0(1.8)

with some kf > 0, Kf > 0 and α > 1, and that moreover

gLloc1(Ω¯×[0,T))(1.9)

and

u0L1(Ω)isnonnegative.(1.10)

Main results I: Constructing very weak solutions without need for L1 compactness properties of uα.

In view of the above observations on precedent studies, our first objective will consist in examining how far solutions can be obtained even despite possibly lacking estimates ensuring compactness features that allow for standard limit passages in classical weak formulations associated with (1.4). For this purpose, in a first step we shall further develop an approach from [40] by resorting to a solution concept which in its most crucial part concentrates on the function ln (u + 1) and merely requires this quantity to satisfy an integral inequality reflecting a certain supersolution property of ln (u + 1) with respect to its parabolic problem formally corresponding to (1.4); along with a suitable additional mass control from above, this yields a concept which for smooth functions is indeed consistent with classical solvability. The main advantage of this relaxation consists in the circumstance that in comparison to standard notions of weak solvability, such as formulated e.g. in Definition 3.1 below, with respect to the decisive nonlinear parts this will here require significantly reduced integrability and compactness properties only, which we will see to indeed be available in quite a general framework.

More precisely, in this first part we shall adapt a concept originally introduced in [40] for a particular chemotaxis problem, and later on extended to various relatives thereof (see e.g. [3, 34]), in the following manner.

Definition 1.1

Let T ∈ (0, ∞], and suppose that (1.5), (1.6), (1.7), (1.9) and (1.10) hold with some kA > 0 and KA > 0. Then a nonnegative function uLloc1(Ω × [0, T)) will be called a very weak solution of (1.4) in Ω × (0, T) if f(,,u)u+1Lloc1(Ω × [0, T)) and

ln(u+1)Lloc2(Ω¯×[0,T);Rn),(1.11)

if the inequality

0TΩln(u+1)φtΩln(u0+1)φ(,0)0TΩ{(A(x,t)ln(u+1))ln(u+1)}φ0TΩ(A(x,t)ln(u+1))φ+0TΩuu+1(b(x,t)ln(u+1))φ0TΩuu+1b(x,t)φ0TΩf(x,t,u)u+1φ+0TΩg(x,t)u+1φ(1.12)

is valid for each nonnegative φC0(Ω × [0, T)), and if

Ωu(x,t0)dx+0t0Ωf(x,t,u(x,t))dxdtΩu0(x)dx+0t0Ωg(x,t)dxdtfora.e.t0(0,T).(1.13)

Indeed, by straightforward modification of the arguments from [40, Lemma 2.1] and [18, Lemma 2.5], one can readily verify that this concept is consistent with that of classical solvability in the sense that if A, f, g and u are suitably smooth and u solves (1.4) in the very weak sense described below, then in fact u already must be a classical solution.

Now to substantiate our approach toward solvability in the context of a convenient approximation to (1.4), let us further specify our setting by imposing the hypothesis, forming a standing assumption in this general part, that from whatever source we are given nonnegative classical solutions uεC0(Ω × [0, T)) ∩ C2,1(Ω × (0, T)) to the regularized variants of (1.4) specified by

uεt=(Aε(x,t)uε)+(bε(x,t)uε)f(x,t,uε)+gε(x,t),xΩ,t(0,T),(Aε(x,t)uε)ν+bε(x,t)uεν=0,xΩ,t(0,T),uε(x,0)=u0ε(x),xΩ,(1.14)

where ε ∈ (εj)j∈ℕ with some sequence (εj)j∈ℕ ⊂ (0, 1) fulfilling εj ↘ 0 as j → ∞. As for the ingredients herein, in line with the above we will assume that with positive constants kA, KA, kf, KF and s0, without loss of generality coinciding with those introduced above, we have

AεC1(Ω¯×(0,T);Rn×n)is such that(Aε)ij(x,t)=(Aε)ji(x,t)for all(x,t)Ω×(0,T)and i,j{1,...,n}with(Aε(x,t)ξ)ξkA|ξ|2for all (x,t,ξ)Ω×(0,T)×Rnand |(Aε)ij(x,t)|KAfor all (x,t)Ω×(0,T) and i,j{1,...,n},(1.15)

with

AεAa.e. in Ω×(0,T)as ε=εj0,(1.16)

that

bεC1(Ω¯×(0,T);Rn)Lloc2(Ω¯×[0,T);Rn)(1.17)

approaches b in the sense that

bεbin Lloc2(Ω¯×[0,T))as ε=εj0,(1.18)

and that the functions

gεC1(Ω¯×(0,T))Lloc1(Ω¯×[0,T))(1.19)

satisfy

gεgin Lloc1(Ω¯×[0,T))asε=εj0.(1.20)

Finally, the initial data in (1.14) will be subject to the assumptions that

u0εC0(Ω¯)is nonnegative(1.21)

with

u0εu0in L1(Ω)as ε=εj0.(1.22)

The first of our main results, to be achieved in Section 2, then asserts that these approximation properties, and especially the crucial L2 convergence requirement in (1.18), ensure solvability in the considered very weak framework, indeed assuming no more regularity of u0 than merely integrability:

Theorem 1.2

Suppose that (1.5), (1.6), (1.7), (1.8), (1.9) and (1.10) hold for some T ∈ (0, ∞], kA > 0, KA > 0, kf > 0, Kf > 0, s0 > 0 and α > 1, and for ε ∈ (εj)j∈ℕ with some sequence (εj)j∈ℕ ⊂ (0, 1) such that εj ↘ 0 as j → ∞, assume that uεC0(Ω × [0, T)) ∩ C2,1(Ω × (0, T)) is a classical solution of (1.14) with certain Aε, bε, gε and u0ε satisfying (1.15), (1.16), (1.17), (1.18), (1.19), (1.20), (1.21) and (1.22). Then there exist a subsequence (εjk)k∈ℕ and a very weak solution u of (1.4) in Ω × (0, T), in the sense of Definition 1.1 below, such that

uεuinLloc1(Ω¯×[0,T))anda.e.inΩ×(0,T),(1.23)

uεuinLlocα(Ω¯×[0,T))and(1.24)

ln(uε+1)ln(u+1)inLloc2(Ω¯×[0,T))(1.25)

as ε = εjk ↘ 0.

Main results II: Construction of genuine weak solutions by turning weak into strong Lα convergence for sufficiently regular b.

The major step in our analysis thereafter consists in investigating how far despite the mentioned obstacles the solution gained above in fact solves (1.4) in the standard weak sense. In view of (1.8), this essentially amounts to identifying conditions under which the weak convergence statement in (1.24) can be turned into a corresponding strong compactness property, where in accordance with the above discussion, our ambition to avoid further regularity requirements on the initial data apparently reduces the availability of well-established techniques which in related situations have provided equi-integrability features of, say, some family (hj)j∈ℕ by deriving L1 bounds for (Ψ(hj))j∈ℕ with certain superlinearly growing Ψ:ℝ → ℝ ([7, 17, 24, 27]).

In our key step toward circumventing this, we will purely concentrate on the weak supersolution property satisfied by the limit function u due to Theorem 1.2, and the main challenge here will be to create an appropriate testing procedure in the corresponding integral inequality which allows for a rigorous justification of the mass evolution relation

Ωu(,t0)+0t0Ωf(x,t,u)Ωu0+0t0Ωg(x,t),(1.26)

as formally associated with (1.4) even as an identity. Combined with (1.8) and (2.2) this will readily imply that 0t0Ωuαlim infε=εj00t0Ωuεα, and that hence f(⋅, ⋅, uε) → f(⋅, ⋅, u) in L1(Ω × (0, t0)), for suitably many t0 ∈ (0, T). We underline already here that developing (1.26) from the inequality (1.12) will go along with considerable efforts, especially due to the circumstance that (1.12) addresses ln (u + 1) rather than u itself, and that according to the poor regularity information available for u, quite restrictive requirements for the corresponding test functions are in order.

It will turn out in Section 3, however, that under slightly sharpened assumptions on α and the integrability properties of b this can successfully be accomplished, thus leading to the following result.

Theorem 1.3

Suppose that the assumptions from Theorem 1.2 hold, and that furthermore α ≥ 2 and

bLlocq(Ω¯×[0,T);Rn)withsomeq2αα1.(1.27)

Then the limit function obtained in Theorem 1.2 is a weak solution of (1.4) in the sense of Definition 3.1 below.

Application to logistic Keller-Segel systems. To indicate how the above general theory can be employed in the construction of solutions to concrete cross-diffusion systems involving couplings to further quantities, in Sections 4 and 5 we will focus on the two examples (1.2) and (1.3) introduced above; in order to avoid to become too extensive here, we only mention that further applications to several models of biological relevance are possible, including chemotaxis-haptotaxis systems for tumor invasion or coupled chemotaxis-fluid systems, for instance ([2, 6]).

Let us firstly consider the Neumann problem for the relative of (1.2) given by

ut=Δu(uv)+F(u),xΩ,t>0,vt=Δvv+u,xΩ,t>0,uν=vν=0,xΩ,t>0,u(x,0)=u0(x),v(x,0)=v0(x),xΩ,(1.28)

where

FC1([0,)) is such that kFsαF(s)KFsα for allss0(1.29)

with some kF > 0, KF > 0, s0 > 0 and α > 1, and where u0L1(Ω) and v0L2(Ω) are nonnegative, with a particular representative constituted by the classical logistic Keller-Segel system with quadratic degradation, as given by

ut=Δu(uv)+λuμu2,xΩ,t>0,vt=Δvv+u,xΩ,t>0,uν=vν=0,xΩ,t>0,u(x,0)=u0(x),v(x,0)=v0(x),xΩ,(1.30)

for λ ∈ ℝ and μ > 0. It is known from the literature that for initial data additionally satisfying u0C0(Ω) and v0W1,∞(Ω), the latter problem admits global classical solutions when either n ≤ 2 and μ > 0 is arbitrary ([23]), or n ≥ 3 and μ > μ0(λ, Ω) with some μ0(λ, Ω) > 0 ([36]); for arbitrary values of μ > 0 and suitably regular data, global weak solutions have been obtained in [17]. Analytic studies focusing on solvability issues in presence of smaller powers α in the degradation term F from (1.28) and (1.29) apparently go back to [35] where some global generalized solutions could be constructed for a parabolic-elliptic relative under the assumption that

α>21n,(1.31)

with a recent extension to the fully parabolic case (1.28) for smooth initial data achieved in [33].

Now based on an application of Theorem 1.2, some considerable relaxation with regard to both the condition (1.31) and the initial regularity becomes possible, thus leading to a result on solvability in the fully parabolic problem (1.28) not only for initial data merely belonging to L1 × L2, but apart from that also for a range of degradation parameters α apparently not addressed by any existence result in the literature so far:

Theorem 1.4

Let Ω ⊂ ℝn be a bounded domain with smooth boundary, let F satisfy (1.29) with some positive constants kF > 0, KF > 0 and

α>2n+4n+4,(1.32)

and suppose that u0L1(Ω) and v0L2(Ω) are nonnegative. Then there exist nonnegative functions defined on Ω × (0, ∞) which for all T > 0 have the properties that

uL((0,T);L1(Ω))Lα(Ω×(0,T))andvL((0,T);L2(Ω))L2((0,T);W1,2(Ω))L2n+4n(Ω×(0,T)),(1.33)

and that (u, v) forms a very weak solution of (1.28) in Ω × (0, ∞) in the sense that u is a very weak solution on (1.4) in the style of Definition 1.1 with Aij = δij, i, j ∈ {1, …, n}, b := −∇ v, f(⋅, ⋅, s) := −F(s), s ≥ 0, and g := 0, and that

0ΩvφtΩv0φ(,0)=0Ωvφ0Ωvφ+0Ωuφ(1.34)

for all φC0(Ω × [0, ∞)). This solution can be obtained as the limit of classical solutions (uε, vε) to (4.3) below in the sense that there exists (εj)j∈ℕ ⊂ (0, 1) such that εj ↘ 0 as j → ∞ and that uεu and vεv a.e. in Ω × (0, ∞) as ε = εj ↘ 0.

