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Advances in Nonlinear Analysis

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


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Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems

Davide Addona
  • Dipartimento di Scienze Matematiche, Fisiche ed Informatiche, Edificio di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy
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  • Dipartimento di Scienze Matematiche, Fisiche ed Informatiche, Edificio di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy
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Published Online: 2017-02-04 | DOI: https://doi.org/10.1515/anona-2016-0166

Abstract

We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over d and in Lp-spaces with respect to tight evolution systems of measures. Here, the linear part of the equation is a nonautonomous second-order elliptic operator with unbounded coefficients defined in I×d, (I being a right-halfline). To the above Cauchy problem we associate a nonlinear evolution operator, which we study in detail, proving some summability improving properties. We also study the stability of the null solution to the Cauchy problem.

Keywords: Nonautonomous second-order elliptic operators; semilinear parabolic equations; unbounded coefficients; hypercontractivity; supercontractivity; ultraboundedness; stability

MSC 2010: 35K58; 37L15

1 Introduction

This paper is devoted to continuing the analysis started in [5]. We consider a family of linear second-order differential operators 𝒜(t) acting on smooth functions ζ as

(𝒜(t)ζ)(x)=i,j=1dqij(t,x)Dijζ(x)+i=1dbi(t,x)Diζ(x),tI,xd,(1.1)

where I is either an open right halfline or the whole . Then, given T>sI, we are interested in studying the nonlinear Cauchy problem

{Dtu(t,x)=(𝒜(t)u)(t,x)+ψu(t,x),(t,x)(s,T]×d,u(s,x)=f(x),xd,(1.2)

where ψu(t,x)=ψ(t,x,u(t,x),xu(t,x)). We assume that the coefficients qij and bi (i,j=1,,d), possibly unbounded, are smooth enough, the diffusion matrix Q=[qij]i,j=1,,d is uniformly elliptic and there exists a Lyapunov function φ for 𝒜(t) (see Hypothesis 2.1 (iii)). These assumptions yield that the linear part 𝒜(t) generates a linear evolution operator {G(t,s):tsI} in Cb(d). More precisely, for every fCb(d) and sI, the function G(,s)f belongs to Cb([s,+)×d)C1,2((s,+)×d), it is the unique bounded classical solution of the Cauchy problem (1.2), with ψu0, and satisfies the estimate

G(t,s)ff,t>sI,fCb(d).(1.3)

We refer the reader to [9] for the construction of the evolution operator G(t,s) and for further details.

Classical arguments can be adapted to our case to prove the existence of a unique local mild solution uf of problem (1.2) for any fCb(d), i.e., a function u:[s,τ]×d (for some τ>s) such that

u(t,x)=(G(t,s)f)(x)+st(G(t,r)ψu(r,))(x)𝑑r,t[s,τ],xd.(1.4)

Under reasonable assumptions such a mild solution uf is classical, defined in the whole [s,+) and satisfies the condition

uf+supt(s,T)t-sxuf(t,)<+

for any T>s. Hence, setting 𝒩(t,s)f=uf(t,) for any t>s we deduce that 𝒩(t,s) maps Cb(d) into Cb1(d) and, from the uniqueness of the solution to (1.2), it follows that it satisfies the evolution law

𝒩(t,s)f=𝒩(t,r)𝒩(r,s)f

for any r(s,t) and fCb(d).

As in the linear case we are also interested to set problem (1.2) in an Lp-context. However, as it is already known from the linear case, the most natural Lp-setting where problems with unbounded coefficients can be studied is that related to the so-called evolution systems of measures [6], that is one-parameter families of Borel probability measures {μt:tI} such that

dG(t,s)f𝑑μt=df𝑑μs,fCb(d),t>sI.(1.5)

When they exist, evolution families of measures are in general infinitely many, even the tight ones, where, roughly speaking, tight means that all the measures of the family are essentially concentrated on the same large ball (see Section 2 for a rigorous definition of tightness). Under additional assumptions on the coefficients of the operator 𝒜(t) (see Section 2), there exists a unique tight evolution system of measures {μt:tI}, which has the peculiarity to be the unique system related to the asymptotic behavior of G(t,s) as t tends to +. We also mention that, typically, even if for ts, the measures μt and μs are equivalent (being equivalent to the restriction of the Lebesgue measure to the Borel σ-algebra in d), the corresponding Lp-spaces differ.

Formula (1.5) and the density of Cb(d) in Lp(d,μs) allow to extend G(t,s) to a contraction from Lp(d,μs) to Lp(d,μt) for any t>s and any p[1,+) and to prove very nice properties of G(t,s) in these spaces.

In view of these facts, it is significant to extend 𝒩(t,s) to an operator from Lp(d,μs) to Lp(d,μt) for any Is<t. This can be done if pp0 (see Hypothesis 2.1 (v)), ψ(t,x,,) is Lipschitz continuous in d+1 uniformly with respect to (t,x)(s,T]×d and, in addition, supt(s,T]t-sψ(t,,0,0)Lp(d,μt)<+. In particular, each operator 𝒩(t,s) is continuous from Lp(d,μs) to W1,p(d,μt).

We stress that the first condition on ψ may seem too restrictive, but in fact it is not. Indeed, the Sobolev embedding theorems fail to hold, in general, when the Lebesgue measure is replaced by any of the measures μt. This can be easily seen in the particular case of the one-dimensional Ornstein–Uhlenbeck operator, where the evolution system of measures is replaced by a time-independent measure μ (the so-called invariant measure), which is the Gaussian centered at zero with covariance 1/2. For any ε>0, the function xexp(2(2p+ε)-1|x|2) belongs to Wk,p(,μ) for any k but it does not belong to Lp+ε(d,μ).

Under the previous assumptions, for any fLp(d,μs), 𝒩(,s) can be identified with the unique mild solution to problem (1.2) which belongs to Lp((s,T)×d,μ)Wp0,1(J×d,μ), for any J(s,T], such that uf(t,)W1,p(d,μt) for almost every t(s,T]. Here, μ is the unique Borel measure on the σ-algebra of all the Borel subsets of I×d which extends the map defined on the product of a Borel set AI and a Borel set Bd by

μ(A×B):=Aμt(B)𝑑t.

Since, as it has been stressed, in this context the Sobolev embedding theorems fail to hold in general, the summability improving properties of the nonlinear evolution operator 𝒩(t,s) are not immediate and true in all the cases. For this reason in Section 4 we investigate properties such as hypercontractivity, supercontractivity, ultraboundedness of the evolution operator 𝒩(t,s) and its spatial gradient. Differently from [5], where ψ=ψ(t,u) and the hypercontractivity of 𝒩(t,s) is proved assuming ψ(t,0)=0 for any t>s, here we consider a more general case. More precisely we assume that there exist ξ00 and ξ1,ξ2 such that uψ(t,x,u,v)ξ0|u|+ξ1u2+ξ2|u||v| for any ts, x,vd, u. Under some other technical assumptions on the growth of the coefficients qij and bi (i,j=1,,d) as |x|+, we show that as in the linear case, (see [3, 4]), the hypercontractivity and the supercontractivity of 𝒩(t,s) and x𝒩(t,s) are related to some logarithmic Sobolev inequalities with respect to the tight system {μt:tI}. These estimates are the natural counterpart of the Sobolev embedding theorems in the context of invariant measures and evolution systems of measures.

For what concerns the ultraboundedness of 𝒩(t,s) and x𝒩(t,s) we first prove an Harnack-type estimate which establishes a pointwise estimate of |𝒩(t,s)f|p in terms of G(t,s)|f|p for any fCb(d), p>p0 and t>s. This estimate, together with the evolution law and the ultraboundedness of G(t,s), allow us to conclude that, for any fLp(d,μs) and any t>s, the function 𝒩(t,s)f belongs to W1,(d,μt) and to prove an estimate of 𝒩(t,s)fW1,(d,μt) in terms of fLp(d,μs).

Finally, assuming that ψ(t,x,0,0)=0 for every t(s,+) and xd, we prove that the trivial solution to the Cauchy problem (1.2) is exponentially stable both in W1,p(d,μt) and in Cb1(d). This means that uf(t,)XCXe-ωXt as t+ for some constants CX>0 and ωX<0, both when X=W1,p(d,μt) and X=Cb1(d). In the first case, the space X depends itself on t. We stress that, under sufficient conditions on the coefficients of the operators 𝒜(t), which include their convergence at infinity, in [2, 11] it has been proved that the measure μt weakly* converges to a measure μ, which turns out the invariant measure of the operator 𝒜, whose coefficients are the limit as t+ of the coefficients of the operator 𝒜(t). This gives more information on the convergence to zero of uf(t,)W1,p(d,μt) at infinity. We refer the reader also to [12] for the case of T-time periodic coefficients.

To get the exponential stability of the trivial solution in Cb(d), differently from [5] where a nonautonomous version of the principle of linearized stability is used and more restrictive assumptions on ψ are required, we let p tend to + in the decay estimate of uf(t,)W1,p(d,μt), since all the constants appearing in this estimate admit finite limit as p tends to +. In particular, we stress that we do not need any additional assumptions on the differentiability of ψ but, on the other hand, we require that the mild solution uf of (1.2) is actually classical.

Notations

For k0, by Cbk(d) we mean the space of the functions in Ck(d) which are bounded together with all their derivatives up to the [k]-th order. Cbk(d) is endowed with the norm

fCbk(d)=|α|[k]Dαf+|α|=[k][Dαf]Cbk-[k](d),

where [k] denotes the integer part of k. When k, we use the subscript “loc” to denote the space of all fC[k](d) such that the derivatives of order [k] are (k-[k])-Hölder continuous in any compact subset of d. Given an interval J, we denote by B(J×d;Lip(d+1)) and Cα/2,α(J×d) (α(0,1)), respectively, the set of all functions f:J×d××d such that f(t,x,,) is Lipschitz continuous in d+1, uniformly with respect to (t,x)J×d, and the usual parabolic Hölder space. The subscript “loc” has the same meaning as above.

We use the symbols Dtf, Dif and Dijf to denote respectively the time derivative ft and the spatial derivatives fxi and 2fxixj for any i,j=1,,d.

The open ball in d centered at 0 with radius r>0 and its closure are denoted by Br and B¯r, respectively. For any measurable set A, contained in or in d, we denote by 𝟙A the characteristic function of A. Finally, we write AB when A is compactly contained in B.

2 Assumptions and preliminary results

Let {𝒜(t):tI} be the family of linear second-order differential operators defined by (1.1).

Hypotheses 2.1.

Our standing assumptions on the coefficients of the operators 𝒜(t) are as follows.

  • (i)

    The coefficients qij,bi belong to Clocα/2,1+α(I×d) for any i,j=1,,d and some α(0,1).

  • (ii)

    For every (t,x)I×d, the matrix Q(t,x)=[qij(t,x)]ij is symmetric and there exists a function

    κ:I×d,

    with positive infimum κ0, such that

    Q(t,x)ξ,ξκ(t,x)|ξ|2

    for any (t,x)I×d and any ξd.