Under slightly stronger assumptions on α and the initial regularity of v, yet retaining the mere requirement u0L1(Ω), we shall next derive from Theorem 1.3 the following result on genuine weak solvability. Here and below, we let A denote the setorial realization of −Δ + 1 under homogeneous Neumann boundary conditions in L2(Ω) with its domain of definition accordingly given by D(A) = {ϕW2,2(Ω) | ϕν = 0 on Ω}.

Theorem 1.5

Let Ω ⊂ ℝn be a bounded domain with smooth boundary, and let (1.29) be valid with some kF > 0, KF > 0 and

αn+22.(1.35)

Then given any nonnegative functions u0 : Ω → ℝ and v0 : Ω → ℝ fulfilling

u0L1(Ω)andv0D(Aβ)withsomeβ(n+24α,12]ifα>n+22,β=12ifα=n+22,(1.36)

one can find nonnegative functions u and v defined on Ω × (0, ∞) which are such that for all T > 0,

uL((0,T);L1(Ω))Lα(Ω×(0,T))L1((0,T);W1,1(Ω))andvL((0,T);D(Aβ))Lq((0,T);W1,q(Ω))foreachq[1,2(n+2)n+24β)ifβ<12,forq=2(n+2)nifβ=12,(1.37)

and which form a weak solution of (1.28) in Ω × (0, ∞) in the sense that (1.34) holds and that u solves (1.4) with A, b, f and g as specified in Theorem 1.4; in particular,

0ΩuφtΩu0φ(,0)=0Ωuφ+0Ωuvφ+0ΩF(u)φ(1.38)

holds for all φC0(Ω × [0, ∞)).

In the particular context of the system (1.30) with quadratic degradation, the latter implies the following.

Corollary 1.6

Let n = 2, λ ∈ ℝ and μ > 0, and suppose that 0 ≤ u0L1(Ω) and 0 ≤ v0W1,2(Ω). Then there exist nonnegative functions u and v on Ω × (0, ∞) such that for any T > 0 we have

uL((0,T);L1(Ω))L2(Ω×(0,T))L1((0,T);W1,1(Ω))andvL((0,T);W1,2(Ω))L4((0,T);W1,4(Ω)),

and that (u, v) solves (1.30) in the weak sense specified in Theorem 1.5.

Application to a Shigesada-Kawasaki-Teramoto type system.

Finally, we briefly address a specific version of the comprehensive model (1.3), reducing the full complexity therein by resorting to a tridiagonal case in which cross-diffusion enters only one of the equations. Up to the exceptional approach based on exploiting global entropies ([7]), such simplifications have been an essential prerequisite in most previous studies on global solvability in the context of (1.3), mainly in frameworks of smooth solutions for smooth initial data ([8, 12, 19, 21, 22, 32]). Specifically, we will focus on the system

ut=d1Δu+a12Δ(uv)+μ1u(1ua1v),xΩ,t>0,vt=d2Δv+a22Δv2+μ2v(1va2u),xΩ,t>0,uν=vν=0,xΩ,t>0,u(x,0)=u0(x),v(x,0)=v0(x),xΩ,(1.39)

and firstly derive from Theorem 1.2 the following existence result for data in L1 × L.

Theorem 1.7

Let Ω ⊂ ℝn be a bounded domain with smooth boundary, let d1, d2 and μ1 be positive and a12, a22, μ2, a1 and a2 be nonnegative, and let 0 ≤ u0L1(Ω) and 0 ≤ v0L(Ω). Then one can find nonnegative functions u and v on Ω × (0, ∞) such that for all T > 0,

uL((0,T);L1(Ω))L2(Ω×(0,T))andvL(Ω×(0,T))L2((0,T);W1,2(Ω)),(1.40)

and such that u is a very weak solution of (1.4) in Ω × (0, ∞) in the sense of Definition 1.1 with Aij(x, t) := (d1 + a12 v(x, t))δij, i, j ∈ {1, …, n}, b(x, t) := a12v(x, t), f(x, t, s) := μ1 sμ1 s2 and g(x, t) := − μ1 a1 u(x, t)v(x, t) for (x, t) ∈ Ω × (0, ∞) and s ≥ 0, and that

0ΩvφtΩv0φ(,0)=d20Ωvφ2a220Ωvvφ+μ20Ωvφμ20Ωv2φμ2a20Ωuvφ(1.41)

for all φC0(Ω × [0, ∞)). Furthermore, letting (uε, vε) denote classical solutions of the approximate problem (4.3) below for ε ∈ (0, 1), with initial data fulfilling (5.2), then with some (εj)j∈ℕ ⊂ (0, 1) satisfying εj ↘ 0 as j → ∞ we have uεu and vεv a.e. in Ω × (0, ∞) as ε = εj ↘ 0.

In order to identify this very weak solution as an actually weak solution by means of Theorem 1.3, we here only need to invest the additional hypothesis that v0 belong to W1,2(Ω).

Theorem 1.8

Let Ω ⊂ ℝn be a bounded domain with smooth boundary, and let d1 > 0, d2 > 0 and μ1 > 0 as well as a12, a22, μ2, a1 and a2 be nonnegative. Then whenever u0L1(Ω) and v0W1,2(Ω) ∩ L(Ω) are nonnegative, there exist nonnegative functions u and v defined on Ω × (0, ∞) which for all T > 0 satisfy (1.40) as well as

uL1((0,T);W1,1(Ω))andvL((0,T);W1,2(Ω))L4((0,T);W1,4(Ω)),(1.42)

and which constitute a weak solution of (1.28) in Ω × (0, ∞) in that (1.40) holds for all φC0(Ω × [0, ∞)), and that u is a weak solution of (1.4) in the sense of Definition 3.1 with A, b, f and g as specified in Theorem 1.7; in particular, we have

0ΩuφtΩu0φ(,0)=d10Ωuφa120Ωvuφa120Ωuvφ+μ10Ωuφμ10Ωu2φμ1a10Ωuvφ(1.43)

for all φC0(Ω × [0, ∞)).

2 Solvability despite lacking strong compactness. Proof of Theorem 1.2

In order to construct a very weak solution by means of a limit procedure involving supposedly given classical solutions of the regularized problems (1.14), let us assume throughout this section that (1.5), (1.6), (1.7), (1.8), (1.9) and (1.10) hold for some T > 0, kA > 0, KA > 0, kf > 0, Kf > 0, s0 > 0 and α > 1, and that furthermore the boundedness and approximation properties formulated in (1.15), (1.17), (1.19), (1.21) and (1.22) are satisfied.

Then a basic but important property can immediately be seen.

Lemma 2.1

For each ε ∈ (εj)j∈ℕ, we have

Ωuε(,t)+0t0Ωf(x,t,uε)=Ωu0ε+0t0Ωgε(x,t)forallt0(0,T).(2.1)

Proof

Thanks to the no-flux boundary condition in (1.14), integrating the first equation therein yields

ddtΩuε=Ωf(x,t,uε)+Ωgε(x,t)for all t(0,T),

which directly leads to (2.1). □

As a consequence of (1.8), under an additional assumption on the positive part of gε, actually weaker than our hypothesis (1.19) on L1 convergence needed later on, Lemma 2.1 entails a first set of yet quite basic a priori estimates.

Lemma 2.2

Assume that beyond the above hypotheses we have

supε(εj)jN0T0Ω(gε)+<forallT0(0,T).(2.2)

Then for any T0 ∈ (0, T) there exists C(T0) > 0 such that

Ωuε(,t)C(T0)forallt(0,T0)andε(εj)jN(2.3)

and

0T0ΩuεαC(T0)forallε(εj)jN(2.4)

as well as

0T0Ω(gε)C(T0)forallε(εj)jN.(2.5)

Proof

To adequately exploit (3.29), on splitting the spatial integral of f(x, t, uε) we use (1.8) to estimate

Ωf(x,t,uε)={uε<s0}f(x,t,uε)+{uεs0}f(x,t,uε)c1(T0)+kf{uεs0}uεα=c1(T0)+kfΩuεαkf{uε<s0}uεαc1(T0)+kfΩuεαc2for all t(0,T0)

with c1(T0) := ∥fL(Ω×(0,T0)×(0,s0)) ⋅ |Ω| and c2 := kf s0α |Ω|. Therefore, (3.29) implies that

Ωuε(,t)+kf0tΩuεα+0tΩ(gε)Ωu0ε+(c1+c2)t+0tΩ(gε)+u0ε+(c1(T0)+c2)T0+0T0Ω(gε)+for all t(0,T0),

whence (2.3), (2.4) and (2.5) result in view of (1.22) and (2.2). □

To achieve further regularity information, especially on spatial gradients, besides the above we will make substantial use of a boundedness assumption on the flux coefficient functions bε which is yet weaker than the hypothesis (1.18) to be imposed in Theorem 1.2, but which already refers to essentially the same topology as the one addressed therein.

Lemma 2.3

Assume that (2.2) holds, and that

supε(εj)jN0T0Ω|bε|2<forallT0(0,T).(2.6)

Then for each T0 ∈ (0, T) there exists C(T0) > 0 such that

0T0Ω|ln(uε+1)|2C(T0)forallε(εj)jN.(2.7)

Proof

On testing (1.14) against 1uε+1 we see that

ddtΩln(uε+1)=Ω1(uε+1)2(Aεuε)uε+Ωuε(uε+1)2bεuεΩf(x,t,uε)uε+1+Ωgεuε+1,(2.8)

and due to (1.15) we know that herein

Ω1(uε+1)2(Aεuε)uεkAΩ|uε|2(uε+1)2.

Since Young’s inequality warrants that

|Ωuε(uε+1)2bεuε|kA2Ω|uε|2(uε+1)2+12kAΩuε2(uε+1)2|bε|2kA2Ω|uε|2(uε+1)2+12kAΩ|bε|2for all t(0,T),

and since again writing c1(T0) := ∥fL(Ω×(0,T0)×(0,s0)) ⋅ |Ω|, by (1.8) we have

Ωf(x,t,uε)uε+1Kf{uεs0}uεαuε+1+c1(T0)KfΩuεα+c1(T0)for all t(0,T0)

and, clearly, also

Ωgεuε+1Ω(gε)for all t(0,T),

from (2.8) it follows that

Ωln(u0ε+1)+kA20tΩ|uε|2uε+1Ωln(uε(,t)+1)+12kA0tΩ|bε|2+Kf0tΩuεα+c1(T0)t+0tΩ(gε)for all t(0,T0).

As evidently ∫Ω ln (u0ε + 1) ≥ 0 and ∫Ω ln (uε(⋅, t) + 1) ≤ ∫Ω uε(⋅, t) for all t ∈ (0, T), by making use of (2.3), (2.6), (2.4) and (2.5) we immediately infer (2.7) from this.

Together with Lemma 2.2, this also entails some regularity in time of ln (uε + 1): □

Lemma 2.4

If (2.2) and (2.6) hold, then for all T0 ∈ (0, T) and each m ∈ ℕ such that m > n2 there exists C(T0, m) > 0 such that

0T0tln(uε(,t)+1)(Wm,2(Ω))dtC(T0,m)forallε(εj)jN.(2.9)

Proof

For fixed t ∈ (0, T) and ϕC(Ω), from (1.14), (1.15) and (1.8) we obtain that

|Ωt(uε(,t)+1)ϕ|=|Ω1(uε+1)2{(Aεuε)uε}ϕΩ1uε+1(Aεuε)ϕ+Ωuε(uε+1)2(bεuε)ϕΩuεuε+1bεϕΩf(x,t,uε)uε+1ϕΩgεuε+1ϕ|KA{Ω|uε|2(uε+1)2}ϕL(Ω)+KA{Ω|uε|2(uε+1)2}12ϕL2(Ω)+{Ω|uε|2(uε+1)2}12{Ω|bε|2}12ϕL(Ω)+{Ω|bε|2}12ϕL2(Ω)+{fL(Ω×(0,T)×(0,s0))|Ω|+Kf{uεs0}uεα}ϕL(Ω)+{Ω(gε)++Ω(gε)}ϕL(Ω).

As Wm,2(Ω) ↪ L(Ω) by assumption on m, by using Young’s inequality we therefore see that with some c1 > 0 we have

tln(uε(,t)+1)(Wm,2(Ω))c1Ω|uε|2(uε+1)2+c1Ω|bε|2+c1Ωuεα+c1Ω(gε)++c1Ω(gε)+c1for allt(0,T)and each ε(εj)jN,

whence (2.9) results upon integrating and applying Lemma 2.3, (2.6), (2.4), (2.2) and (2.5). □

Now the extraction of suitably converging subsequences essentially reduces to applying an Aubin-Lions lemma.

Lemma 2.5

Assume (2.2) and (2.6). Then one can find a subsequence (εjk)k∈ℕ of (εj)j∈ℕ and a nonnegative function uLloc1(Ω × [0, T)) such that (1.23), (1.24) and (1.25) hold as ε = εjk ↘ 0.

Proof

From Lemma 2.3 and Lemma 2.2 it follows that ( ln(uε + 1))ε ∈ (εj)j∈ℕ is bounded in L2((0, T0); W1,2(Ω)) for all T0 ∈ (0, T), while for any fixed integer m > n2, Lemma 2.4 states boundedness of (t ln(uε + 1))ε ∈ (εj)j∈ℕ in L1((0, T0); (Wm,2(Ω))) for any such T0. Therefore, employing an appropriate Aubin-Lions lemma ([31]) yields precompactness of (ln(uε + 1))ε ∈ (εj)j∈ℕ in L2(Ω × (0, T0)) for all T0 ∈ (0, T), whence extracting a suitable subsequence (εjk)k∈ℕ of (εj)j∈ℕ we obtain a nonnegative function u:Ω × (0, T) → ℝ for which both (1.25) and, by strict monotonicity of 0 ≤ ξ ↦ ln (ξ + 1), also uεu a.e. in Ω × (0, T) hold as ε = εjk ↘ 0. As α > 1 and (uε)ε ∈ (εj)j∈ℕ is bounded in Lα(Ω × (0, T0)) for all T0 ∈ (0, T) by Lemma 2.2, it firstly follows from Egorov’s theorem that also (1.24) is valid along this subsequence, and secondly we may conclude from the Vitali convergence theorem that moreover uεu in L1(Ω × (0, T0)) as ε = εjk ↘ 0 for all T0 ∈ (0, T). □

It thus remains to be shown that the obtained limit function solves (1.4) in the spirit of Definition 1.1). By arguments based on Fatou’s lemma and lower semicontinuity of Hilbert space norms with respect to weak convergence, however, the properties asserted by Lemma 2.5 can indeed be identified as sufficient for guaranteeing the integral inequalities (1.12) and (1.13):

Proofc of Theorem 1.2

Taking (εjk)k∈ℕ and u as provided by Lemma 2.5, from the latter we directly obtain that (1.23), (1.24) and (1.25) hold, and that in view of (1.8) also the regularity requirements imposed in Definition 1.1 are satisfied.

For the verification of (1.13), according to (1.23) and the Fubini-Tonelli theorem we fix a null set N ⊂ (0, T) such that for all t0 ∈ (0, T) ∖ N we have uε(⋅, t0) → u(⋅, t0) a.e. in Ω as ε = εjk ↘ 0, whence by Fatou’s lemma and Lemma 2.1,

Ωu(,t0)+0t0Ωf(x,t,u)=Ωu(,t0)+0t0Ωf+(x,t,u)0t0Ωf(x,t,u)lim infε=εjk0{Ωuε(,t0)+0t0Ωf+(x,t,uε)}0t0Ωf(x,t,u)=lim infε=εjk0{Ωuε(,t0)+0t0Ωf(x,t,uε)+0t0Ωf(x,t,uε)}0t0Ωf(x,t,u)=lim infε=εjk0{Ωu0ε+0t0Ωgε+0t0Ωf(x,t,uε)}0t0Ωf(x,t,u)for all t0(0,T)N.(2.10)

Here by (1.22) and (1.20),

Ωu0εΩu0and0t0Ωgε0t0Ωgas ε=εjk0,

and combining (1.23) with the continuity of f we find that

0t0Ωf(x,t,uε)0t0Ωf(x,t,u)as ε=εjk0

by the dominated convergence theorem, because f is bounded in Ω × (0, t0) × (0, ∞) thanks to (1.8). Therefore, (1.13) is a consequence of (2.10), so that it remains to derive (1.12).

To this end, we fix a nonnegative φC0(Ω × [0, T)) and then obtain from (1.14) that the identity

0TΩln(uε+1)φtΩln(u0ε+1)φ(,0)=0TΩ{(Aεln(uε+1))ln(uε+1)}φ0TΩ(Aεln(uε+1))φ+0TΩuεuε+1(bεln(uε+1))φ0TΩuεuε+1bεφ0TΩf(x,t,uε)uε+1φ+0TΩgεuε+1φ(2.11)

is valid for each ε ∈ (εj)j∈ℕ. Here since |ln (ξ1 + 1) − ln(ξ2 + 1)| ≤ |ξ1ξ2| for all ξ1 ≥ 0 and ξ2 ≥ 0, from (1.23) and (1.22) it follows that

0TΩln(uε+1)φtΩln(u0ε+1)φ(,0)0TΩln(u+1)φtΩln(u0+1)φ(,0)(2.12)

as ε = εjk ↘ 0, and (1.23) together with (1.20) ensures that furthermore

0TΩgεuε+1φ0TΩgu+1φas ε=εjk0,(2.13)

because with T0 ∈ (0, T) taken such that φ ≡ 0 on Ω × (T0, T) we have

0T0Ω|gεuε+1gu+1|=0T0Ω|gεguε+1(uε+u)g(uε+1)(u+1)|0T0Ω|gεg|+0T0Ω|uεu|(uε+1)(u+1)|g|

for all ε ∈ (εj)j∈ℕ, and because the majorization |uεu|(uε+1)(u+1)uε(uε+1)(u+1)+u(uε+1)(u+1) ≤ 2 along with the dominated convergence theorem warrants that

0T0Ω|uεu|(uε+1)(u+1)|g|0as ε=εjk0.

Moreover, since

0T0Ω|f(x,t,uε)uε+1|α{uε<s0}|f|αα1+Kfαα1{uεs0}uεαfL(Ω×(0,T0)×(0,s0))αα1|Ω|T0+Kfαα10T0Ωuεαfor all ε(εj)jN

due to (1.8), using Lemma 2.2 and that αα1 > 1 we infer from the accordingly implied equi-integrability property of (f(,,uε)uε1)ε(εj)jN that again thanks to (1.23), f(,,uε)uε+1f(,,u)u+1 in L1(Ω × (0, T0)) and hence

0TΩf(x,t,uε)uε+1φ0TΩf(x,t,u)u+1φas ε=εjk0.(2.14)

We next rely on (1.25) to firstly see that the second summand in (2.11) satisfies

0TΩ(Aεln(uε+1))φ0TΩ(Aln(u+1))φas ε=εjk0,(2.15)

for clearly (1.15) together with (1.16) ensures that AεA in L2(Ω × (0, T0)) as ε = εjk ↘ 0.

We secondly combine (1.25) with the fact that

uεuε+1bεuu+1bin Lloc2(Ω¯×[0,T))as ε=εjk0,(2.16)

the latter resulting from (1.18) and the circumstance that 0 ≤ uεuε+11anduεuε+1uu+1 a.e. in Ω × (0, T) as ε = εjk ↘ 0 by (1.23), through a well-known stabilization feature of products involving uniformly bounded and a.e. convergent function sequences as well as strongly L2-convergent factors ([40, Lemma 10.4]). By (1.25), namely, (2.16) guarantees that

0TΩuεuε+1(bεln(uε+1))φ0TΩuu+1(bln(u+1))φas ε=εjk0,(2.17)

whereas another application of (2.16) shows that

0TΩuεuε+1bεφ0TΩuu+1bφas ε=εjk0.(2.18)

Finally, as the matrices Aε are symmetric and positive definite, and hence possess self-adjoint square roots Aε, the limiting behavior of the first summand on the right of (2.11) can be made accessible to a standard argument based on lower semicontinuity with respect to weak convergence: Indeed, from (1.25), (1.23) and (1.15) it follows that also (Aεln(uε+1))φ(Aln(u+1))φ in L2(Ω × (0, T)) as ε = εjk ↘ 0, and that therefore

0TΩ{(Aln(u+1))ln(u+1)}φ=0TΩ|(Aln(u+1))φ|2lim infε=εjk00TΩ|(Aεln(uε+1))φ|2=lim infε=εjk00TΩ{(Aεln(uε+1))ln(uε+1)}φ.

In conjunction with (2.12), (2.13), (2.14), (2.15), (2.17) and (2.18), this shows that (1.12) is a consequence of (2.11). □

3 Turning weak into strong convergence. Proof of Theorem 1.3

Next approaching the core of our analysis, we intend to make sure that under the assumptions from Theorem 1.3, the very weak solutions obtained above are indeed weak solutions in the natural sense specifies as follows.

Definition 3.1

Let T ∈ (0, ∞], and let A, b, f, g and u0 be such that (1.5), (1.6), (1.7), (1.9) and (1.10) are satisfied with some kA > 0 and KA > 0. Then by a weak solution of (1.4) in Ω × (0, T) we mean a nonnegative function

uLloc1([0,T);W1,1(Ω))(3.1)

which is such that

ubLloc1(Ω¯×[0,T);Rn)andf(,,u)Lloc1(Ω¯×[0,T)),(3.2)

and that

0TΩuφtΩlnu0φ(,0)=0TΩ(A(x,t)u)φ0TΩub(x,t)φ0TΩf(x,t,u)φ+0TΩg(x,t)φ(3.3)

for all φC0(Ω × [0, T)).

Here a crucial step will consist in passing to the limit ε ↘ 0 in the respective second last summand in (3.3), which in view of (1.8) essentially amounts to turning the weak convergence feature in (1.24) into an appropriate statement on strong convergence. Our method of approaching this is in principle inspired by a strategy already pursued in previous studies (see e.g. [24, 34, 40]), namely intending to derive inequalities of the form

0t0Ωuαlim infε=εjk00t0Ωuεα,t0(0,T),(3.4)

by estimating the left-hand side therein directly through the weak inequality (1.12); in contrast to virtually all precedent cases, however, a major challenge in the present context stems from the circumstance that the integral inequality (1.12) merely addresses ln (u + 1) rather than u itself, which seems to substantially impede appropriate testing procedures.

As a preparation for our main argument in this direction, to be detailed in the proof of Lemma 3.2, let us recall (cf. e.g. [40] for statements quite precisely covering the present situation) the well-known fact that for T0 > 0 and ψLp(Ω × (−1, T0);ℝN) with p ∈ [1, ∞] and N ∈ ℕ, the Steklov averages Sh ψLp(Ω × (0, T0);ℝN), h ∈ (0, 1), as defined by letting

(Shψ)(x,t):=1hthtψ(x,s)ds,xΩ,t(0,T0),h(0,1),(3.5)

in the limit h ↘ 0 satisfy Sh ψψ in Lp(Ω × (0, T0)) whenever p ∈ [1, ∞) and Sh ψ ψ in L(Ω × (0, T0)) if p = ∞, and that clearly ∇ Sh ψ = Sh [∇ψ] a.e. in Ω × (0, T0) for all h ∈ (0, 1) if ψL1((− 1, T0); W1,1(Ω)).

By adequately exploiting (1.12) with carefully chosen test functions, we can achieve our main technical step toward th derivation of Theorem 1.3 in the following.