  • (iii)

    There exists a nonnegative function φC2(d), diverging to + as |x|+, such that

    (𝒜(t)φ)(x)a-cφ(x)

    for any (t,x)I×d and some positive constants a and c.

  • (iv)

    There exists a locally bounded function ρ:I+ such that

    |xqij(t,x)|ρ(t)κ(t,x)

    for any (t,x)I×d and any i,j=1,,d, where κ is defined in (ii).

  • (v)

    There exists a function r:I×d such that

    xb(t,x)ξ,ξr(t,x)|ξ|2

    for any ξd and (t,x)I×d. Further, there exists p0(1,2] such that

    +>σp0=sup(t,x)I×d(r(t,x)+d3(ρ(t))2κ(t,x)4min{p0-1,1}).(2.1)

Under Hypotheses 2.1 (i)–(iii) (actually even under weaker assumptions) it is possible to associate an evolution operator {G(t,s):tsI} to the operator 𝒜(t) in Cb(d), as described in the Introduction. The function G(,)f is continuous in {(s,t,x)I×I×d:st} and

(G(t,s)f)(x)=df(y)p(t,s,x,dy),Is<t,xd,(2.2)

where p(t,s,x,dy) are probability measures for any Is<t, x,yd. This implies that

|G(t,s)f|pG(t,s)(|f|p)

for any Is<t, fCb(d) and p1. Moreover, Hypotheses 2.1 (iv) and (v) yield the pointwise gradient estimates

|(xG(t,s)f)(x)|pepσp(t-s)(G(t,s)|f|p)(x),fCb1(d),(2.3)|(xG(t,s)f)(x)|p{Cεpep(σp+ε)(t-s)(t-s)-p2(G(t,s)|f|p)(x),if σp<0,C0p(1+(t-s)-p2)(G(t,s)|f|p)(x),otherwise,(2.4)

for any fCb(d), t>s, xd, p[p0,+), ε>0 and some positive constants C0 and Cε, where σp is given by (2.1), with p instead of p0. We stress that the pointwise estimates (2.3) and (2.4) have been proved with the constants C0 and Cε also depending of p. Actually, these constants may be taken independent of p. Indeed, consider for instance estimate (2.4). If pp0, then using the representation formula (2.2) we can estimate

|xG(t,s)f|p=(|xG(t,s)|p0)pp0(Cp0p0(1+(t-s)-p02)G(t,s)|f|p0)pp02p-p0p0Cp0p(1+(t-s)-p2)(G(t,s)|f|p0)pp02p-p0p0Cp0p(1+(t-s)-p2)G(t,s)|f|p

for any t>sI and fCb(d), and, hence, estimate (2.4) holds true with a constant which can be taken independent of p.

Remark 2.2.

The case p=1 in estimate (2.3) is much more delicate and requires stronger assumptions. Indeed, as [1] shows, the algebraic condition Dhqij+Diqjh+Djqih=0 in I×d for any i,j,h{1,,d} with ijh is a necessary condition for (2.3) (with p=1) to hold. For this reason, if the diffusion coefficients are bounded and independent of x, then the pointwise gradient estimate (2.3) holds true also with p=1 and σ1=r0, where r0 is the supremum over I×d of the function r in Hypothesis 2.1 (v).

Under Hypotheses 2.1 we can also associate an evolution system of measures {μt:tI} with the operators 𝒜(t). Such a family of measures is tight, namely for every ε>0 there exists r>0 such that μs(dBr)<ε for any sI. The invariance property (1.5) and the density of Cb(d) in Lp(d,μs), sI, allows to extend G(t,s) to a contraction from Lp(d,μs) to Lp(d,μt) for any t>s. As it has been stressed in the Introduction, in general evolution systems of measures are infinitely many, but, under suitable assumptions, there exists a unique tight evolution system of measures. This is, for instance, the case when Hypotheses 2.1 are satisfied as well as the following two conditions:

  • (i)

    qij and bi belong to Clocα/2,1+α([a,+)×d) for any i,j=1,,d and some aI. Moreover, qij belongs to Cb([a,+)×BR) and Dkqij,bj belong to Cb([a,+);Lp(BR)) for any i,j,k{1,,d}, R>0 and some p>d+2.

  • (ii)

    There exists a constant c>0 such that either |Q(t,x)|c(1+|x|)φ(x) and b(t,x),xc(1+|x|2)φ(x) for any (t,x)[a,+)×d, or the diffusion coefficients are bounded in [a,+)×d.

For more details and the proofs of the results that we have mentioned, we refer the reader to [9, 10, 11, 13].

3 The semilinear problem in a bounded time interval

Given Is<T, we are interested in studying the Cauchy problem (1.2) both in the case when fCb(d) and in the case when fLp(d,μs).

Hypotheses 3.1.

Our standing assumptions on ψ are as follows.

  • (i)

    The function ψ:[s,T]×d××d is continuous. Moreover, there exists β[0,1) such that for any R>0 and some constant LR>0

    |ψ(t,x,u1,(t-s)-12v1)-ψ(t,x,u2,(t-s)-12v2)|LR(t-s)-β(|u1-u2|+|v1-v2|)(3.1)

    for any t(s,T], xd, u1,u2[-R,R], v1,v2B¯R.

  • (ii)

    The function ψ(,,0,0) belongs to Cb([s,T]×d).

Theorem 3.2.

Under Hypotheses 2.1 and 3.1, for any f¯Cb(Rd) there exist constants r0,δ(0,T-s] such that, if fCb(Rd) and f-f¯r0, then the nonlinear Cauchy problem (1.2) admits a unique mild solution ufCb([s,s+δ]×Rd)C0,1((s,s+δ]×Rd) which satisfies the estimate

uf+supt(s,s+δ]t-sxuf(t,)2(1+2C0+C0δ)(fCb(d)+2δψ(,,0,0)Cb([s,s+δ]×d)),(3.2)

where C0 is the constant in (2.4). Moreover, for any R>0, θ(0,1) and t(s,s+δ], uf(t,) belongs to C1+θ(BR) and there exists a positive constant CR,T-s such that

supt(s,s+δ](t-s)1+θ2uf(t,)C1+θ(BR)CR,T-sf.

Finally, if gCb(Rd) is such that g-f¯r0, then

uf-ug+supt(s,s+δ]t-sxuf(t,)-xug(t,)2(1+C0+C0δ)f-g.(3.3)

Proof.

Even if the proof is quite standard, for the reader’s convenience we provide some details.

Fix f¯Cb(d). Let R0>0 be such that R0/(1+K0)8f¯, where K0=C0(1+T-s) and C0 is the constant in (2.4). Further, for any δ(0,T-s], let Yδ be the set of all uCb([s,s+δ]×d)C0,1((s,s+δ)×d) such that uYδ=uCb((s,s+δ]×d)+supt(s,s+δ]t-sxu(t,)<+.

Step 1. We prove that there exists δ>0 such that, for any fCb(d) satisfying the condition

f-f¯r0:=R04+4K0,

there exists a mild solution to problem (1.2) defined in the time interval [s,s+δ]. For this purpose, we consider the operator Γ, defined by the right-hand side of (1.4) for any uBYδ(R0) (the ball of Yδ centered at zero with radius R0). Clearly, the function ψu is continuous in (s,s+δ]×d and ψu(t,) is bounded in d for any t(s,s+δ]. Moreover, estimating |ψu(t,x)||ψu(t,x)-ψ(t,x,0,0)|+|ψ(t,x,0,0)| and taking (3.1) into account, we can easily show that the function t(t-s)βψu(t,) is bounded in (s,s+δ). Hence, Proposition A.1 and estimates (1.3) and (2.4) show that Γ(u)Yδ for any t(s,s+δ] and uBYδ(R0). To show that, for a suitable δ(0,1], Γ is a 1/2-contraction in BYδ(R0), we observe that, using again (3.1), it follows that

ψu(t,)-ψv(t,)LR0(t-s)-βu-vYδ,t(s,s+δ],(3.4)

for any u,vBYδ(R0), where LR0 is the constant in Hypothesis 3.1 (i). From this inequality and estimates (1.3) and (2.4) we conclude that Γ(u)-Γ(v)Yδc1δ1-βu-vYδ for any u,vBYδ(R0), where c1, as the forthcoming constants, is independent of δ and u, if not otherwise specified. Hence, choosing δ properly, we can make Γ a 1/2-contraction in BYδ(R0).

It is also straightforward to see that Γ maps BYδ(R0) into itself, up to replacing δ with a smaller value if needed. It suffices to split Γ(u)=(Γ(u)-Γ(0))+Γ(0), use the previous result and estimate

Γ(0)Yδ(1+C0+C0δ)f+δ(1+2C0+C0δ)ψ(,,0,0)Cb([s,T]×d).

As a consequence, Γ has a unique fixed point in BYδ(R0), which is a mild solution of (1.2) and satisfies (3.2).

Step 2. We prove the uniqueness of the mild solution uf. For this purpose, let u1,u2Yδ be two mild solutions. By Lemma A.2, the function rh(r):=u1(r,)-u2(r,)+r-sxu1(r,)-xu2(r,) is measurable in (s,s+δ). Moreover, using (3.4), we easily deduce that

Dxju1(t,)-Dxju2(t,)c2(M)st(t-r)-j2(r-s)-βh(r)𝑑r(3.5)

for j=0,1, any t[s,s+δ], where M=max{u1Yδ,u2Yδ}. Estimating t-s with t-r+r-s for any r(s,t), from (3.5), with j=1, it follows that

t-sxu1(t,)-xu2(t,)c2(M)st(r-s)-βh(r)𝑑r+c2(M)st(t-r)-12(r-s)12-βu1(r,)-u2(r,)𝑑r+c2(M)st(t-r)-12(r-s)1-βxu1(r,)-xu2(r,)𝑑r.(3.6)

Using (3.5), we estimate the last two integral terms in the right-hand side of (3.6), which we denote by (t) and 𝒥(t). Replacing (3.5), with j=0, in (t), we get

(t)c3(M)δ1-βst(σ-s)-βh(σ)𝑑σ.(3.7)

The same arguments show that 𝒥(t) can be estimated pointwise in [s,s+δ] by the right-hand side of (3.7), with c3(M) being possibly replaced by a larger constant c4(M). Summing up, we have proved that

t-sxu1(t,)-xu2(t,)c5(M)δ1-βst(σ-s)-βh(σ)𝑑σ.(3.8)

From (3.5) and (3.8) we conclude that

h(t)c6(M,δ)st(r-s)-βh(r)𝑑r,t(s,s+δ].

The generalized Gronwall lemma (see [7]) yields h(t)0 for any t(s,s+δ), i.e., u1u2 in (s,s+δ)×d.

Step 3. We prove (3.2) and (3.3). Since uf=Γ(0)+(Γ(uf)-Γ(0)) and Γ is a 1/2-contraction in BYδ(R0), we conclude that ufYδ2Γ(0)Yδ and (3.2) follows from the estimate on Γ(0)Yδ proved above. Estimate (3.3) can be proved in the same way.

Step 4. We prove that uf(t,)C1+θ(BR) for any t(s,s+δ], R>0, θ(0,1), and

supt(s,s+δ](t-s)1+θ2uf(t,)C1+θ(BR)c7f

for some constant c7, independent of f. For this purpose, we observe that the results in the previous steps show that the function ψu satisfies the estimate (t-s)βψu(t,)c8f for any t(s,s+δ], the constant c8 being independent of f. Applying Proposition A.1 and estimate (A.5), we complete the proof. ∎

Corollary 3.3.

In addition to the assumption of Theorem 3.2 suppose that there exist β[0,1) and γ(0,1) such that 2β+γ<2 and

|ψ(t,x,u,(t-s)-12v)-ψ(t,y,u,(t-s)-12v)|CR(t-s)-β|x-y|γ(3.9)

for any t(s,T], x,y,vBR, u[-R,R], any R>0 and some positive constant CR. Then, for any fCb(Rd), the mild solution uf to problem (1.2) belongs to C1,2((s,s+δ]×Rd) and it is a classical solution to (1.2).

Proof.

Fix R>0. Theorem 3.2 shows that uf(t,) belongs to C1+γ(BR) and

xuf(t,)Cγ(BR)CR(t-s)-1+γ2f

for any t(s,s+δ]. Moreover, by interpolation from (3.2) it follows that uf(t,)Cbγ(d)C(t-s)-γ/2f for any t(s,s+δ]. From these estimates, adding and subtracting ψ(t,y,u(t,x),xu(t,x)), we deduce that |ψu(t,x)-ψu(t,y)|Cf(t-s)-β-γ2|x-y|γ for any t(s,s+δ], x,yd such that |x-y|R and some positive constant C, depending on R and u. As a byproduct, ψu(t,)Cγ(BR)C~(t-s)-β-γ2f for any t(s,s+δ] and some positive constant C~, depending on R and u. Now, using Proposition A.1, we conclude that uC1,2((s,s+δ]×d)Cloc0,2+θ((s,s+δ]×d) for any θ<γ if γα, and for θ=γ otherwise. ∎

Remark 3.4.

Suppose that (3.1) is replaced by the condition

|ψ(,,u1,v1)-ψ(,,u2,v2)|LR(|u1-u2|+|v1-v2|)

in [s,T]×d, for any R>0, u1,u2[-R,R], v1,v2BR and some positive constant LR. Then, the proof of the previous theorem can be repeated verbatim with Yδ=Cb0,1([s,s+δ]×d), endowed with the natural norm, and we can show that the mild solution to problem (1.2) belongs to Cb0,1([s,s+δ]×d) and ufCb0,1([s,s+δ]×d)C~δfCb1(d) for some positive constant C~δ, independent of f.

We now provide some sufficient conditions for the mild solution to problem (1.2) to exist in the large. Such conditions will be crucial to define the nonlinear evolution operator associated with the Cauchy problem (1.2).

Hypotheses 3.5.

We introduce the following assumptions.

  • (i)

    For any R>0 there exists a positive constant LR such that

    |ψ(t,x,u1,v1)-ψ(t,x,u2,v2)|LR(|u2-u1|+|v2-v1|)

    for any t[s,T], xd, u1,u2[-R,R] and v1,v2d.

  • (ii)

    For any τ>sI there exist positive constants k0, k1 and a, and a function φ~C2(d) with nonnegative values and blowing up at infinity such that uψ(t,x,u,v)k0(1+u2)+k1|u||v| and 𝒜φ~+k1|φ~|aφ~ in d for any t[s,τ], x,vd and u.

In the rest of this section, for any p[p0,+) and T>s we denote by [ψ]p,T the supremum over (s,T) of the function t-sψ(t,,0,0)Lp(d,μt); [ψ],T is defined similarly, replacing Lp(d,μt) by Cb(d).

Theorem 3.6.

Assume that Hypotheses 2.1, 3.1(ii), 3.5 and condition (3.9) are satisfied. Then, for any fCb(Rd), the classical solution uf to problem (1.2) exists in [s,T]. If, further, the constant in Hypothesis 3.5(i) is independent of R, then for any p[p0,+],

supt(s,T)(uf(t,)Lp(d,μt)+t-sxuf(t,)Lp(d,μt))CT-s(fLp(d,μs)+(T-s+1)[ψ]p,T),(3.10)supt(s,T)(uf(t,)-ug(t,)Lp(d,μt)+t-sxuf(t,)-xug(t,)Lp(d,μt))CT-sf-gLp(d,μs)(3.11)

for every f,gCb(Rd), where Cτ=(τ+1)ed1τ3/2+d2 for some positive constants d1 and d2.

Proof.

We split the proof into two steps.

Step 1. We prove that, for any fCb(d), uf is defined in the whole [s,T]. To this end, we fix fCb(d), denote by [s,τf) the maximal time domain where uf is defined and assume, by contradiction, that τf<T. We are going to prove that uf is bounded in [s,τf)×d. Once this is proved, we can use Hypotheses 3.5 (i) to deduce, adding and subtracting ψ(t,x,0,0), that |ψ(t,x,uf(t,x),v)|C(1+|v|) for t[s,T], x,vd and some constant C>0, which depends on ufCb([s,τf)×d) and ψ(,,0,0)Cb([s,T]×d). Applying the same arguments as in Step 2 of the proof of Theorem 3.2, we can show that also the function tt-sxuf(t,) is bounded in [s,τf)×d. This is enough to infer that uf can be extended beyond τf, contradicting the maximality of the interval [s,τf).

To prove that uf is bounded in (s,τf)×d, we fix b(0,τf-s), λ>a+k0 and we set

vn(t,x):=e-λ(t-s)uf(t,x)-n-1φ~(x)

for any (t,x)[s,s+b]×d. A straightforward computation shows that

Dtvn-𝒜vn=e-λ(-s)ψ(,,eλ(-s)(vn+n-1φ~),eλ(-s)(xvn+n-1φ~))-λ(vn+n-1φ~)+n-1𝒜φ~(3.12)

in (s,s+b]×d. Since uf is bounded in [s,s+b]×d and φ~ blows up at infinity, the function vn admits a maximum point (tn,xn). If vn(tn,xn)0 for any n, then uf0 in [s,s+b]×d. Assume that vn(tn,xn)>0 for some n. If tn=s, then vn(tn,xn)supdf. If tn>s, then Dtvn(tn,xn)-𝒜(tn)vn(tn,xn)0, so that, multiplying both the sides of (3.12) by vn(tn,xn)+n-1φ~(xn)>0 and using Hypotheses 3.5 (ii) we get 0(-λ+k0+a)(vn(tn,xn)+n-1φ~(xn))2+k0, which clearly implies that, also in this case, u is bounded from above in [s,s+b] by a constant, independent of b.

Repeating the same arguments with uf being replaced by -uf, we conclude that uf is bounded also from below by a positive constant independent of b. Since b is arbitrary, it follows that ufCb((s,τf)×d)<+ as claimed.

Step 2. Fix f,gCb(d), pp0 and let ||||||p be the norm defined by

|||v|||p=supt(s,T)e-ω(t-s)(v(t,)Lp(d,μt)+t-sxv(t,)Lp(d,μt))

on smooth functions v, where ω is a positive constant to be chosen later on and to fix the ideas we assume that p<+. From Hypothesis 3.5 (i), where LR is replaced by a constant L, it follows that

ψug(r,)-ψuf(r,)Lp(d,μr)Lug(r,)-uf(r,)W1,p(d,μr)

for any r(s,T]. Hence, recalling that each operator G(t,r) is a contraction from Lp(d,μr) to Lp(d,μt) and using the second pointwise gradient estimate in (2.4) and the invariance property of the family {μt:tI}, we conclude that

|||uf-ug|||p|||G(,s)(f-g)|||p+L|||uf-ug|||psupt(s,T)steω(r-t)(1+1r-s)𝑑r+LC0|||uf-ug|||psupt(s,T)t-ssteω(r-t)(1+1t-r)(1+1r-s)𝑑r[1+C0(1+T-s)]f-gLp(d,μs)+L|||uf-ug|||p[(1+C0T-s)(1ω+steω(r-t)r-sdr)+πωC0T-s+C0supt(s,T)t-ssteω(r-t)drt-rr-s].(3.13)

To estimate the integral terms in the last side of (3.13), we fix δ>0 and observe that

steω(r-t)r-s𝑑r=ss+δeω(r-t)r-s𝑑r+(s+δteω(r-t)r-s𝑑r)+2δ+1δω.(3.14)

Hence, minimizing over δ>0, we conclude that the left-hand side of estimate (3.14) is bounded from above by 8ω-1/2. Splitting t-st-r+r-s and arguing as above, also the last term in square brackets in the last side of (3.13) can be estimated by (8+π)ω-1/2. It thus follows that

|||uf-ug|||p[1+C0(1+T-s)]f-gLp(d,μs)+L(cT-sω-12+(1+C0T-s)ω-1)|||uf-ug|||p,

where cτ=(8+π)C0(τ+1)+8. Choosing ω such that cT-sω-1/2+(1+C0T-s)ω-1(2L)-1, we obtain

|||uf-ug|||p2[1+C0(1+T-s)]f-gLp(d,μs)

and estimate (3.11) follows at once.

Estimate (3.10) can be proved likewise. Hence, the details are omitted. ∎

As a consequence of Theorem 3.6 we prove the existence of a mild solution to problem (1.2) in the time domain (s,T) when fLp(d,μs), that is a function ufLp((s,T)×d,μ)Wp0,1(J×d,μ), for any J(s,T], such that uf(t,)W1,p(d,μt) for almost every t(s,T] and, for such values of t, the equality

uf(t,x)=(G(t,s)f)(x)+st(G(t,r)ψu(r,))(x)𝑑r

holds true in dAt, where At is negligible with respect to the measure μt (or, equivalently, with respect to the restriction of the Lebesgue measure to the Borel σ-algebra in d).

Corollary 3.7.

Under all the assumptions of Theorem 3.6, for any fLp(Rd,μs) (pp0) there exists a unique mild solution to the Cauchy problem (1.2). The function uf satisfies estimates (3.10) and (3.11) with the supremum being replaced by the essential supremum and, as a byproduct, ufWp0,1((s,T)×Rd,μ) if p<2, and ufWq0,1((s,T)×Rd,μ) for any q<2 otherwise. Finally, if there exists γ(0,1) such that

|ψ(t,x,ξ,η)-ψ(t,y,ξ,η)|CJ,R(1+|ξ|+|η|)|x-y|γ(3.15)

for any tJ, x,yBR, ηRd, ξR, J(s,T], R>0 and a constant CJ,R>0, then, for any fLp(Rd,μs) and almost every t(s,T), uf(t,) belongs to Wloc2,p(Rd). Moreover, ufWp,loc1,2((s,T)×Rd) and satisfies the equation Dtuf=Auf+ψuf.

Proof.