Lemma 3.2

Under the assumptions of Theorem 1.2, there exists a null set N ⊂ (0, T) such that the function from Theorem 1.2 has the property that for all t0 ∈ (0, T) ∖ N,

Ωu(,t0)+11+u(,t0)+1kln(1+u(,t0)+1k)+Ωk1+u(,t0)+1kln(1+u(,t0)+1k)Ωu0+11+u0+1kln(1+u0+1k)Ωk1+u0+1kln(1+u0+1k)0t0Ω(u+1)2k(1+u+1k)2{(A(x,t)ln(u+1))ln(u+1)}+0t0Ωu(u+1)k(1+u+1k)2b(x,t)ln(u+1)0t0Ωf(x,t,u)u+1+0t0Ωg(x,t)u+1forallkN.(3.6)

Proof

Without loss of generality we may assume that T be finite. For k ∈ ℕ, we then let

ψk(x,t):=u(x,t)+11+u(x,t)+1k,xΩ,t(0,T),(3.7)

and using that ψ0k:=u0+11+u0+1k belongs to L(Ω) with 0 ≤ ψ0kk a.e. in Ω we can fix (ψ0kl)l∈ℕC1(Ω) such that

0ψ0kl2k inΩfor alllNandψ0klψ0k a.e. inΩas l,(3.8)

and extend ψk to a function ψkl defined on all of Ω × ℝ by letting

ψkl(x,t):=ψ0kl(x)ifxΩand t0,ψk(x,t)ifxΩand t(0,T),0ifxΩand tT.(3.9)

We furthermore abbreviate

k(ξ):=lnξ1ξk,ξ[0,k),kN,(3.10)

and

Lk(ξ):=0ξk(σ)dσ,ξ[0,k),kN,(3.11)

so that actually Lk is explicitly given by

Lk(ξ)=ξlnξ+k(1ξk)ln(1ξk)for allξ[0,k)and kN.(3.12)

Then

Lk(ψk)=u+11+u+1klnu+11+u+1k+k1+u+1kln(1+u+1k)in Ω×(0,T),

whence using that uLloc1(Ω × [0, T)) and that ln (1 + ξ) ≤ ξ for all ξ ≥ 0 we conclude that besides the inclusion ln (u + 1) ⋅ ψkLloc1(Ω × [0, T)) we also have Lk(ψk) ∈ Lloc1(Ω × [0, T)), whereby it becomes possible to find a null set N ⊂ (0, T) such that

u(,t0)L1(Ω)for all t0(0,T)N,(3.13)

and that moreover each t0 ∈ (0, T) ∖ N is a common Lebesgue point of all the countably many mappings (0, T) ∋ t ↦ ∫Ω ln (u(x, t) + 1) ψk(x)dx and (0, T) ∋ t ↦ ∫Ω Lk(ψk(x, t)) dx for k ∈ ℕ.

Now given any t0 ∈ (0, T) ∖ N, we let

φ(x,t)φδ,h(t0)(x,t):=ζδ(t)(Shψkl)(x,t),xΩ,t(0,T),δ(0,Tt0),h(0,1),

with Sh as determined through (3.5), and with

ζδ(t)ζδ(t0)(t):=1if t[0,t0],1tt0δif t(t0,t0+δ),0if tt0+δ,(3.14)

noting that then ζδ belongs to W1,∞(ℝ) and satisfies

ζδ(t)=0if t[0,)[t0,t0+δ],1δif t(t0,t0+δ).(3.15)

Then moreover observing that

ψk=u+1(1+u+1k)2ln(u+1)a.e. in Ω×(0,T),(3.16)

on the basis of the regularity property (1.11) one can readily verify that φ beongs to L(Ω × (0, T)) with ∇φL2(Ω × (0, T);ℝn) and φtL2(Ω × (0, T)), and that φ = 0 a.e. in Ω × (t0 + δ, T). By means of a standard approximation argument, we therefore conclude that the integral inequality in (1.12) extends so as to remain valid for any such φ = φδ,h(t0), and that accordingly, by (3.15),

1δt0t0+δΩln(u+1)Shψkl0TΩζδ(t)ln(u+1)ψkl(,t)ψkl(,th)hΩln(u0+1)ψ0kl0TΩζδ(t){(A(x,t)ln(u+1))ln(u+1)}Shψkl0TΩζδ(t)(A(x,t)ln(u+1))Sh[ψkl]+0TΩζδ(t)uu+1(b(x,t)ln(u+1))Shψkl0TΩζδ(t)uu+1(bSh[ψkl])0TΩζδ(t)f(x,t,u))u+1Shψkl+0TΩζδ(t)g(x,t)u+1Shψklfor allδ(0,Tt0),h(0,1)and lN.(3.17)

Here since (3.10) and (3.11) ensure that k is increasing and hence Lk is convex on $[0, k), we obtain the pointwise inequality

Lk(ψkl(x,t))Lk(ψkl(x,th))hLk(ψkl(x,t))ψkl(x,t)ψkl(x,th)h=k(ψkl(x,t))ψkl(x,t)ψkl(x,th)h

for a.e. (x, t) ∈ Ω × (0, T), whence on the left-hand side of (3.17) we can estimate

=0TΩζδ(t)k(ψkl(,t))ψkl(,t)ψkl(,th)h0TΩζδ(t)Lk(ψkl(,t))Lk(ψkl(,th))hfor allδ(0,Tt0),h(0,1)and lN.(3.18)

Since according to our definition (3.9) of ψkl a substitution shows that

0TΩζδ(t)Lk(ψkl(,t))Lk(ψkl(,th))h=0TΩζδ(t+h)ζδ(t)hLk(ψkl(,t))+ΩLk(ψ0kl)for allδ(0,Tt0),h(0,min{1,Tt0δ})and lN,

by using that clearly ζδ(+h)ζδhζδ in L((0, ∞)) as h ↘ 0 due to (3.14), we obtain that

lim suph0{0TΩζδ(t)ln(u+1)ψkl(,t)ψkl(,th)h}1δt0t0+δΩLk(ψkl)+ΩLk(ψ0kl)for allδ(0,Tt0)and lN(3.19)

thanks to (3.15).

Now in the remaining seven integrals in (3.17) we only need to recall that as a consequence of the inclusions ψklL(Ω × ℝ) and ∇ψklL2(Ω × (−1, t0 + δ); ℝn), as for each fixed δ ∈ (0, Tt0) asserted by (3.7), (3.9) and (3.16), we have Sh ψkl ψklψk in L(Ω × (0, t0 + δ)) and Sh[∇ψkl] → ∇ψkl ≡ ∇ψk in L2(Ω × (0, t0 + δ)) as h ↘ 0. Since

{ln(u+1),(Aln(u+1))ln(u+1),uu+1(bln(u+1)),f(,,u)u+1,gu+1}L1(Ω×(0,t0+δ)),(3.20)

and since

{Aln(u+1),uu+1b}L2(Ω×(0,t0+δ);Rn)(3.21)

for any such δ, namely, these properties enable us to take h ↘ 0 in the first integral in the left and each of the summands on the right of (3.17) to infer by using (3.19) that

1δt0t0+δΩln(u+1)ψk1δt0t0+δΩLk(ψk)+ΩLk(ψ0kl)Ωln(u0+1)ψ0kl0TΩζδ(t){(A(x,t)ln(u+1))ln(u+1)}ψk0TΩζδ(t)(A(x,t)ln(u+1))ψk+0TΩζδ(t)uu+1(b(x,t)ln(u+1))ψk0TΩζδ(t)uu+1b(x,t)ψk0TΩζδ(t)f(x,t,u)u+1ψk+0TΩζδ(t)g(x,t)u+1ψkfor allδ(0,Tt0)and lN.(3.22)

Here the Lebesgue point properties of t0 apply so as to guarantee that on the left-hand side we have

1δt0t0+δΩln(u+1)ψkΩln(u(,t0)+1)ψk(,t0)as δ0

and

1δt0t0+δΩLk(ψk)ΩLk(ψk(,t0))as δ0,

while on the right-hand side we may use the evident fact that ζδ ζ in L((0, ∞)), with ζ(t) := 1 for t ∈ (0, t0) and ζ(t) := 0 for tt0, which when combined with (3.20), (3.21) and the inclusion ∇ψkL2(Ω × (0, t0 + 1);ℝn) ensures that each of the integrals approach their expected limit as δ ↘ 0. In conclusion, (3.22) entails that

Ωln(u(,t0)+1)ψk(,t0)ΩLk(ψk(,t0))+ΩLk(ψ0kl)Ωln(u0+1)ψ0kl0t0Ω{(A(x,t)ln(u+1))ln(u+1)}ψk0t0Ω(A(x,t)ln(u+1))ψk+0t0Ωuu+1(b(x,t)ln(u+1))ψk0t0Ωuu+1b(x,t)ψk0t0Ωf(x,t,u)u+1ψk+0t0Ωg(x,t)u+1ψkfor all lN.(3.23)

In a last limiting step, we recall the approximation property (3.8) of (ψ0kl)l∈ℕ, which through two arguments based on the dominated convergence theorem, namely, asserts that

ΩLk(ψ0kl)ΩLk(ψ0k)as l(3.24)

and that

Ωln(u0+1)ψ0klΩln(u0+1)ψ0kas l,(3.25)

because for each fixed k ∈ ℕ and all l ∈ ℕ we have 0 ≤ Lk(ψ0kl) ≤ Lk(2k) and 0 ≤ ln (u0 + 1) ψ0kl ≤ 2k ln (u0 + 1) a.e. in Ω due to (3.8), with the majorants Lk(2k) and 2k ln (u0 + 1) being integrable thanks to our assumption that u0L1(Ω). We finally observe that according to (3.7) and the representation (3.16), on the right of (3.23) we can simplify

0t0Ω{(A(x,t)ln(u+1))ln(u+1)}ψk0t0Ω(A(x,t)ln(u+1))ψk=0t0Ωu+11+u+1k{(A(x,t)ln(u+1))ln(u+1)}0t0Ωu+1(1+u+1k)2{(A(x,t)ln(u+1))ln(u+1)}=0t0Ωu+1k(1+u+1k)2{(A(x,t)ln(u+1))ln(u+1)}(3.26)

and

0t0Ωuu+1(b(x,t)ln(u+1))ψk0t0Ωuu+1b(x,t)ψk=0t0Ωu1+u+1kb(x,t)ln(u+1)0t0Ωu(1+u+1k)2b(x,t)ln(u+1)=0t0Ωu(u+1)(1+u+1k)b(x,t)ln(u+1)(3.27)

as well as

0t0Ωf(x,t,u)u+1ψk+0t0Ωg(x,t)u+1ψk=0t0Ωf(x,t,u)1+u+1k+0t0Ωg(x,t)1+u+1k.(3.28)

Similarly inserting (3.7) into (3.24) and (2.5), in view of the definition (3.12) of Lk we immediately conclude that (3.6) is a consequence of (3.23)-(3.28). □

Now if b complies with the regularity assumptions from Theorem 1.3, then the above can be combined with the convergence statements from Theorem 1.2 to deduce (3.4), and hence the desired strong approximation property, in the following sense.

Lemma 3.3

In addition to the assumptions from Theorem 1.2, suppose that

bLlocq(Ω¯×[0,T);Rn)forsomeq2αα1.(3.29)

Then there exists a null set N ⊂ (0, T) such that with u and (εjk)k∈ℕ as given by Theorem 1.2 we have

uεuinLα(Ω×(0,t0))forallt0(0,T)N(3.30)

as ε = εjk ↘ 0.

Proof

Since Lemma 2.5 especially entails that for a.e. t0 ∈ (0, T) we have

uε(,t0)u(,t0)in L1(Ω)as ε=εjk0,(3.31)

according to Lemma 3.2 we can pick a null set N ⊂ (0, T) with the property that both (3.31) and (3.6) hold for each t0 ∈ (0, T) ∖ N and all k ∈ ℕ. Using that (3.31) in particular warrants that for any such t0 we know that u(⋅, t0) + 1 belongs to L1(Ω) and hence is finite a.e. in Ω, we see that

u(,t0)+11+u(,t0)+1kln(1+u(,t0)+1k)0a.e. in Ωas k,

whereas the validity of

0ln(1+ξ)ξfor all ξ0(3.32)

asserts the majorization

0u(,t0)+11+u(,t0)+1kln(1+u(,t0)+1k)u(,t0)+11+u(,t0)+1ku(,t0)+1ku(,t0)+1a.e. in Ω.