Fix fLp(d,μs) and let (fn)Cb(d) be a sequence converging to f in Lp(d,μs). By (3.11), (ufn(t,)) is a Cauchy sequence in W1,p(d,μt) for any t(s,T]. Hence, there exists a function v such that ufn(t,) converges to v(t,) in W1,p(d,μt) for any t(s,T]. Moreover, writing (3.10), with f being replaced by fn, and letting n tend to + we deduce that v satisfies (3.10) as well.

Next, using (3.11) we can estimate

ufn-ufmLp((s,T)×d,μ)p=sTufn(t,)-ufm(t,)Lp(d,μt)p𝑑tCT-sp(T-s)fn-fmLp(d,μs)p

and

xufn-xufmLq((s,T)×d,μ)q=sTxufn(t,)-xufm(t,)Lq(d,μt)q𝑑t2CT-sq2-q(T-s)1-q2fn-fmLq(d,μs)q

for any q[1,2) if p2 and for p=q otherwise. Hence, recalling that Lp(d,μt)Lq(d,μt) for any tI, we conclude that the sequence (ufn) converges in Lp((s,T)×d,μ)Wq0,1((s,T)×d,μ) to a function, which we denote by uf. Clearly, v(t,)=uf(t,) almost everywhere in d for almost every t(s,T). Letting n tend to + in formula (1.4), with fn replacing f, we deduce that uf is a mild solution to problem (1.2). The uniqueness follows, arguing as in the proof of Theorem 3.2 with the obvious changes.

Let us now prove the last part of the statement. We again use an approximation argument. Fix t>sI and R>0. At a first step, we estimate the norm of the operator G(t,r) in (Lp(d,μr),Lp(BR+1)) and in (Lp(d,μr),W2,p(BR+1)), for any r[s,t). In the rest of the proof, we denote by c a positive constant, possibly depending on R, but being independent of t, r and fLp(d,μr), which may vary from line to line. Since there exists a positive and continuous function ρ:I×d such that μr=ρ(r,)dx, the spaces Lp(BM) and Lp(BM,μr) coincide and their norms are equivalent for any M>0. From this remark, the interior Lp-estimates in Theorem A.3, with u=G(,s)f and the contractiveness of G(t,r) from Lp(d,μr) to Lp(d,μt), imply that

G(t,r)fW2,p(BR+1)c(t-r)-1fLp(d,μr),s<r<t<T,(3.16)

first for any fCb(d), and then, by density, for any fLp(d,μr). Since, for θ(0,1),

(Lp(d,μr),Lp(d,μr))θ,p=Lp(d,μr)and(W1,p(BR+1),W2,p(BR+1))θ,p=W1+θ,p(BR+1),

with equivalence of the corresponding norms, by an interpolation argument and (3.16) we deduce that G(t,r)(Lp(d,μr),W1+θ,p(BR+1))c(t-r)-1+θ2 for any s<r<t<T. Hence, if for any n we consider the function zn, which is the integral term in (1.4), with u being replaced by ufn, and use (3.11) and the fact that ψB([s,T]×d;Lip(d+1)), then we get

xzn(t,)-xzm(t,)Wθ,p(BR+1)cst(t-r)-1+θ2ufn(r,)-ufm(r,)Lp(d,μr)𝑑r+cst(t-r)-1+θ2xufn(r,)-xufm(r,)Lp(d,μr)𝑑rc(t-s)-θ2fn-fmLp(d,μs)

for any n. We have so proved that, for any θ(0,1) and almost every t(s,T], the function uf(t,) belongs to W1+θ(BR+1) and

ufn(t,)-ufm(t,)W1+θ,p(BR+1)c(t-s)-1+θ2fn-fmLp(d,μs),m,n.

Similarly,

ufn(t,)W1+θ,p(BR+1)c(t-s)-1+θ2(fLp(d,μs)+ψ(,,0,0)),n.

Using these estimates, we can now show that ψuf(r,)Wθ,p(BR+1), for any θ<γ. For this purpose, we add and subtract ψ(t,y,ufn(t,x),xufn(t,x)), use condition (3.15) and the Lipschitz continuity of ψ with respect to the last two variables to infer that

|ψufn(t,x)-ψufn(t,y)|c|ufn(t,x)-ufn(t,y)|+c|xufn(t,x)-xufn(t,y)|+c|x-y|γ(1+|ufn(t,x)|+|xufn(t,x)|)(3.17)

and

|ψufn(t,x)-ψufm(t,x)|c(|ufn(t,x)-ufm(t,x)|+|xufn(t,x)-xufm(t,x)|)(3.18)

for any t(s,T), x,yd and m,n. Hence, using (3.17) we obtain

|ψufn(t,x)-ψufn(t,y)-ψufm(t,x)+ψufm(t,y)|c[|ufn(t,x)-ufn(t,y)|+|xufn(t,x)-xufn(t,y)|+|ufm(t,x)-ufm(t,y)|+|xufm(t,x)-xufm(t,y)|+|x-y|γ(1+|ufn(t,x)|+|ufm(t,x)|+|xufn(t,x)|+|xufm(t,x)|)]=:(t,x,y)

and, using (3.18),

|ψufn(t,x)-ψufn(t,y)-ψufm(t,x)+ψufm(t,y)|c[|ufn(t,x)-ufm(t,x)|+|xufn(t,x)-xufm(t,x)|+|ufn(t,y)-ufm(t,y)|+|xufn(t,y)-xufm(t,y)|]=:𝒥(t,x,y).

From these two estimates we conclude that

|ψufn(t,x)-ψufn(t,y)-ψufm(t,x)+ψufm(t,y)|p((t,x,y))βp(𝒥(t,x,y))(1-β)p

for any (t,x)(s,T)×d, any β(0,1) and any m,n. Hence, for any θ<γ and β, such that (0,1)θ=θ/β+d(1-β)/(pβ), a long but straightforward computation reveals that

[ψufn(t,)-ψufm(t,)]Wθ,p(BR+1)cufn(t,)-ufm(t,)W1,p(d,μt)(1-β)(ufn(t,)W1+θ,p(BR+1)β+ufm(t,)W1+θ,p(BR+1)β+1)

and, consequently,

ψufn(t,)-ψufm(t,)Wθ,p(BR+1)c(t-s)β-θ2-1fn-fmLp(d,μs)1-β

for any t(s,T). We are almost done. Indeed, by interpolation from Proposition A.3 we deduce that G(t,r)(Wθ,p(BR+1),W2,p(BR))c(t-r)-1+θ/2. From this and the previous estimate we conclude that

zn(t,)-zm(t,)W2,p(BR)c(t-s)β+θ-θ2-1fn-fmLp(d,μs)1-β,m,n,

for any t(s,T] and β>θ, so that

ufn(t,)-ufm(t,)W2,p(BR)c(t-s)-1fn-fmLp(d,μs)1-β

for any m,n, thanks to (3.16). From this estimate it is easy to deduce that (ufn) is a Cauchy sequence in Wp,loc0,2((s,T)×d). Since ufn is a classical solution to problem (1.2), we conclude that (Dtufn) is a Cauchy sequence in Llocp((s,T)×d). It thus follows that ufWp,loc1,2((s,T)×d) and it solves the equation

Dtuf=𝒜uf+ψuf

in (s,T)×d. ∎

The arguments in the proof of Theorem 3.6 and Corollary 3.7 allow us to prove the following result.

Proposition 3.8.

Under Hypotheses 2.1, the following properties are satisfied.

  • (i)

    Let ψC((s,T]×d××d) with [ψ],T+sup(t,x)(s,T]×d[ψ(t,x,,)]Lip(d+1)<+ . Then, for any fCb(d) , the Cauchy problem ( 1.2 ) admits a unique mild solution ufC([s,T]×d)C0,1((s,T]×d) which satisfies ( 3.10 ) and ( 3.11 ) for any p[p0,+].

  • (ii)

    Let ψC((s,T]×d××d) and [ψ]p,T+sup(t,x)(s,T]×d[ψ(t,x,,)]Lip(d+1)<+ for some pp0 . Then, for any fLp(d,μs) , the Cauchy problem ( 1.2 ) admits a unique mild solution uf which belongs to Wp0,1((s,T)×d) , if p0p<2 , and to Wp0,1(J×d) for any J(s,T] , if p2 . Further, uf satisfies ( 3.10 ) and ( 3.11 ), with the supremum being replaced by the essential supremum.

Proof.

To prove property (i), it suffices to apply the Banach fixed point theorem in the space of all the functions vCb([s,T]×d)C0,1((s,T]×d) such that |||v|||<+, where |||||| is defined in Step 2 of the proof of Theorem 3.6, with p=+. The uniqueness of the so obtained solution follows from the condition sup(t,x)(s,T]×d[ψ(t,x,,)]Lip(d+1)<+, in a standard way.

To prove property (ii), one can argue by approximation. We fix fLp(d,μs), approximate it by a sequence (fn)Cb(d), converging to f in Lp(d,μs), and introducing a standard sequence (ϑn) of cut-off functions. If we set ψn=ϑnψ for any n, then each function ψn satisfies the assumptions in property (i) and [ψn]p,T[ψ]p,T. Therefore, the Cauchy problem (1.2), with fn and ψn replacing f and ψ admits a unique mild solution uCb([s,T]×d)C0,1((s,T]×d), which satisfies (3.10) and (3.11) with fn replacing f. The arguments in the first part of the proof of Corollary 3.7 allow us to prove the existence of a mild solution uf to the Cauchy problem (1.2) with the properties in the statement of the proposition. The uniqueness of the solution follows also in this case from the condition sup(t,x)(s,T]×d[ψ(t,x,,)]Lip(d+1)<+. ∎

4 The evolution operator and its summability improving properties

Suppose that, besides Hypotheses 2.1, the assumptions on ψ in Theorem 3.6 hold true for any Is<T or ψC(I×d××d)B(J×d;Lip(d+1)) for each JI and ψ(,,0,0)Cb(d+1). Then, for any fCb(d) and sI the mild solution to problem (1.2) exists in the whole of [s,+). Hence, we can set 𝒩(t,s)f=uf(t,) for any t>s. Each operator 𝒩(t,s) maps Cb(d) into Cb1(d). Moreover, the uniqueness of the solution to problem (1.2) yields the evolution law 𝒩(t,s)f=𝒩(t,r)𝒩(r,s)f for any r(s,t) and fCb(d). Hence {𝒩(t,s):Is<t} is a nonlinear evolution operator in Cb(d). It can be extended to the Lp-setting, for any pp0, using the same arguments as in the first part of the proof of Corollary 3.7. Clearly, if ψ(t,x,,) is Lipschitz continuous in d+1, uniformly with respect to (t,x)×J×d, for any JI, then by density, we still deduce that 𝒩(t,s) satisfies the evolution law and, moreover, each operator 𝒩(t,s) is bounded from Lp(d,μs) to W1,p(d,μt) and

𝒩(t,s)fLp(d,μt)+t-sx𝒩(t,s)fLp(d,μt)CT-s(fLp(d,μs)+(T-s+1)ψ(,,0,0)).(4.1)

4.1 Continuity properties of the nonlinear evolution operator

In the following theorem, assuming the above conditions on ψ, we prove an interesting continuity property of the operator 𝒩(t,s).

Theorem 4.1.

Let (fn)Cb(Rd) be a bounded sequence converging to some function fCb(Rd) pointwise in Rd. Then, for any sI, N(,s)fn and xN(,s)fn converge to N(,s)f and xN(,s)f, respectively, locally uniformly in (s,+)×Rd.

Proof.

Let (fn) and f be as in the statement. To ease the notation, we write ufn and uf for 𝒩(,s)fn and 𝒩(,s)f, respectively. Moreover, we set hn(r,)=G(t,r)(|ufn(r,)-uf(r,)|p+|x(ufn(r,)-uf(r,))|p) for any n, t>s and r(s,t], and we denote by LR,T any constant such that

|ψ(t,x,u2,v2)-ψ(t,x,u1,v1)|LR,T(|u2-u1|+|v2-v1|)(4.2)

for any t[s,s+T], x,v1,v2d, u1,u2[-R,R] and T>0. As a first step, formula (3.2) shows that, for any T>0, there exists a positive constant MT such that uf+ufnMT. Fix p(1,2). Using formula (1.4), we can estimate

|Dxjufn(t,x)-Dxjuf(t,x)|p2p-1|(DxjG(t,s)(fn-f))(x)|p+2p-1|st(DxjG(t,r)(ψufn(r,)-ψuf(r,))(x)dr|p

for any (t,x)(s,+)×d and j=0,1. By the representation formula (2.2), Hölder inequality, estimates (2.4) and (4.2), we deduce that

|ufn(t,)-uf(t,)|p2p-1G(t,s)|fn-f|p+(4T)p-1LMTpsth(r,)𝑑r

and

|xufn(t,)-xuf(t,)|p2p-1(t-s)-p2cTG(t,s)|fn-f|p+(4T)p-1cTLMT,Tpst(t-r)-p2h(r,)𝑑r

in d, for any t(s,s+T) and some positive constant cT. Hence, the function hn(,x) satisfies the differential inequality

hn(t,x)Cp,T(t-s)-p2(G(t,s)|fn-f|p)(x)+Cp,Tst(t-r)-p2hn(r,x)𝑑r

for any t(s,s+T) and xd. Since hn(,x) is continuous in (s,t] and hn(r,x)C~T(r-s)-p/2 for some positive constant C~T, independent of n, and any r(s,t), we can apply [8, Lemma 7.1] and conclude that

hn(t,x)Cp,T(t-s)-p2(G(t,s)|fn-f|p)(x)+Cp,Tst(t-r)-p2(r-s)-p2(G(r,s)|fn-f|p)(x)𝑑r

for any t(s,s+T). Hence,

hn(t,)Cb(BR)Cp,T(t-s)-p2G(t,s)|fn-f|pCb(BR)+Cp,Tst(t-r)-p2(r-s)-p/2G(r,s)|fn-f|pCb(BR)𝑑r

for any R>0. By [9, Proposition 3.1 (i)], G(r,s)|fn-f|pCb(BR) vanishes as n+ for any r>s. Hence, by dominated convergence, hn(t,)Cb(BR) vanishes as n+ for any t(s,s+T), which means that, for any t(s,s+T), ufn(t,) and xufn(t,) converge uniformly in BR to uf(t,) and xufn(t,), respectively. The arbitrariness of R and T yields the assertion. ∎

4.2 Hypercontractivity

Throughout this and the forthcoming subsections we set

(ζ)=|Qxζ|2,𝒢(ζ)=i=1d|QxDiζ|2

for any smooth enough function ζ. To begin with, we recall the following crucial result.

Lemma 4.2 ([4, Lemma 3.1]).

Assume that Hypotheses 2.1 hold true and fix [a,b]I. If fCb1,2([a,b]×Rd) and f(r,) is constant outside a compact set K for every r[a,b], then the function rRdf(r,)𝑑μr is continuously differentiable in [a,b] and

Drdf(r,)𝑑μr=dDrf(r,)𝑑μr-d𝒜(r)f(r,)𝑑μr,r[a,b].

Hypotheses 4.3.

We introduce the following assumptions.

  • (i)

    ψB(I×d;Lip(d+1))C(I×d××d), condition (3.9) is satisfied in [s,T], for any T>sI and some constant which may depend also on s and T, and there exist two constants ξ00 and ξ1 such that uψ(t,x,u,v)ξ0|u|+ξ1u2+ξ2|u||v| for any ts, x,vd and u.

  • (ii)

    There exists a nonnegative function φ~:d, blowing up at infinity such that 𝒜φ~+k1|φ~|aφ~ in d for some locally bounded functions a,k1.

  • (iii)

    There exist locally bounded functions C0,C1,C2:I+ such that

    |Q(t,x)x|C0(t)|x|3φ~(x),Tr(Q(t,x))C1(t)(1+|x|2)φ~(x),b(t,x),xC2(t)|x|2φ~(x)

    for any tI and any xd.

  • (iv)

    There exists a positive constant K such that

    d|f|qlog(|f|)𝑑μtfLq(d,μt)qlog(fLq(d,μt))+Kqd|f|q-2|f|2𝟙{f0}𝑑μt(4.3)

    for any t>s, fCb1(d) and q(1,+).

Remark 4.4.

(i) Hypothesis 4.3 (i) implies that ψ(,,0,0) is bounded in [s,+)×d and

ψ(,,0,0)Cb([s,+)×d)ξ0.

(ii) Sufficient conditions for (4.3) to hold are given in [4]. In particular, (4.3) holds true when (2.3) is satisfied with p=1 (see Remark 2.2).

We can now prove the main result of this subsection.

Theorem 4.5.

Let Hypotheses 2.1 and 4.3 be satisfied. Then, for any fLp(Rd,μs) (pp0) and t>s, the function N(t,s)f belongs to W1,pγ(t)(Rd,μt) and satisfies the estimates

𝒩(t,s)fLpγ(t)(d,μt)eωp,γ(t-s)[fLp(d,μs)+ξ0(t-s)],(4.4)x𝒩(t,s)fLpγ(t)(d,μt)c0(t-s)eωp,γ(t-s)[fLp(d,μs)+ξ0(t-s)]+c1(t-s)ξ0,(4.5)

where pγ(t):=γ-1(p-1)(eκ0K-1(t-s)-1)+p for any γ>1, κ0 being the ellipticity constant in Hypothesis 2.1(ii) and K being the constant in (4.3), ωp,σ=ξ1+(ξ2+)2σ[(σ-1)(p-1)κ0]-1 and the functions c0,c1:(0,+)R+ are continuous and blow up at zero.

Proof.

To begin with, we observe that it suffices to prove (4.4) and (4.5) for functions fCb1(d). Indeed, in the general case, the assertion follows approximating f with a sequence (fn)Cb1(d) which converges to f in Lp(d,μs). By (3.10), 𝒩(t,s)fn converges to 𝒩(t,s)f in W1,p(d,μt) for almost every t>s. Hence, writing (4.4) and (4.5) with f being replaced by fn and letting n tend to +, the assertion follows at once by applying Fatou lemma.

We split the rest of the proof into two steps. In the first one we prove (4.4) and in the latter one (4.5).

Step 1. Fix fCb1(d), n, ε>0 and set

βn,ε(t):=vn,ε(t,)pγ(t)

for any t>s, where vn,ε=(ϑn2𝒩(,s)f+ε)1/2 and ϑn=ζ(n-1|x|) for any xd and n. Here, ζ is a smooth function such that 𝟙B1ζ𝟙B2. Moreover, we set

φ1,n=|ζ(||n)|φ~,φ2,n=|ζ′′(||n)|φ~

for any n. We recall that in [9, Theorem 5.4] it has been proved that suptIφ~L1(d,μt)<+. Hence, the functions tφj,nLp(d,μt) (j=1,2) are bounded in I and pointwise converge to zero as n+.

By definition, the function u=𝒩(,s)f belongs to Cb0,1([s,τ]×d) for any τ>s and is a classical solution to problem (1.2). Moreover, Lemma 4.2 shows that βn,ε is differentiable in (s,+) and a straightforward computation reveals that

βn,ε(t)=-pγ(t)pγ(t)βn,ε(t)log(βn,ε(t))+1pγ(t)(βn,ε(t))1-pγ(t)d{Dt[(vn,ε(t,))pγ(t)]-𝒜(t)[(vn,ε(t,)pγ(t)]}dμt.

Taking into account that

Dt[(vn,ε(t,))pγ(t)]-𝒜[(vn,ε(t,)pγ(t)]=pγ(t)(vn,ε(t,))pγ(t)log(vn,ε(t,))-pγ(t)(pγ(t)-1)(vn,ε(t,))pγ(t)-2((vn,ε))(t,)+pγ(t)(vn,ε(t,))pγ(t)-1(Dtvn,ε(t,)-𝒜(t)vn,ε(t,))

and

Dtvn,ε-𝒜vn,ε=ϑn2vn,ε-1uψu-εϑn2(u)vn,ε-3-Tr(QD2ϑn)vn,ε-1ϑnu2-εu2vn,ε-3(ϑn)-b,ϑnvn,ε-1ϑnu2-2(2εu+ϑn2u3)Qϑn,xuϑnvn,ε-3=:ϑn2vn,ε-1uψu-εϑn2(u)vn,ε-3+gn,ε(t,),

we deduce

βn,ε(t)=(βn,ε(t))1-pγ(t)d(vn,ε(t,))pγ(t)-1gn,ε(t,)𝑑μt-pγ(t)pγ(t)βn,ε(t)log(βn,ε(t))-(pγ(t)-1)(βn,ε(t))1-pγ(t)d(vn,ε(t,))pγ(t)-2((vn,ε))(t,)𝑑μt+pγ(t)pγ(t)(βn,ε(t))1-pγ(t)d(vn,ε(t,))pγ(t)log(vn,ε(t,))𝑑μt+(βn,ε(t))1-pγ(t)d(vn,ε(t,))pγ(t)-2ϑn2u(t,)ψu(t,)𝑑μt-ε(βn,ε(t))1-pγ(t)dϑn2(vn,ε(t,))pγ(t)-4((u))(t,))dμt.