Therefore, the dominated convergence theorem ensures that

Ωu(,t0)+11+u(,t0)+1kln(1+u(,t0)+1k)0as kfor all t0(0,T)N,(3.33)

and quite a similar reasoning based on (1.22) shows that

Ωu0+11+u0+1kln(1+u0+1k)0as k.(3.34)

Next, once more relying on (3.31), by means of the ľHospital rule we readily find that

k1+u(,t0)+1kln(1+u(,t0)+1k)u(,t0)+1a.e. in Ωas k,

while thanks to (3.32),

0k1+u(,t0)+1kln(1+u(,t0)+1k)u(,t0)+11+u(,t0)+1ku(,t0)+1a.e. in Ω.

Again by the dominated convergence theorem, we thus obtain that

Ωk1+u(,t0)+1kln(1+u(,t0)+1k)Ω(u(,t0)+1)as kfor all t0(0,T)N,(3.35)

and that, similarly,

Ωk1+u0+1kln(1+u0+1k)Ω(u0+1)as k.(3.36)

Now on the right-hand side in (3.6), in order to adequately cope with the second summand we first recall (1.15) and invoke Young’s inequality to estimate

0t0Ω(u+1)2k(1+u+1k)2{(Aln(u+1))ln(u+1)}+0t0Ωu(u+1)k(1+u+1k)2bln(u+1)kA0t0Ω(u+1)2k(1+u+1k)2|ln(u+1)|2+0t0Ωu(u+1)k(1+u+1k)2bln(u+1)14kA0t0Ωu2k(1+u+1k)2|b|2for all t0(0,T).(3.37)

Since

u2k(1+u+1k)2u2k(1+u+1k)=u2u+1+ku2u+ka.e. in Ω×(0,T),

using the Hölder inequality we see that here

0t0Ωu2k(1+u+1k)2|b|2{0t0Ω(u2u+k)qq2}q2q{0t0Ω|b|q}2qfor all t0(0,T),(3.38)

and observe that the first integrand on the right satisfies (u2u+k)qq2 → 0 a.e. in Ω × (0, t0) as k → ∞, and is majorized according to (u2u+k)qq2uqq2 a.e. in Ω × (0, t0) with uqq2L1(Ω × (0, t0)) due to Lemma 2.5 and the fact that qq2=112q112α12α = α by hypothesis. As a further consequence of the dominated convergence theorem, from (3.38) we thus infer that

0t0Ωu2k(1+u+1k)2|b|20as kfor all t0(0,T),

and that hence, by (3.37),

lim infk{0t0Ω(u+1)2k(1+u+1k)2{(Aln(u+1))ln(u+1)}+0t0Ωu(u+1)k(1+u+1k)2bln(u+1)}0for all t0(0,T).(3.39)

Finally, two further arguments based on dominated convergence show that thanks to (1.9) and the inclusion f(⋅, ⋅, u) ∈ Llocα(Ω × [0, T)), as asserted by (1.24) in view of (1.7) and (1.8),

0t0Ωg(x,t)1+u+1k0t0Ωg(x,t)as kfor all t0(0,T)(3.40)

and

0t0Ωf(x,t,u)1+u+1k0t0Ωf(x,t,u)as kfor all t0(0,T).(3.41)

In summary, upon collecting (3.33)-(3.36) and (3.39)-(3.41) we obtain from Lemma 3.2 that

0t0Ωf(x,t,u)0t0Ωg(x,t)Ωu(,t0)+Ωu0for all t0(0,T)N,(3.42)

where now making full use of (3.31) we see that due to (1.20) and (1.22), the right-hand side appears as a limit of the corresponding expressions associated with (1.14) in the sense that for all t0 ∈ (0, T) ∖ N,

0t0Ωgε(x,t)Ωuε(,t0)+Ωu0ε0t0Ωg(x,t)Ωu(,t0)+Ωu0as ε=εjk0.

Since moreover, again by dominated convergence,

0t0Ωf(x,t,uε)0t0Ωf(x,t,u)as ε=εjk0for all t0(0,T)

thanks to (1.23) and the boundedness of f in Ω × (0, t0) × [0, ∞) for t0 ∈ (0, T), as implied by (1.7) and (1.8), from (3.42) and Lemma 2.1 we infer that

0t0Ωf+(x,t,u)=0t0Ωf(x,t,u)+0t0Ωf(x,t,u)0t0Ωg(x,t)Ωu(,t0)+Ωu0+0t0Ωf(x,t,u)=limε=εjk0{0t0Ωgε(x,t)Ωuε(,t0)+Ωu0ε+0t0Ωf(x,t,uε)}=limε=εjk0{0t0Ωf(x,t,uε)+0t0Ωf(x,t,uε)}=limε=εjk00t0Ωf+(x,t,uε)for all t0(0,T)N.

As furthermore 0t0Ωf+(x,t,u)lim infε=εjk00t0Ωf+(x,t,uε) for all t0 ∈ (0, T) due to (1.23) and Fatou’s lemma, this means that actually

0t0Ωf+(x,t,uε)0t0Ωf+(x,t,u)as ε=εjk0for all t0(0,T)N,

which again in view of (1.23) implies that for any such t0, f+(⋅, ⋅, uε) → f+(⋅, ⋅, u) in L1(Ω × (0, t0)) as ε = εjk ↘ 0. Since (1.8) entails that

uεαmax{s0α,f+(,,uε)kf}a.e. in Ωfor all ε(εj)jN,

one final application of a dominated convergence principle reveals that again by (1.23),

0t0Ωuεα0t0Ωuαas ε=εjk0for all t0(0,T)N.

Together with the weak convergence statement in (1.24), by uniform convexity of Lα(Ω × (0, t0)) for all t0 > 0 this yields (3.30). □

3.1 Proof of Theorem 1.3

As a last preliminary for Theorem 1.3, let us state a chain rule type statement which should be essentially well-known, but for which we include a brief argument as we could not find a precise reference in the literature.

Lemma 3.4

Let w : Ω → ℝ be measurable and nonnegative and such that ewL2(Ω) as well aswL2(Ω;ℝn). Then ewW1,1(Ω) withew = eww a.e. in Ω.

Proof

For k ∈ ℕ letting ρk(ξ) := min{eξ, ek}, due to the Lipschitz continuity of ρk we may invoke a well-known version of the chain rule in W1,2(Ω) to infer from the inclusions ewL2(Ω) and ∇ wL2(Ω;ℝn) that ρk(w) belongs to W1,2(Ω) with

ρk(w)=χ{w<k}ewwa.e. in Ω.(3.43)

Accordingly, for integers k and l with l > k we can estimate

Ω|ρl(w)ρk(w)|={kw<l+1}ew|w|{{wk}e2w}12{{wk}|w|2}12,

whence again by hypothesis we conclude that (∇ ρk(w))k∈ℕ forms a Cauchy sequence in L1(Ω;ℝn). Since, on the other hand, clearly ρk(w) → ew in L2(Ω) as k → ∞ by Beppo Levi’s theorem, we thus must have ∇ ρk(w) → ∇ ew in L1(Ω) as k → ∞, so that the claim results on observing that an application of the dominated convergence theorem to (3.43) directly shows that ∇ ρk(w) → eww in L1(Ω) as k → ∞. □

We are now in the position to verify our main result on genuine weak solvability in (1.4).

Proofc of Theorem 1.3

We fix φC0(Ω × [0, T)) and then obtain on integrating by parts in (1.14) that

0TΩuεφtΩu0εφ(,0)=0TΩ(Aε(x,t)uε)φ0TΩuεbε(x,t)φ0TΩf(x,t,uε)φ+0TΩgε(x,t)φ(3.44)

for all ε ∈ (εj)j∈ℕ. Here (1.22) and (1.20) directly yield

Ωu0εφ(,0)Ωu0φ(,0)as ε=εj0(3.45)

and

0TΩgε(x,t)φ0TΩg(x,t)φas ε=εj0,(3.46)

while relying on (1.23) and (1.24) we see that with (εjk)k∈ℕ as provided by Theorem 1.2 we have

0TΩuεφt0TΩuφtas ε=εjk0(3.47)

and

0TΩuεbε(x,t)φ0TΩub(x,t)φas ε=εjk0,(3.48)

because due to (1.24) our hypothesis α ≥ 2 in particular implies that uεu in Lloc2(Ω × [0, T)) as ε = εjk ↘ 0, and because bεb in Lloc2(Ω × [0, T)) as ε = εj ↘ 0 by (1.18).

In appropriately passing to the limit in the crucial first and third summand on the right of (3.44), we now make essential use of Lemma 3.3 by fixing the null set N ⊂ (0, T) as given there, and taking t0 ∈ (0, T) ∖ N sufficiently close to T such that φ ≡ 0 in Ω × (t0, T), so that Lemma 3.3 guarantees that

uεuin Lα(Ω×(0,t0))L2(Ω×(0,t0))as ε=εjk0.(3.49)

Therefore, namely, we firstly obtain that according to the dominated convergence theorem,

(uε+1)(Aε)ij(u+1)Aijin L2(Ω×(0,t0))as ε=εjk0,(3.50)

because in the majorization {(uε+1)(Aε)ij}2Ka2(uε+1)2 asserted by (1.15) the right-hand side is convergent in L1(Ω × (0, t0)) as ε = εjk ↘ 0 by (3.49), and because (uε + 1)(Aε)ij → (u + 1) Aij a.e. in Ω × (0, T) as ε = εjk ↘ 0 due to (1.23) and (1.16).

Combining (3.50) with the weak convergence statement in (1.25), by means of the chain rule-type result from Lemma 3.4 we thus conclude that

0TΩ(Aε(x,t)uε)φ=0t0Ω(uε+1){Aε(x,t)ln(uε+1)}φ0t0Ω(u+1){A(x,t)ln(u+1)}φ=0TΩ(A(x,t)u)φas ε=εjk0.(3.51)

We secondly make full use of the strong convergence property (3.49) in the space Lα(Ω × (0, t0)), actually possibly smaller than L2(Ω × (0, t0)), to treat the superlinear nonlinearity in (3.44): Since from (1.8) we know that

|f(x,t,uε)|fL(Ω×(0,T)×(0,s0)+Kfuεαin Ω×(0,T)for all ε(εj)jN,

and since herein uεαuα in L1(Ω × (0, t0)) as ε = εjk ↘ 0 by (3.49), once more employing the dominated convergence theorem we see that thanks to (1.23) and the continuity of f,

f(,,uε)f(,,u)in L1(Ω×(0,t0))as ε=εjk0,(3.52)

and that thus

0TΩf(x,t,uε)φ=0t0Ωf(x,t,uε)φ0t0Ωf(x,t,u)φ=0TΩf(x,t,u)φ

as ε = εjk ↘ 0. In conjunction with (3.45) and (3.48) and (3.51), this yields (3.3) as a consequence of (3.44) upon taking ε = εjk ↘ 0, so that the proof becomes complete by noting that the regularity requirements in (3.1) and (3.2) are direct consequences of the integrability propertis implied by (1.23)-(1.25) and (3.52) when combined with Lemma 3.4. □

4 Application to logistic Keller-Segel systems

As our first concrete example, we here consider the logistic Keller-Segel system (1.28) under the permanent assumption that the reaction term F therein satisfies (1.29) with some kF > 0, KF > 0, s0 > 0 and α > 1, and that the initial data are such that

u0L1(Ω)andv0L2(Ω)are nonnegative.(4.1)

Then adapting well-established arguments ([1, 13, 17, 36]) readily shows that if we fix (u0ε)ε∈(0, 1)C1(Ω) and (v0ε)ε∈(0, 1)C2(Ω) such that v0εν=0 on ∂Ω, and that

0u0εu0in L1(Ω)andv0εv0in L2(Ω)as ε0,(4.2)

each of the problems

uεt=Δuε(uεvε)+λuεμuεα,xΩ,t>0,vεt=Δvεvε+uε1+εuε,xΩ,t>0,uν=vν=0,xΩ,t>0,u(x,0)=u0(x),v(x,0)=v0(x),xΩ,(4.3)

admits a global classical solution (uε, vε), ε ∈ (0, 1), with 0 ≤ uεC0(Ω × [0, ∞)) ∩ C2,1(Ω × (0, ∞)) and 0 ≤ vε ∈ ⋂q>n C0([0, ∞);W1,q(Ω)) ∩ C2,1(Ω × (0, ∞)).

In order to make our general results derived above applicable to the present particular setting, for ε ∈ (0, 1) we let Aij = (Aε)ij := δij, i, j ∈ {1, …, n}, and bε := –∇vε as well as f(x, t, s) := –F(s) and g(x, t) = gε(x, t) := 0 for xΩ, t ≥ 0 and s ≥ 0. Then (1.5), (1.15) and (1.16) as well as (1.9), (1.19) and (1.20) are trivially satisfied, while (1.17), (1.19) (1.10), (1.21) and (1.22) are asserted by the regularity properties of uε and vε and the requirements on u0 and u0ε in (4.1) and (4.2); furthermore, our choice of f is compatible with (1.7) and (1.8) due to (1.29).

4.1 Very weak solutions. Proof of Theorem 1.4

In light of the above observations, for an application of Theorem 1.2 it will thus be sufficient to find (εj)j∈ℕ ⊂ (0, 1) and vLloc2([0, ∞);W1,2(Ω)) such that εj ↘ 0 as j → ∞, and that

vεvin Lloc2(Ω¯×[0,))as ε=εj0.(4.4)

This will be achieved through an analysis of the specific systems (1.28) and (4.3), particularly focusing on the second equation therein as the main additional ingredient in comparison to (1.4) and (1.14), but in some places as well resorting to statements derived for the latter general setting in Section 2. A fundamental property of (4.3), for instance, has been achieved in Lemma 2.2 already:

Lemma 4.1

Suppose that (1.29) holds with some kF > 0, KF > 0 and α > 1. Then for all T > 0 there exists C(T) > 0 such that

0TΩuεα(x,t)dxdtC(T)forallε(0,1).(4.5)

Proof

Noting that our above selections warrant applicability of Lemma 2.2, we immediately obtain (4.5) from (2.4).□

By relying on appropriate smoothing properties of the inhomogeneous heat equation satisfied by vε, the previous lemma firstly entails a uniform spatial L2 bound for vε whenever α complies with the largeness assumption from Theorem 1.4.

Lemma 4.2

If (1.29) is valid with some kF > 0, KF > 0 and

α>2n+4n+4,(4.6)

then for all T > 0 one can find C(T) > 0 such that for any ε ∈ (0, 1),

vε(,t)L2(Ω)C(T)forallt(0,T).(4.7)

Proof

By a well-known Lp-Lq estimate for the Neumann heat semigroup (e)t≥0 on Ω([37, Lemma 1.3]), there exists c1 > 0 such that

etΔϕL2(Ω)c1(1+tn2(1α12)+)ϕLα(Ω)for allt>0and each ϕLα(Ω).

Since e acts as a contraction on L2(Ω) for all t > 0, according to a Duhamel representation associated with the second equation in (4.3) we can therefore estimate

vε(,t)L2(Ω)=et(Δ1)v0ε+0te(ts)Δuε(,s)1+εuε(,s)dsL2(Ω)etv0εL2(Ω)+c10t(1+(ts)n2(1α12)+)e(ts)uε(,s)1+εuε(,s)Lα(Ω)dsc2+c10t(1+(ts)n2(1α12)+)uε(,s)Lα(Ω)ds(4.8)

for all t > 0 and ε ∈ (0, 1), with c2 := supε∈(0, 1)v0εL2(Ω) being finite according to (4.2). Here using Young’s inequality, given T > 0 we see that for all t ∈ (0, T) and ε ∈ (0, 1),

0t(1+(ts)n2(1α12)+)uε(,s)Lα(Ω)ds0t(1+(ts)n2(1α12)+)αα1ds+0tuε(,s)Lα(Ω)αds0T(1+σn2(1α12)+)αα1dσ+0TΩuεα,(4.9)

where by Lemma 4.1 we can find c3(T) > 0 such that

0TΩuεαc3(T)for all ε(0,1),(4.10)

and where our hypothesis (4.6) ensures that moreover

c4(T):=0T(1+σn2(1α12)+)αα1dσ<:

Indeed, again by means of Young’s inequality we obtain that

c4(T)21α10T(1+σn2(1α12)+αα1)dσ,

where if α ≥ 2 we trivially have n2(1α12)+αα1=0, and where in the case when α < 2 we may rely on (4.6) in estimating

n2(1α12)+αα11=2n+4(n+4)α4(α1)<0

and in thus concluding finiteness of c4(T) also for such α. As (4.8), (4.9) and (4.10) imply that

vε(,t)L2(Ω)c2+c1(c4(T)+c3(T))for allt(0,T)and ε(0,1),

we thereby arrive at (4.7).□

Two straightforward testing procedures let us conclude further regularity properties of vε from the latter and Lemma 4.1.

Lemma 4.3

Let (1.29) hold with positive constants kF, KF and α fulfilling (4.6). Then for all T > 0 there exists C(T) > 0 such that

0TΩ|vε|2C(T)forallε(0,1)(4.11)

and

0TΩvε2n+4nC(T)forallε(0,1)(4.12)

as well as

0Tvεt(,t)(W1,2(Ω))min{α,2}dtC(T)forallε(0,1).(4.13)

Proof

We abbreviate p := 2n+4n and then obtain on combining the Gagliardo-Nirenberg inequality with the uniform L2 bound for vε from Lemma 4.2 to find c1 > 0 and c2(T) > 0 such that

0TΩvεpc10T{vε(,t)L2(Ω)2vε(,t)L2(Ω)4n+vε(,t)L2(Ω)p}dtc2(T)0TΩ|vε|2+c2(T)for all ε(0,1).(4.14)

As our assumption (4.6) precisely asserts that p>αα1, through an application of Young’s inequality this especially entails the existsnce of c3(T) > 0 such that

0TΩvεαα112c2(T)0TΩvεp+c3(T)120TΩ|vε|2+12+c3(T)for all ε(0,1),

whence testing the second equation in (4.3) by vε in a standard manner we see, again by Young’s inequality, that for all ε ∈ (0, 1),

12Ωvε2(,T)12Ωv0ε2+0TΩ|vε|2+0TΩvε2=0TΩuε1+εuεvε0TΩuεα+0TΩvεαα10TΩuεα+120TΩ|vε|2+12+c3(T).

According to (4.2) and the outcome of Lemma 4.1, this firstly entails (4.11) and therefore, after another application of (4.14), also establishes (4.12).

The estimate in (4.13) can be achieved in a straightforward way by taking ϕC(Ω) and again using the second equation in (4.3) to find that for fixed t > 0 and arbitrary ε ∈ (0, 1),

|Ωvεt(,t)ϕ|=|ΩvεϕΩvεϕ+Ωuε1+εuεϕ|vεL2(Ω)ϕL2(Ω)+vεL2(Ω)ϕL2(Ω)+uεLα(Ω)ϕLαα1(Ω).

Since αα1<2n(n2)+ and hence W1,2(Ω)Lαα1(Ω) by (4.11), we thus obtain c3 > 0 such that writing q := min{α, 2} we have

vεt(,t)(W1,2(Ω))qc3{vεL2(Ω)q+vεL2(Ω)q+uεLα(Ω)q}c3{Ω|vε|2+Ωvε2+Ωuεα+3}for all ε(0,1)

due to Young’s inequality. In view of (4.11), Lemma 4.2 and Lemma 4.1, an integration over t ∈ (0, T) yields (4.13).□

The following statements on convergence of both uε and vε are thus rather evident.

Lemma 4.4

Assume (1.29) with some kF > 0, KF > 0 and α satisfying (4.6). Then there exist (εj)j∈ℕ ⊂ (0, 1) such that εj ↘ 0 as j → ∞, and nonnegative functions u and v which are defined on Ω × (0, ∞) and such that for all T > 0,

uLα(Ω×(0,T))andvL((0,T);L2(Ω))L2((0,T);W1,2(Ω))L2n+4n(Ω×(0,T)),(4.15)

and that for all T > 0 we have (1.23) and (1.24) as well as

vεva.e.inΩ×(0,T)andinLp(Ω×(0,T))forallp[1,2n+4n)and(4.16)

vεvinL2(Ω×(0,T))(4.17)

as ε = εj ↘ 0.

Proof

In view of (4.11), Lemma 2.5 applies so as to yield the statements concerning uε along an appropriate sequence. Relying on the boundedness properties derived in Lemma 4.2) and Lemma 4.3, as well as on the Vitali convergence theorem, a straightforward further subsequence extraction based on the Aubin-Lions lemma thereafter enables us to achieve also (4.16) and (4.17) with some nonnegative v fulfilling (4.15).□

As an application of Theorem 1.2 will require strong, rather than merely weak, L2 convergence of bε = –∇vε, an additional consideration concerning this will be necessary:

Lemma 4.5

Let (1.29) be valid with positive parameters kF > 0, KF > 0 and α such that (4.6) holds, and let T > 0. Then with (εj)j∈ℕ and (u, v) as in Lemma 4.4, we have

vεvinL2(Ω×(0,T))asε=εj0.(4.18)

Proof

We fix ε ∈ (0, 1) and ε′ ∈ (0, 1) and then obtain on taking differences in the respective second equations from (4.3) that

12Ω(vε(,T)vε(,T))2+0TΩ|vεvε|2+0TΩ(vεvε)2=12Ω(v0εv0ε)2+0TΩ(uε1+εuεuε1+εuε)(vεvε).(4.19)

Here since (uε1+εuε)ε(εj)jN is bounded in Lα(Ω × (0, T)) by Lemma 4.1 and a.e. in Ω × (0, T) convergent to u according to (1.23), from Egorov’s theorem it follows that

uε1+εuεuε1+εuεuε1+εuεuin Lα(Ω×(0,T))as (εj)jNε0.

Apart from that, from (4.16) we know that

vεvεvεvin Lαα1(Ω×(0,T))as (εj)jNε0,(4.20)

once more because the hypothesis (4.6) warrants that αα1<2n+4n.. In view of (4.2) and (4.17) we hence infer by employing the Hölder inequality that for all ε ∈ (0, 1),

0TΩ|vεv|2lim inf(εj)jNε00TΩ|vεvε|2lim inf(εj)jNε0{12Ω(v0εv0ε)2+0TΩ(uε1+εuεuε1+εuε)(vεvε)}=12Ω(v0εv0)2+0TΩ(uε1+εuεu)(vεv)12Ω(v0εv0)2+uε1+εuεuLα(Ω×(0,T))vεvLαα1(Ω×(0,T))12Ω(v0εv0)2+(uε1+εuεLα(Ω×(0,T))+uLα(Ω×(0,T)))vεvLαα1(Ω×(0,T)),(4.21)

so that again relying on (4.2) and (4.20), and on the boundedness of (uε1+εuε)ε(0,1) in Lα(Ω × (0, T)), as resulting from Lemma 4.1, we see that (4.18) is a consequence of (4.21).□

Thus having at hand all ingredients necessary for an application of Theorem 1.2, we can utilize the latter to obtain our main results on global very weak solvability in (1.28).

Proof of Theorem 1.4

Taking (εj)j∈ℕ, u and v as provided by Lemma 4.4, on the basis of the strong convergence result from Lemma 4.5 we may employ Theorem 1.2 to obtain a subsequence, again denoted by (εj)j∈ℕ for notational convenience, along which for the solutions of (4.3) we have uεũ a.e. in Ω × (0, ∞) as ε = εj ↘ 0, so that clearly ũ must coincide with u and hence u must have the claimed solution properties with regard to (1.4). In view of (1.12), the regularity features in (1.33) are therefore immediate by-products of Theorem 1.2 and Lemma 4.4, whereas the derivation of (1.34) can be chieved in a straightforward manner by taking ε = εj ↘ 0 in an accordingly tested version of the second sub-problem from (4.3).□

4.2 Weak solutions. Proof of Theorem 1.5

Next, in order to derive the stronger integrability property (1.27) required for an application of Theorem 1.3, beyond (4.1) and (4.2) we will assume that

v0D(Aβ)and thatsupε(0,1)Aβv0εL2(Ω)<(4.22)

for some β ∈ (0, 1), with A = –Δ + 1 as introduced before the formulation of Theorem 1.5. Here we note that for any nonnegative v0D(Aβ), the requirements in (4.2) and (4.22) can simultaneously be fulfilled with some (v0ε)ε∈(0, 1)C1(Ω;[0, ∞)) by e.g. fixing m ∈ ℕ such that m > n+24 and letting v0ε := (1 + εA)m v0 for ε ∈ (0, 1), for instance: In fact, v0ε then is nonnegative by order preservation of (1 + εA)–1, and the inclusion v0εC1(Ω) is ensured by the fact that m > n+24 warrants continuity of the embeddings D(Am) ↪ W2m,2(Ω) ↪ C1(Ω) ([11]); apart from that, the L2 convergence property in (4.2) can be seen by standard arguments ([26]), whereas the boundedness feature in (4.22) readily results from the inclusion v0D(Aβ) in view of the contractivity of (1 + εA)m on L2(Ω) and the fact that Aβ and (1 + εA)m commute on D(Aβ) ([9]).