Using Hypotheses 4.3 (i), (iv), the expression of the function tpγ(t) and Hypothesis 2.1 (ii), we can estimate

βn,ε(t)(βn,ε(t))1-pγ(t)d(vn,ε(t,))pγ(t)-1gn,ε(t,)𝑑μt+ξ0(βn,ε(t))1-pγ(t)dϑn2|u(t,)|(vn,ε(t,))pγ(t)-2𝑑μt+ξ1βn,ε(t)-εξ1(βn,ε(t))1-pγ(t)d(vn,ε(t,))pγ(t)-2𝑑μt-(p-1)(1-γ-1)(βn,ε(t))1-pγ(t)d(vn,ε(t,))pγ(t)-2((vn,ε))(t,)𝑑μt+ξ2+(βn,ε(t))1-pγ(t)dϑn2(vn,ε(t,))pγ(t)-2|u(t,)||xu(t,)|𝑑μt-ε(βn,ε(t))1-pγ(t)dϑn2(vn,ε(t,))pγ(t)-4(u(t,))𝑑μt.(4.6)

Further, since (vn,ε)=ϑn2(ϑn)u4vn,ε-2+ϑn4u2(u)vn,ε-2+2ϑn2u3vn,ε-2Qxu,ϑn and

d(v(t,))pγ(t)-4ϑn2(u(t,))3Q(t,)xu(t,),ϑn𝑑μtδd(v(t,))pγ(t)-4ϑn4(u(t,))2((u))(t,)𝑑μt+1δd(v(t,))pγ(t)-4(u(t,))4((ϑn))(t,)𝑑μt,

it follows that

d(vn,ε(t,))pγ(t)-2((vn,ε))(t,)𝑑μt(1-δ)dϑn4(u(t,))2(vn,ε(t,))pγ(t)-4((u))(t,)𝑑μt-Cε,δ(t)dφ1,n𝑑μt

for any δ>0 and some continuous function Cε,δ:[s,+)+. Moreover, applying Hölder and Young inequalities and Hypothesis 2.1 (ii) we can infer that

dϑn2(vn,ε(t,))pγ(t)-2|u(t,)||xu(t,)|𝑑μtδ1κ0dϑn4(vn,ε(t,))pγ(t)-4|u(t,)|2((u))(t,)𝑑μt+14δ1(βn,ε(t,))pγ(t)

for any δ1>0 and

dϑn2u(t,)(vn,ε(t,))pγ(t)-2𝑑μtd(vn,ε(t,))pγ(t)-1𝑑μt(βn,ε(t,))pγ(t)-1.

Hence,

βn,ε(t)ξ0+(ξ1+ξ2+4δ1)βn,ε(t)+(βn,ε(t))1-pγ(t)d(vn,ε(t,))pγ(t)-1gn,ε(t,)𝑑μt-[(p-1)(1-γ-1)(1-δ)-κ0-1ξ2+δ1](βn,ε(t))1-pγ(t)dϑn4|u(t,)|2(vn,ε(t,))pγ(t)-4((u))(t,)𝑑μt+C~ε,δ,p,γ(t)(βn,ε(t))1-pγ(t)dφ1,n𝑑μt-ε(βn,ε(t))1-pγ(t)dϑn2(vn,ε(t,))pγ(t)-4((u))(t,)𝑑μt-εξ1(βn,ε(t))1-pγ(t)d(vn,ε(t,))pγ(t)-2𝑑μt

for some continuous function C~ε,δ,p,γ:[s,+)+. Now, we estimate the integral term containing gn. We begin by observing that

-2d(2εu(t,)+ϑn2(u(t,))3)Q(t,)ϑn,xuϑn(vn,ε(t,))p(t)-4𝑑μt4εδ2dϑn2(vn,ε(t,))pγ(t)-4((u))(t,)𝑑μt+εδ2-1d|u(t,)|2(vn,ε(t,))pγ(t)-4((ϑn))(t,)𝑑μt+δ2dϑn4|u(t,)|2(vn,ε(t,))pγ(t)-4((u))(t,)𝑑μt+δ2-1dϑn2|u(t,)|4(vn,ε(t,))pγ(t)-4((ϑn))(t,)𝑑μt4εδ2dϑn2(vn,ε(t,))pγ(t)-4((u))(t,)𝑑μt+C~ε,δ2(t)dφ1,n𝑑μt+δ2dϑn4|u(t,)|2(vn,ε(t,))pγ(t)-4((u))(t,)𝑑μt

for some continuous function C~ε,δ2:[s,+)+. Moreover,

-d(vn,ε(t,))pγ(t)-4u(t,)[ϑn𝒜(t)ϑn-εu(t,)((ϑn))(t,)]𝑑μtC¯ε(t)d(φ1,n+φ2,n)𝑑μt

for some positive and continuous function C¯ε:[s,+)+. Hence, replacing these estimates in (4.6), we get

βn,ε(t)ξ0+(ξ1+ξ2+4δ1)βn,ε(t)+C^ε,δ,δ2,p(t)(βn,ε(t))1-pγ(t)d(φ1,n+φ2,n)𝑑μt-[(p-1)(1-γ-1)(1-δ)-κ0-1ξ2+δ1-δ2](βn,ε(t))1-pγ(t)×dϑn4|u(t,)|2(vn,ε(t,))pγ(t)-4((u))(t,)𝑑μt-ε(1-4δ2)(βn,ε(t))1-pγ(t)dϑn2(vn,ε(t,))pγ(t)-4((u))(t,)𝑑μt-εξ1(βn,ε(t))1-pγ(t)d(vn,ε(t,))pγ(t)-2𝑑μt,(4.7)

where, again, C^ε,δ,δ2,p:(s,+)+ is a continuous function. Choosing δ=12, δ1=(p-1)(1-γ-1)κ04ξ2 if ξ2>0, δ1=0 otherwise, and then δ2 small enough we obtain

βn,ε(t)ξ0+ωp,γβn,ε(t)+C^ε,1/2,δ2,p(t)(βn,ε(t))1-pγ(t)φ1,n+φ2,nL1(d,μt)-εξ1(βn,ε(t))1-pγ(t)d(vn,ε(t,))pγ(t)-2𝑑μt.(4.8)

Hence, integrating (4.8) between s and t and letting first n+ and then ε0+, by dominated convergence we get

u(t,)Lpγ(t)(d,μt)fLp(d,μs)+ξ0(t-s)+ωp,γstu(r,)Lp(r)(d,μr)𝑑r.

Applying the Gronwall lemma, we conclude the proof of (4.4).

Step 2. To check estimate (4.5), we arbitrarily fix γ(1,+), t>s and we take

ε=K2κ0log(γeκ0K-1(t-s)γ+eκ0K-1(t-s)-1),γ=γeκ0K-1(t-s-ε)-1eκ0K-1(t-s)-1.(4.9)

With these choices of ε and γ, we have

pγ(t-ε)=pγ(t).

From Step 1, we know that 𝒩(t-ε,s)fLpγ(t-ε)(d,μt-ε) and

𝒩(t-ε,s)fLpγ(t)(d,μt-ε)eωp,γ(t-s-ε)(fLp(d,μs)+ξ0(t-s)).(4.10)

By the evolution law and estimates (4.10) and (4.1) we get

supτ(t-ε,T)τ-t+εx𝒩(τ,s)fLpγ(t)(d,μτ)CT-t+ε{eωp,γ(t-s-ε)(fLp(d,μs)+ξ0(t-s))+(T-t+ε+1)ψ(,,0,0)}

for any T>t-ε. In particular, taking T=t and using Remark 4.4 (i) to estimate ψ(,,0,0)ξ0, we get

x𝒩(t,s)fLpγ(t)(d,μt)Cεεe-εωp,γeωp,γ(t-s)[fLp(d,μs)+ξ0(t-s)]+(ε+1ε)Cεξ0.(4.11)

Replacing the value of ε in the expression of γ (see (4.9)), we deduce that

γinfδ1γ(δ-1)-1[(δ(γ+δ-1)γ)1/2-1]=γ

and, since the function σωp,σ is decreasing, ωp,γωp,γ. Finally, observing that e-εωp,γ is bounded in (s,+), ε<(2κ0)-1Klog(γ) (which follows from (4.9) recalling that γγ) and ε(2γ)-1(γ-1)(t-s) as t-s0+, formula (4.5) follows immediately replacing in (4.11) the value of ε given by (4.9). ∎

Remark 4.6.

As the proof of Theorem 4.5 shows, if ξ20, then we can take γ=1 and ωp,1=ξ1 in (4.4).

4.3 Supercontractivity

In the next theorem we prove a stronger result than Theorem 4.5, i.e., we prove that the nonlinear evolution operator 𝒩(t,s) satisfies a supercontractivity property. For this purpose, we introduce the following additional assumption.

Hypothesis 4.7.

There exists a decreasing function ν:(0,+)+ blowing up as σ tends to 0+ such that

d|f|plog(|f|)𝑑μr-fLp(d,μr)plog(fLp(d,μr))ν(σ)pfLp(d,μr)p+σpd|f|p-2|f|2𝟙{f0}𝑑μr(4.12)

for any rI, σ>0 and fCb1(d).

Remark 4.8.

Sufficient conditions for (4.12) to hold are given in [3]. In particular, it holds true when (2.3) is satisfied with p=1 (see Remark 2.2) and there exist K>0 and R>1 such that b(t,x),x-K|x|2log|x| for any tI and |x|R.

Theorem 4.9.

Let Hypotheses 2.1, 4.3(i)–(iii) and 4.7 be satisfied. Then, for any t>sI, p0p<q<+ and any fLp(Rd,μs), N(t,s)f belongs to W1,q(Rd,μt) and

𝒩(t,s)fLq(d,μt)c2(t-s)(fLp(d,μs)+ξ0(t-s)),(4.13)x𝒩(t,s)fLq(d,μt)c3(t-s)fLp(d,μs)+c4(t-s)ξ0.(4.14)

Here, c2,c3,c4:(0,+)R+ are continuous functions such that limr0+ck(r)=+ (k=2,3,4).

Proof.

The proof of this result follows the same lines of the proof of Theorem 4.5. For this reason we use the notation therein introduced and we limit ourselves to sketching it in the case when fCb1(d).

Step 1. Here, we prove (4.13). For any σ>0 and any ts, we set

p(t)=eκ0(2σ)-1(t-s)(p-1)+1,m(t)=ν(σ)(p-1-(p(t))-1),ζn,ε(t)=e-m(t)βn,ε(t).

The function ζn,ε is differentiable in (s,+) and arguing as in the proof of the quoted theorem, using (4.12) instead of (4.3) and the definition of m(t) and p(t), we deduce that

ζn,ε(t)=[(βn,ε(t))1-p(t)d(vn,ε(t,))p(t)-1gn,ε(t,)dμt-p-12(βn,ε(t))1-p(t)d(vn,ε(t,))p(t)-2((vn,ε))(t,)𝑑μt+(βn,ε(t))1-p(t)d(vn,ε(t,))p(t)-2ϑn2u(t,)ψu(t,)𝑑μt-ε(βn,ε(t))1-p(t)dϑn2(vn,ε(t,))p(t)-4((u))(t,)dμt]e-m(t)

and the same arguments used to prove (4.7) show that, if δ2<14, then

ζn,ε(t)ξ0e-m(t)+(ξ1+ξ2+4δ1)ζn,ε(t)+(C^ε,δ,δ2,p(t)(βn,ε(t))1-p(t)d(φ1,n+φ2,n)dμt-[2-1(p-1)(1-δ)-κ0-1ξ2+δ1-δ2](βn,ε(t))1-p(t)dϑn4|u(t,)|2(vn,ε(t,))p(t)-4((u))(t,)𝑑μt-εξ1(βn,ε(t))1-p(t)d(vn,ε(t,))p(t)-2dμt)e-m(t).

Choosing δ=12, δ1=(p-1)κ0(8ξ2)-1 if ξ2>0, δ1=0 otherwise, and δ2=[(p-1)2]/8 we get

ζn,ε(t)ω~pζn,ε(t)+e-m(t)[ξ0+C^ε,δ2,p(t)(βn,ε(t))1-p(t)φ1,n+φ2,nL1(d,μt)-εξ1(βn,ε(t))1-p(t)d(vn,ε(t,))p(t)-2dμt],(4.15)

where ω~p=ξ1+2(ξ2+)2(κ0(p-1))-1. Hence, integrating (4.15) between s and t and letting first n+ and then ε0+, by dominated convergence we get

e-m(t)uf(t,)Lp(t)(d,μt)ξ0(t-s)+fLp(d,μs)+ω~pste-m(r)u(r,)Lp(r)(d,μr)𝑑r,

which yields

uf(t,)Lp(t)(d,μt)eω~p(t-s)+m(t)(ξ0(t-s)+fLp(d,μs)).

Now, for any q>p and t>s, we fix σ=κ0(t-s)(2log(q-1)-2log(p-1))-1. We get p(t)=q and from the previous inequality the claim follows with

c2(r)=exp(ω~pr+(p-1-q-1)ν(κ0r(2log(q-1)-2log(p-1))-1)).

Step 2. Fix q>p. By Step 1, 𝒩((t+s)/2,s)f belongs to Lq(d,μ(t+s)/2) and

𝒩(t+s2,s)fLq(d,μ(t+s)/s)c2(t-s2)(fLp(d,μs)+ξ0t-s2).

The same arguments used in Step 2 of the proof of Theorem 4.5 show that 𝒩(t,s)fW1,q(d,μτ) for any τ>t+s2 and

t-s2x𝒩(t,s)fLq(d,μτ)C(t-s)/2[c2(t-s2)(fLp(d,μs)+ξ0t-s2)+(t-s2+1)ξ0].

Estimate (4.14) follows with c3(r)=2rCr/2c2(2r), c4(r)=Cr/2[c2(2r)2r+2r+2r]. ∎

4.4 Ultraboundedness

To begin with, we prove a sort of Harnack inequality, which besides the interest in its own will be crucial to prove the ultraboundedness of the nonlinear evolution operator 𝒩(t,s).

Proposition 4.10.

Let Hypotheses 2.1(i)–(iii) and 4.3(i)–(iii) be satisfied. Further, suppose that estimate (2.3) holds, with p=1 and some constant σ1R. Then, for any fCb(Rd), p>1, t>s and x,yRd, the following estimate holds true:

|(𝒩(t,s)f)(x)|pexp(p(1+ξ1+)(t-s)+pΘ(t-s)(|x-y|+ξ2+(t-s))24κ0(t-s)2(p-1))[(G(t,s)|f|p)(y)+ξ0p],(4.16)

where Θ(r)=(e2σ1r-1)/(2σ1) if σ1>0 and Θ(r)=r otherwise.

Proof.

To begin with, we observe that it suffices to prove (4.16) for functions in Cb1(d). Indeed, if fCb(d), we can determine a sequence (fn)Cb1(d), bounded with respect to the sup-norm and converging to f locally uniformly in d. Writing (4.16) with f replaced by fn and using Theorem 4.1 and [9, Proposition 3.1 (i)], we can let n tend to + and complete the proof.

So, let us fix fCb1(d) and set Φn(r):=[G(t,r)(ϑn2vε(r,))](ϕ(r))+ξ0p for any n and r(s,t), where vε=(uf2+ε)p/2, uf=𝒩(,s)f (see Theorem 3.6), ϕ(r)=(r-s)(t-s)-1x+(t-r)(t-s)-1y and (ϑn) is a standard sequence of cut-off functions. We note that Φn(r)CΦ>0 for any r[s,t] and any nn0. This is clear if ξ0>0. Suppose that ξ0=0. If r<t, then Φn(r) is positive since vε>0. If r=t, then Φn(t)=(ϑn(x))2vε(t,x) which is positive if we choose n large enough such that xsupp(ϑn). Moreover, ΦnC1((s,t)). Hence log(Φn)C1((s,t)) and we have

ddrlog(Φn(r))=1Φn(r){[G(t,r)(ϑn2Dtvε(r,))-𝒜(ϑn2vε(r,)))](ϕ(r))   +(t-s)-1[xG(t,r)(ϑn2vε(r,))](ϕ(r)),x-y}.

We observe that

Dt(ϑn2vε)-𝒜(ϑn2vε)=pϑn2(uf2+ε)p2-1ufψuf-pϑn2vε1-4p((p-1)uf2+ε)(uf)-4pvε1-2pϑnufQϑn,xuf-2ϑnvε𝒜ϑn-2vε(ϑn),

and

|xG(t,r)(ϑn2vε(r,))|eσ1(t-r)G(t,r)|x(ϑn2vε(r,))|peσ1(t-r)G(t,r)(ϑn2(vε(r,))1-2p|uf(r,)|κ0-12(((uf))(r,))12)+eσ1(t-r)G(t,r)(2ϑn|ϑn|vε(r,)).

Hence, we get

ddrlogΦn(r)1Φn(r){p|x-y|t-seσ1(t-r)G(t,r)[ϑn2(vε(r,))1-2p|uf(r,)|κ0-12(((uf))(r,))12]   -G(t,r)ζn,ε(r,)+|x-y|t-seσ1(t-r)G(t,r)(2ϑn|ϑn|vε(r,))}(ϕ(r)),

where

ζn,ε=2ϑn(𝒜ϑn)vε+2(ϑn)vε+4pϑnvε1-2pufQϑn,xuf-pϑn2vn,ε1-2pufψuf+pϑn2vε1-4p((p-1)uf+ε)(uf).

From Hypothesis 4.3 (i) it follows that

ddrlogΦn(r)1Φn(r)G(t,r){-2ϑn(𝒜(r)ϑn)vε(r,)+pξ0ϑn2vε1-1p+ξ1+ϑn2pvε(r,)+4p(vε(r,))1-2pϑn|uf(r,)||Q(r,)ϑn,xuf(r,)|-pϑn2vε(r,)[((p-1)(uf(r,))2+ε)(hε(r,))2-|uf(r,)|hε(r,)eσ1(t-r)|x-y|+ξ2+(t-s)κ0(t-s)]}(ϕ(r))+|x-y|t-seσ1(t-r){G(t,r)[|ϑn|vε(r,)]}(ϕ(r)),(4.17)

where hε=(uf2+ε)-1(uf). Using the Cauchy–Schwarz inequality, we can estimate

vε1-2pϑn|uf||Qϑn,xuf|vε1-2pϑn|uf|(ϑn)(uf)δϑn2vεhε2uf2+14δvε(ϑn).

Moreover, using formula (2.2), we can estimate

(G(t,r)(ϑn2vε1-1/p))(ϕ(r))((G(t,r)(ϑn2vε))(ϕ(r)))1-1p((G(t,r)ϑn)(ϕ(r))p((G(t,r)(ϑn2vε))(ϕ(r)))1-1p(Φn(r))1-1p.

These two estimates replaced in (4.17) give

ddrlogΦn(r)1Φn(r)(G(t,r){[pδ-1((ϑn))(r,)-2ϑn𝒜(r)ϑn]vε(r,)+ξ1+ϑn2pvε(r,)-pϑn2vε(r,)[((p-1-δ)(uf(r,))2+ε)(hε(r,))2-|uf(r,)|hε(r,)eσ1(t-r)|x-y|+ξ2+(t-s)κ0(t-s)]})(ϕ(r))+|x-y|t-seσ1(t-r){G(t,r)[|ϑn|vε(r,)]}(ϕ(r))+p.(4.18)

Straightforward computations show that 𝒜(r)ϑn and ((ϑn))(r,) vanish pointwise in d as n+, for any r(s,t) and there exists a positive constant C such that |𝒜(r)ϑn|+((ϑn))(r,)Cφ~ in d for any n, thanks to Hypothesis 4.3 (iii). By [9, Lemma 3.4] the function G(t,)φ~ is bounded in (s,t)×BR for any R>0. Hence, by dominated convergence we conclude that G(t,r)(ϑn(𝒜(r)ϑn)vε(r,)) vanishes as n+, pointwise in d, for any r(s,t) and

G(t,r)[pδ-1((ϑn))(r,)-2ϑn𝒜(r)ϑn]Cb(BR)Cδ,p,ufsupr(s,t)G(t,r)φ~Cb(BR),(4.19)

where R>max{|x|,|y|}. Similarly, the last but one term in (4.18) vanishes pointwise in d as n+, for any r(s,t) and

|(G(t,r)[|ϑn|vε(r,)])(ϕ(r))|Cp,ufsupr(s,t)G(t,r)φ~Cb(BR).(4.20)

Moreover, using the inequality αβ2-γβ-γ24α for any α>0 and β,γ, and that G(t,s)g1G(t,s)g2 for any t>s and any g1g2, we deduce

ddrlog(Φn(r))1Φn(r)(G(t,r)[pδ-1((ϑn))(r,)-2ϑn𝒜(r)ϑn]vε(r,))(ϕ(r))+p(1+ξ1++e2σ1+(t-r)χδ)+eσ1(t-r)|x-y|(t-s)Φn(r)(G(t,r)[|ϑn|vε(r,)])(ϕ(r)),

where χδ=(|x-y|+ξ2+(t-s))2(4κ0(t-s)2(p-1-δ))-1. Integrating both sides of the previous inequality in (s,t) and taking (4.19) and (4.20) into account to let n+, we get

log(((uf(t,x))2+ε)p2+ξ0p(G(t,s)((f2+ε)p2))(y)+ξ0p)p[(1+ξ1+)(t-s)+Θ(t-s)χδ],

or even

((uf(t,x))2+ε)p/2exp[p(1+ξ1+)(t-s)+pΘ(t-s)χδ][(G(t,s)((f2+ε)p2))(y)+ξ0p].

By formula (2.2) we can let ε and δ tend to zero on both sides of the previous inequality and this yields the assertion. ∎

We can now prove the main result of this subsection. For this purpose, we set φλ(x)=eλ|x|2 for any xd and λ>0, and introduce the following additional assumption.

Hypothesis 4.11.

For any Is<t and λ>0, the function G(t,s)φλ belongs to L(d) and, for any δ>0, +>Mδ,λ:=supt-sδG(t,s)φλ.

Remark 4.12.

A sufficient condition for Hypothesis 4.11 to hold is given in [3, Theorem 4.3]. More precisely, it holds when (2.3) holds with p=1 and there exists K>0, β,R>1 such that b(t,x),x-K|x|2(log(|x|))β for any tI and xdBR.

Theorem 4.13.

Assume that Hypothesis 4.11 and the conditions in Proposition 4.10 are satisfied. Then, for any Is<t, fLp(Rd,μs) (p[p0,+)), the function N(t,s)f belongs to W1,(Rd) and

𝒩(t,s)fc5(t-s)fLp(d,μs)+c6(t-s)ξ0,(4.21)x𝒩(t,s)fc7(t-s)fLp(d,μs)+c8(t-s)ξ0(4.22)

for some continuous functions ck:(0,+)R+ (k=5,6,7,8) which blow up at zero.