Through quite straightforward smoothing properties of the parabolic operator in the second equation from (4.3), the assumption (4.22) indeed has further consequences on the regularity of vε.

Lemma 4.6

Assume (1.29) with some kF > 0, KF > 0, s0 > 0 and α ≥ 2, and suppose that there exists β ∈ (0, 12] such that (4.22) holds. Then for all T > 0 there exists C(T) > 0 such that the solutions of (4.3) have the properties that

Ω|Aβvε(x,t)|2dxC(T)forallt(0,T)andε(0,1)(4.23)

and

0TΩ|A2β+12vε(x,t)|2dxdtC(T)forallε(0,1).(4.24)

Proof

Noting that the assumption β12 along with the regularity features of vε warrant appropriate smoothness of all subsequently appearing quantities, we may use the second equation in (4.3), rewritten in the form vεt + Avε = uε1+εuε, to see that thanks to the self-adjointness of A and all its fractional powers, with some c1 > 0 we have

12ddtΩ|Aβvε|2=ΩA2βvεvεt=Ω|A2β+12vε|2+Ωuε1+εuεA2βvεΩ|A2β+12vε|2+uεLα(Ω)A2βvεLαα1(Ω)Ω|A2β+12vε|2+c1uεLα(Ω)A2βvεL2(Ω)for all t>0(4.25)

according to the Hölder inequality and the fact that αα12. Here we may rely on a standard interpolation result ([9]) to infer from the inequalities β<2β2β+12, as ensured by the restrictions β > 0 and β12, that there exists c2 > 0 fulfilling

A2βvεL2(Ω)c2A2β+12vεL2(Ω)2βAβvεL2(Ω)12βfor all t>0.

When inserted into (4.25) and combined with Young’s inequality, this shows that we can find c3 > 0 such that

12ddtΩ|Aβvε|2Ω|A2β+12vε|2+c1c2uεLα(Ω)A2β+12vεL2(Ω)2βAε\betavL2(Ω)12β12Ω|A2β+12vε|2+c3uεLα(Ω)11βAε\betavL2(Ω)12β1βfor all t>0,(4.26)

where in the case β < 12 we may two more times use Young’s inequality to see that since

α(12β)α(1β)1=12β1β1α12β1β12=2

by assumption on α, we have

uεLα(Ω)11βAβvεL2(Ω)12β1βuεLα(Ω)α+AβvεL2(Ω)α(12β)α(1β)1uεLα(Ω)α+AβvεL2(Ω)2+1for all t>0.

As the resulting inequality evidently extends so as to remain valid also in the borderline case β = 12, from (4.26) we altogether obtain that

ddtΩ|Aβvε|2+Ω|A2β+12vε|22c3Ω|Aβvε|2+2c3Ωuεα+2c3for all t>0,

and that hence both (4.23) and (4.24) follow upon integrating in time and recalling Lemma 4.1.□

By suitable interpolation, the latter indeed entails further integrability properties of the crucial quantity ∇vε.

Lemma 4.7

Let (1.29) be satisfied with some kF > 0, KF > 0, s0 > 0 and α ≥ 2, and suppose that there exists β ∈ (0, 12] such that (4.22) is valid. Then for any T > 0 and each q > 2 fulfilling

q<2(n+2)n+24βifβ<12,q2(n+2)nifβ=12,(4.27)

one can find C(T, q) > 0 such that

0TΩ|vε|qC(T,q)forallε(0,1).(4.28)

Proof

We first consider the case β = 12, in which due to elliptic regularity theory ([9, 10]) it is well-known that A2β+12()L2(Ω)=(Δ+1)()L2(Ω)andAβ()L2(Ω)=(()L2(Ω)2+L2(Ω)2)12 define norms equivalent to ∥⋅∥W2,2(Ω) and ∥⋅∥W1,2(Ω), respectively, so that by a Gagliardo-Nirenberg interpolation we find c1 > 0 and c2 > 0 such that writing q0 := 2(n+2)n we have

ϕLq0(Ω)q0c1ϕW2,2(Ω)2ϕW1,2(Ω)q02c2A2β+12ϕL2(Ω)2AβϕL2(Ω)q02for all ϕD(A).

According to (4.27) and Young’s inequality, we thus obtain that in this case,

0TΩ|vε|q0TΩ|vε|q0+|Ω|Tc20TA2β+12vε(,t)L2(Ω)2Aβvε(,t)L2(Ω)q02dt+|Ω|Tfor all ε(0,1),

and that hence (4.28) results from Lemma 4.6.

If β < 12, however, we first make use of the strivt inequality in (4.27) to fix y > β such that

n4+12n2q<yβ+1q,(4.29)

noting that then a known embedding result ([11]) warrants that D(Ay) ↪ W1,q(Ω). As furthermore our assumption q > 2 ensures that yβ+1q2β+12, once more according to an appropriate interpolation property of fractional powers ([9, Part 2, Theorem 14.1]) we can fix c3 > 0 and c4 > 0 such that

0Tvε(,t)Lq(Ω)qdtc30TAyvε(,t)L2(Ω)qdtc40TA2β+12vε(,t)L2(Ω)2q(yβ)AβvεL2(Ω)q2q(yβ)dtfor all ε(0,1).

Observing that herein 2q(yβ) ≤ 2 thanks to the right inequality in (4.29), again invoking Lemma 4.6 we infer (4.28) from this.□

We can thereby proceed to make sure that our very weak solutions are in fact weak solutions whenever the hypotheses from Theorem 1.5 are met.

Proof of Theorem 1.5

We take u, v and (εj)j∈ℕ as given by Theorem 1.4, and then infer from Lemma 4.6 and Lemma 4.7 that u and v have the regularity properties stated in (1.37). In particular, in the case β = 12 this entails the inclusion ∇vLlocq(Ω × [0, ∞);ℝn) with q = 2(n+2)n complying with the requirement in (1.27) due to the fact that then

q2αα1=2(n+2)n211α2(n+2)n212n+2=0

by (1.35). The claim therefore results by combining Theorem 1.3 with Theorem 1.4.

If β(n+24α,12), and hence α > n+22, observing that then

2(n+2)n+24β>2(n+2)n+24n+24α=2αα1

we may pick any q[2αα1,2(n+2)n+24β) to similarly conclude on the basis of (1.37) that Theorem 1.3 and Theorem 1.4 imply the stated solution properties of (u, v).□

Proof of Corollary 1.6

We only need to apply Theorem 1.5 to α := 2, β := 12 and q := 4, and once more make use of the well-known fact that then D(Aβ)=D(A12)=W1,2(Ω).

5 Application to a Shigesada-Kawasaki-Teramoto system

We will next focus on the Shigesada-Kawasaki-Teramoto system (1.39) under the standing assumptions that d1, d2 and μ1 are positive, that a12, a22, μ2, a1 and a2 are nonnegative, and that

u0L1(Ω)andv0L(Ω)are nonnegative.(5.1)

As approximations of (1.39) convenient for our purposes, for ε ∈ (0, 1) we shall consider

uεt=d1Δuε+a12Δ(uεvε)+μ1uε(1uεa1vε),xΩ,t>0,vεt=d2Δvε+a22Δvε2+μ2vε(1vεuε1+εuε),xΩ,t>0,uεν=vεν=0,xΩ,t>0,uε(x,0)=u0ε(x),vε(x,0)=v0ε(x),xΩ,(5.2)

where we take any (u0ε)ε∈(0, 1)C1(Ω) and (v0ε)ε∈(0, 1)C3(Ω) such that v0εν=0onΩ that

0u0εu0in L1(Ω)and0v0εv0a.e. inΩas ε0,and thatsupε(0,1)v0εL(Ω)<.(5.3)

We shall see that (1.39) and (5.2) fall among the class of problems covered by our general theory if we let A(x, t) := (d1 + a12v(x, t))(δij)i,j=1,…,n, (Aε)(x, t) := (d1 + a12vε(x, t))(δij)i,j=1,…,n, b(x, t) := a12v(x, t), bε(x, t) := a12vε(x, t), f(x, t, s) := –μ1 s + μ1 s2, g(x, t) := –μ1a1u(x, t)v(x, t) and gε(x, t) := –μ1a1uε(x, t)vε(x, t) for xΩ, t > 0, s ≥ 0 and ε ∈ (0, 1).

Several of our overall requirements, and in particular (1.15), are already asserted by the following basic statement on global classical solvability of (5.2) that can be derived by straightforward adaptation of standard arguments:

Lemma 5.1

For any ε ∈ (0, 1), the problem (5.2) admits a global classical solution (uε, vε) with

uεC0(Ω¯×[0,))C2,1(Ω¯×(0,))andvεq>nC0([0,);W1,q(Ω))C2,1(Ω¯×(0,)).(5.4)

Moreover, uε and vε are nonnegative with

vε(x,t)max{1,v0εL(Ω)}forallxΩandt>0.(5.5)

Proof

Standard theory ([1]) asserts local existence of a solution with the indicated regularity properties, extensible up to a maximal existence time Tmax,ε ∈ (0, ∞] such that either Tmax,ε = ∞, or lim suptTmax,εuε(⋅, t)∥L(Ω) = ∞. As three applications of the comparison principle assert nonnegativity of uε and vε as well as the inequality in (5.5), by relying on the boundedness of the reaction term in the second equation from (5.2) we may invoke known results on gradient regularity in scalar parabolic problems ([20]) to see that if Tmax,ε < ∞ then ∇vε is bounded in Ω × (0, Tmax,ε). By means of a straightforward reasoning based on Lp-Lq estimates for the Neumann heat semigroup, this in turn warrants boundedness of uε throughout Ω × (0, Tmax,ε) in this case, by contradiction to the above thus showing that actually Tmax,ε = ∞.□

5.1 Very weak solutions. Proof of Theorem 1.7

As before starting with the construction of very weak solutions, we first collect some basic properties of solutions to (5.2), and especially of the second solution component vε.

Lemma 5.2

Let T > 0. Then there exists C(T) > 0 such that

Ωuε(x,t)dxC(T)forallt(0,T)andε(0,1)(5.6)

and

0TΩuε2(x,t)dxdtC(T)forallε(0,1)(5.7)

as well as

0TΩ|vε(x,t)|2dxdtC(T)forallε(0,1)(5.8)

and

0Tvεt(,t)(W1,2(Ω))2dtC(T)forallε(0,1).(5.9)

Proof

The estimates in (5.6) and (5.8) directly result from Lemma 2.2. To verify (5.8), we test the second equation in (5.2) by vε and thereby obtain that for all t > 0,

12ddtΩvε2+d2Ω|vε|2=2a22Ωvε|vε|2+μ2Ωvε2μ2Ωvε3μ2a2Ωuε1+εuεvε2μ2Ωvε2,

so that

d20TΩ|vε|212Ωv0ε2+μ20TΩvε2,

from which (5.8) follows due to (5.5).

Finally, (5.9) can be derived from this in a standard manner by using (5.2) to see that for all ϕC1(Ω) with ϕL2(Ω)2+ϕL2(Ω)21

|Ωvεt(,t)ϕ|=|d2Ωvεϕ2a22Ωvεvεϕ+μ2Ωvεϕμ2Ωvε2ϕμ2a2Ωuε1+εuεvεϕ|d2vεL2(Ω)+2a22vεL(Ω)vεL2(Ω)+μ2vεL2(Ω)+μ2vεL4(Ω)2+μ2a2uεL2(Ω)vεL(Ω)

for all t > 0 and ε ∈ (0, 1), whence by Young’s inequality,

vεt(,t)(W1,2(Ω))25(d22+a222vεL(Ω)2)Ω|vε|2+5μ22Ωvε2+5μ22Ωvε4+5μ22a22vεL(Ω)2Ωuε2for allt>0andε(0,1),

implying (5.9) upon integrating and using (5.5), (5.8) and (5.7).□

Again, some approximation properties of (5.2) thereby become quite obvious.

Lemma 5.3

There exists (εj)j∈ℕ ⊂ (0, 1) such that εj ↘ 0 as j → ∞, and nonnegative functions u and v on Ω × (0, ∞) which for each T > 0 satisfy

uL((0,T);L1(Ω)L2(Ω×(0,T))andvL(Ω×(0,T))L2((0,T);W1,2(Ω)),(5.10)

and which are such that for all T > 0,

uεuinL1(Ω×(0,T))anda.e.inΩ×(0,T),(5.11)

uεuinL2(Ω×(0,T)),(5.12)

vεvinL2(Ω×(0,T))anda.e.inΩ×(0,T),andthat(5.13)

vεvinL2(Ω×(0,T))(5.14)

as ε = εj ↘ 0. Moreover, the second equation in (1.39) is satisfied in the sense that (1.41) holds for all φC0(Ω × [0, ∞)).

Proof

The existence of a sequence (εj)j∈ℕ and limit functions u and v with the properties in (5.11)-(5.14) immediately results from a straighforward extraction process based on Lemma 2.5, Lemma 5.2, (5.5) and the Aubin-Lions lemma. The verification of (1.41) can thereupon be achieved on testing the second equation in (5.2) by φC0(Ω × [0, ∞)) and observing that (5.12)-(5.14) are especially sufficient for passing to the limit in each of the respective nonlinear contributions in the sense that

0Ωvεvεφ0Ωvvφand0Ωvε2φ0Ωv2φ

as well as

0Ωuε1+εuεvεφ0Ωuvφ

as ε = εj ↘ 0, the latter because clearly also uε1+εuεuinLloc2(Ω¯×[0,)) according to (5.11), (5.12) and Egorov’s theorem.□

In comparison with the corresponding statement from the previous section in Lemma 4.5, due to the presence of nonlinear diffusion in the second equation from (5.2) the derivation of a strong convergence feature of ∇vε here requires an additional argument.

Lemma 5.4

Let (εj)j∈ℕ and v be as in Lemma 5.3. Then there exist a null set N ⊂ (0, ∞) and a subsequence, again denoted by (εj)j∈ℕ, such that

vεvinL2(Ω×(0,t0))forallt0(0,)Nasε=εj0.(5.15)

Proof

As v belongs to L(Ω × (0, T)) ∩ L2((0, T);W1,2(Ω)) for each T > 0, by means of a standard approximation procedure (see e.g. [40, Lemma 8.2]) it can be verified that v can be used as a test function in (1.41) in the sense that if we fix a null set N1 ⊂ (0, ∞) such that (0, ∞) ∖ N1 exclusively consists of Lebesgue points of (0, ∞) ∋ t ↦ ∫Ω v2(⋅, t), then

12Ωv2(,t0)12Ωv02d20t0Ω|v|22a220t0Ωv|v|2+μ20t0Ωv2μ20t0Ωv3μ2a20t0Ωuv2(5.16)

for all t0 ∈ (0, ∞) ∖ N1, where actually even equality can be achieved but will not be needed here. To further exploit this, according to (5.13) we fix a second null set N2 ⊂ (0, ∞) such that for all t0 ∈ (0, ∞) ∖ N2 we have Ωvε2(,t0)Ωv2(,t0)asε=εj0. Apart from that, we note that in the most complex case when a22 is positive, writing ρ(s):=13a22d2+2a22s,s0, we see that due to (5.5) and (5.8) the family (∇ρ(vε))ε∈(0, 1)(d2+2a22vεvε)ε(0,1) is bounded in L2(Ω × (0, T)) for all T > 0, which in view of (5.13) means that for any such T,

ρ(vε)ρ(v)in L2(Ω×(0,T))as ε=εj0.(5.17)

By lower semicontinuity of L2 norms with respect to weak convergence, this implies that

0t0Ω|ρ(v)|2lim infε=εj00t0Ω|ρ(vε)|2=lim infε=εj0{d20t0Ω|vε|2+2a220t0Ωvε|vε|2}for all t0>0,(5.18)

while on the other hand, by definition of N2 and by (5.3), (5.13), (5.12), (5.5) and the dominated convergence theorem, testing the second equation in (5.2) by vε we readily find that for all t0 ∈ (0, ∞) ∖ N2,

d20t0Ω|vε|2+2a220t0Ωvε|vε|2=12Ωv0ε212Ωvε2(,t0)+μ20t0Ωvε2μ20t0Ωvε3μ2a20t0Ωuε1+εuεvε212Ωv0212Ωv2(,t0)+μ20t0Ωv2μ20t0Ωv3μ2a20t0Ωuv2

as ε = εj ↘ 0. In conjunction with (5.16) and (5.18), this shows that if we let N := N1N2, then

0t0Ω|ρ(vε)|20t0Ω|ρ(v)|2for all t0(0,)Nas ε=εj0,

and that hence, by (5.17),

ρ(vε)ρ(v)in L2(Ω×(0,t0))for all t0(0,)Nas ε=εj0(5.19)

and therefore also

|ρ(vε)|2|ρ(v)|2in L1(Ω×(0,t0))for all t0(0,)Nas ε=εj0.(5.20)

On particularly choosing t0 = t0k here, with (t0k)k∈ℕ ⊂ (0, ∞) ∖ N fulfilling t0k ↗ ∞ as k → ∞, we easily infer from (5.19) that passing to a conveniently relabeled subsequence we can achieve that also ∇ρ(vε) → ∇ρ(v) a.e. in Ω × (0, ∞)) and thus, by (5.13) and positivity of ρ(s)=d2+2a22son[0,),

vεva.e. in Ω×(0,)(5.21)

as ε = εj ↘ 0. Since furthermore

|vε|2=1d2+2a22vε|ρ(vε)|21d2|ρ(vε)|2in Ω×(0,)for all ε(0,1),

a combination of (5.21) with (5.20) and the dominated convergence theorem shows that

0t0Ω|vε|20t0Ω|v|2for all t0(0,)Nas ε=εj0,

which together with (5.14) entails (5.15) when a22 > 0. In the case a22 = 0 of linear diffusion, the argument actually becomes much simpler and may thus be omitted here.□

We can thus apply Theorem 1.2 in a straightforward manner to achieve the claimed results on very weak solvability in (1.39).

Proof of Theorem 1.7

Thanks to the strong convergence result from Lemma 5.4, in view of Theorem 1.2 and Lemma 5.3 we only need to make sure that gε := –μ1a1uεvε satisfies gεg in Lloc1(Ω × [0, T)) as ε = εj ↘ 0 for each T > 0. However, since gεg a.e. in Ω × (0, ∞) by (5.11) and (5.12), and since |gε| ≤ c1(T) uε in Ω × (0, T) with c1(T) := μ1a1 supε∈(0, 1)vεL(Ω×(0,T)) being finite due to (5.5), by means of the dominated convergence theorem this directly results from the L1 convergence property of (uε)ε∈(εj)j∈ℕ in (5.10).□

5.2 Weak solutions. Proof of Theorem 1.8

As in Section 4, higher regularity of the flux term ∇v will result from suitably strengthened assumptions on the corresponding initial data. Accordingly and in line with the hypotheses from Theorem 1.8, we now assume that beyond (5.1) and (5.3) we have

v0W1,2(Ω)andsupε(0,1)v0εL2(Ω)<.(5.22)

Then using a standard multiplier for the nonlinear diffusion equation for vε in (5.2) yields the following.

Lemma 5.5

Assume (5.22), and let

P(s):=d2s+a22s2,s0.(5.23)

Then for all T > 0 there exists C(T) > 0 such that

Ω|P(vε(,t))|2C(T)forallt(0,T)andε(0,1),(5.24)

and that

0TΩ|ΔP(vε)|2C(T)forallε(0,1).(5.25)

Proof

Integrating by parts and using Young’s inequality in the second equation from (5.2), for all t > 0 we obtain

12ddtΩ|P(vε)|2=ΩΔP(vε)P(vε)vεt=ΩP(vε)|ΔP(vε)|2μ2ΩP(vε)ΔP(vε)vε(1vεuε1+εuε)12ΩP(vε)|ΔP(vε)|2+μ222ΩP(vε)vε2(1vεuε1+εuε)212ΩP(vε)|ΔP(vε)|2+3μ222ΩP(vε)vε2(1+vε2+a22uε2).(5.26)

Here we recall that by (5.5) and (5.3) we can find c1 > 0 such that vεc1 in Ω × (0, T) for all ε ∈ (0, 1), so that in view of (5.23),

d2P(vε)c2:=d2+2a22c1inΩ×(0,T)for allε(0,1).(5.27)

Accordingly,

ΩP(vε)vε2(1+vε2+a22uε2)c12c2|Ω|+c14c2|Ω|+a22c2Ωuε2for all t(0,T),

and thus (5.26) implies that

ddtΩ|P(vε)|2+ΩP(vε)|Δvε|2c3+c4Ωuε2for all t(0,T)

with c3:=3μ22(c12c2|Ω|+c14c2|Ω|)andc4:=3μ22a22c2. Upon integration, again by (5.27) this entails that

Ω|P(vε(,t))|2+d20tΩ|ΔP(vε)|2Ω|P(vε(,t))|2+0tΩP(vε)|ΔP(vε)|2ΩP2(v0ε)|v0ε|2+c3T+c40TΩuε2c22Ω|v0ε|2+c3T+c40TΩuε2for all t(0,T)

and hence establishes (5.24) and (5.25) due to (5.22) and (5.7).□

Once more by interpolation, this has a favorable consequence on integrability of ∇vε.

Lemma 5.6

If (5.22) holds, then for all T > 0 one can find C(T) > 0 fulfilling

0TΩ|vε|4C(T)forallε(0,1).(5.28)

Proof

Again taking P as defined in (5.23), from the Gagliardo-Nirenberg inequality and elliptic regularity theory ([10]) we obtain c1 > 0 such that

0TΩ|P(vε)|4c10TΔP(vε(,t))L2(Ω)2P(vε(,t)L(Ω)2dtc1c20TΩ|ΔP(vε)|2for all ε(0,1),(5.29)

where c1 := supε∈(0, 1)P(vε)∥L(Ω×(0, T)) = supε∈(0, 1) big{d2vεL(Ω×(0, T)) + a22 vεL(Ω×(0,T))2} is finite according to Lemma 5.1. Since |∇P(vε)| = (d2 + 2a22vε)| ∇vε| ≥ d2 |∇vε| in Ω × (0, ∞), due to Lemma 5.5 we directly infer (5.28) from (5.29).□

In conclusion, Theorem 1.3 can be applied so as to yield our claimed results on global existence of weak solutions in (1.39) for initial data merely belonging to L1 × (W1,2L).

Proof of Theorem 1.8

We let (εj)j∈ℕ, u and v be as provided by Theorem 1.7. Then due to the fact that clearly 0TΩ |∇v|4 ≤ lim infε=εj↘0 0TΩ | ∇vε|4 for all T > 0 by Lemma 5.6, we may apply Theorem 1.3 to q := 4 and α := 2 and thereby infer the claimed additional regularity and solution properties, beyond those guaranteed by Theorem 1.7, of u. It thus remains to note that the inclusion vLloc([0,);W1,2(Ω)) is a by-product of (5.24) when once more combined with the fact that P from (5.23) satisfies P′ ≥ d2 > 0 throughout [0, ∞).□

Acknowledgement

The author acknowledges support of the Deutsche Forschungsgemeinschaft in the context of the project Analysis of chemotactic cross-diffusion in complex frameworks.

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About the article

Received: 2018-08-14

Accepted: 2018-10-14

Published Online: 2019-06-16

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 526–566, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0013.

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© 2020 Michael Winkler, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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