Proof.

As usually, we prove the assertion for functions in Cb1(d).

Step 1. We prove (4.21). So, let us fix fCb1(d) and xd. By the invariance property of the family {μt:tI} and inequality (4.16), we can estimate

fLp(d,μs)p=d(G(t,s)|f|p)(y)μt(dy)BR[(G(t,s)|f|p)(y)+ξ0p]μt(dy)-ξ0p|(𝒩(t,s)f)(x)|pe-pϕ(t-s)BRexp(-pΘ(t-s)(|x-y|+ξ2+(t-s))24κ0(t-s)2(p-1))μt(dy)-ξ0p|(𝒩(t,s)f)(x)|pe-pϕ(t-s)exp(-pΘ(t-s)(|x|+R+ξ2+(t-s))24κ0(t-s)2(p-1))μt(BR)-ξ0p,

where ϕ=1+ξ1+. By the tightness of the family {μt:tI} we can fix R>0 such that μt(BR)2-p for any ts and, from the previous chain of inequalities, we conclude that

|(𝒩(t,s)f)(x)|p2p(C~(t-s))pφpΛ(t-s)(x)(fLp(d,μs)p+ξ0p),(4.23)

where

Λ(r)=exp(Θ(r)2κ0r2(p-1)),C~(r)=exp(ϕr+Θ(r)(ξ2+r+R)22κ0r2(p-1)).

Now, using the evolution law and again (4.16), we can write

|(𝒩(t,s)f)(x)|p=|(𝒩(t,t+s2)𝒩(t+s2,s))(x)|p[(G(t,t+s2)|𝒩(t+s2,s)f|p)(y)+ξ0p]×exp(pϕt-s2+pΘ(t-s2)(2|x-y|+ξ2+(t-s))24κ0(t-s)2(p-1))(4.24)

for any yd. From (2.2) and (4.23) we obtain

(G(t,t+s2)|𝒩(t+s2,s)f|p)(y)2p(C~(t-s2))p(fLp(d,μs)p+ξ0p)(G(t,t+s2)φpΛ(t-s)/2)(y)2p(C~(t-s2))p(fLp(d,μs)p+ξ0p)M(t-s)/2,pΛ(t-s)/2.(4.25)

From (4.24), (4.25), choosing y=x in the exponential term, we get

|(𝒩(t,s)f)(x)|[2C~(t-s2)(fLp(d,μs)+ξ0)M(t-s)/2,pΛ(t-s)/21/p+ξ0]exp(ϕt-s2+Θ(t-s2)(ξ2+)24κ0(p-1))

and (4.21) follows with

c5(r)=2C~(r2)Mr/2,pΛr/21/pexp[(1+ξ1+)r2+Θ(r2)(ξ2+)2(4κ0(p-1))-1],c6(r)=(2C~(r2)Mr/2,pΛr/21/p+1)exp[(1+ξ1+)r2+Θ(r2)(ξ2+)2(4κ0(p-1))-1].

Step 2. We fix t>s, fCb1(d). By Theorem 3.2, 𝒩(t,s)fCb1(d) and, by Step 1,

𝒩(t+s2,s)fc5(t-s2)fLp(d,μs)+c6(t-s2)ξ0.

Hence, from (4.1) we get

t-s2x𝒩(t,s)fC~(t-s)/2[c5(t-s2)fLp(d,μs)+c6(t-s2)ξ0+t-s2ξ0+ξ0].

Taking T=t, estimate (4.22) follows with c7(r)=2r-1/2C~r/2c5(r2) and c8(r)=C~r/2[c6(r2)2r+r2+2r]. ∎

5 Stability of the null solution

In this section we study the stability of the null solution to problem (1.2) both in the Cb- and Lp-settings. For this reason, we assume that ψ(,,0,0)=0.

Theorem 5.1.

The following properties are satisfied.

  • (i)

    Let Hypotheses 2.1, 4.3 (i)–(iii) hold true. Further, suppose that the constant ωp=ξ1+(ξ2+)2(4κ0(p-1))-1 is negative, where ξ1 and ξ2 are defined in Hypothesis 4.3 (ii). Then, for any pp0 , there exists a positive constant Kp such that, for any sI, fLp(d,μs) and j=0,1,

    Dxj𝒩(t,s)fLp(d,μt)Kpjeωp(t-s)fLp(d,μs),t>s+j.(5.1)

  • (ii)

    Suppose that the assumptions of Theorem 3.6 are satisfied. Further, assume that Hypotheses 4.3 (i)–(iii) hold with ξ1<0 . Then ( 5.1 ) holds true for any fCb(d) with p=+ and ωp and Kp being replaced, respectively, by ξ1 and C1e-ξ1.

Proof.

(i) Estimate (5.1) can be obtained arguing as in the proof of Theorem 4.5, where now p(t)=p for any ts. As far as the gradient of 𝒩(t,s)f is concerned, we fix t>s+1 and observe that

𝒩(t,s)f=𝒩(t,t-1)𝒩(t-1,s)f.

Hence, from (4.1) we obtain

x𝒩(t,s)fLp(d,μt)C1𝒩(t-1,s)fLp(d,μt-1)Kpeωp(t-s)fLp(d,μs),

where Kp=C1e-ωp.

(ii) The assertion follows easily letting p tend to + in (5.1). ∎

6 Examples

Here, we exhibit some classes of nonautonomous elliptic operators and some classes of nonlinear functions ψ which satisfy the assumptions of this paper.

Throughout this section, we fix a right-halfline I and assume that {𝒜(t):tI} is defined by (1.1) with

qij(t,x)=(1+|x|2)mqij0(t),bi(t,x)=-xib(t)(1+|x|2)r,(t,x)I×d,

for any i,j=1,,d, where r>m, the function bClocα/2(I) is bounded with positive infimum b0 and qij0(t) are the entries of a symmetric and positive definite matrix Q0(t), for any tI, which satisfies the estimate Q0(t)ξ,ξλ0 for any tI, ξB(0,1)d and some positive constant λ0. Then Hypotheses 2.1 are satisfied with φ(x)=1+|x|2 for any xd.

Example 6.1 (Local existence and regularity).

Fix T>sI and let us consider as nonlinear term the function ψ:[s,T]×d××d defined by

ψ(t,x,u,v)=-ϕ(t)g(x)u(1+|v|2)α,t[s,T],x,vd,u.

Here, ϕ:[s,T] and g:d are bounded and continuous functions with positive infimum,

|ϕ(t)|M0(t-s)η

for any t[s,T] and some positive constants M0 and η, and α(0,η+1). We claim that ψ satisfies Hypotheses 3.1 so that Theorem 3.2 can be applied to deduce that for any fCb(d) there exists a unique local mild solution uf to the Cauchy problem (1.2) with these choices of ψ and 𝒜. Hypothesis 3.1 (ii) is trivially satisfied. To prove Hypothesis 3.1 (i), we fix R>1, u1,u2[-R,R] and v1,v2B(0,R)d. Then, for any t(s,(s+1)T] and xd, we can estimate

|ψ(t,x,u1,(t-s)-12v1)-ψ(t,x,u2,(t-s)-12v2)|=ϕ(t)g(x)|u1(1+(t-s)-1|v1|2)α-u2(1+(t-s)-1|v2|2)α|gϕ(t){(1+(t-s)-1R2)α|u1-u2|+R|(1+(t-s)-1|v1|2)α-(1+(t-s)-1|v2|2)α|}Cg(t-s)η-α{(1+R2)α|u1-u2|+R2(1(1+R2)α-1)|v1-v2|}

for some positive constant C, independent of t, x, uj and vj (j=1,2). Hence, Hypothesis 3.1 (i) is satisfied with β=α-η and

LR=Cgmax{(1+R2)α,R2(1(1+R2)α-1)}.

If, in addition, g is γ- Hölder continuous for some γ(0,1) and 2α-2η+γ<2, then, by Corollary 3.3, uf is actually a classical solution to the Cauchy problem (1.2).

Example 6.2 (Global existence and stability).

Fix sI and let ψ:[s,+)×d××d be the function defined by

ψ(t,x,u,v)=ϕ(t)g(x)(-h(u)+χ(v)),ts,x,vd,u,

where ϕCb((s,+)) has positive infimum ϕ0 and gCbγ(d), for some γ(0,1), has positive infimum g0. The function h: is nonnegative, locally Lipschitz continuous in and satisfies the conditions h(0)=0 and uh(u)γ1u2-γ0|u| for some positive constants γ0 and γ1. Finally, the function χ:d is Lipschitz continuous and vanishes at 0. As it is easily seen, ψ satisfies Hypotheses 3.5, condition (3.9) and, clearly, ψ(,,0,0) is a bounded function. Moreover, if we take φ~(x)=1+|x|2 for any xd, then also the condition 𝒜φ~+k1|φ~|aφ~ in I×d is satisfied with some suitable positive constants a and k1. Hence, by Theorem 3.6, problem (1.2) admits a unique global classical solution defined in the whole [s,T]×d.

If h is globally Lipschitz continuous, then ψ satisfies Hypotheses 4.3 (i) with

ξ0=ϕgγ0,ξ1=-ϕ0g0γ1,ξ2=[χ]Lip(d)ϕg.

For any σ>max{m-1,0}, the function φ~:d, defined by φ~(x)=(1+|x|2)σ for any xd, satisfies Hypotheses 4.3 (ii)–(iii) for some suitable locally bounded functions a, k1 and Cj (j=0,1,2). Hence, Theorem 5.1 can be applied to deduce that, if

-ϕ0g0γ1+[s]Lip(d)2ϕ2g2(4λ0(p-1))-1<0,

then the functions tuf(t,)Lp(d,μt) and txuf(t,)Lp(d,μt) exponentially decay to zero as t+ for any fLp(d,μs) and pp0, where λ0 is the constant introduced at the beginning of this section. The same result holds for any fCb(d) if we replace the Lp(d,μt)-norm by the L-norm.

Example 6.3 (Summability improving properties).

We take the same function ψ as in Example 6.2. In view of Remark 4.4, we assume that the diffusion coefficients are independent of x. Since xb(t,x)ξ,ξ-b0|ξ|2 for any tI and x,ξd, and b0 is positive, by [4, Theorem 3.3] Hypothesis 4.3 (iv) is satisfied. Then, by Theorem 4.5, estimates (4.4) and (4.5) are satisfied.

If, in addition, the power r in the drift coefficients is positive, then (4.12) holds true, by Remark 4.12. Indeed, in such a case we can estimate b(t,x),x=-b(t)|x|2(1+|x|2)r-b0|x|2(1+|x|2)r-b0|x|2+2r for any tI and xd, so that, by Theorem 4.13, the nonlinear evolution operator 𝒩(t,s) satisfies estimates (4.21) and (4.22).

A Technical results

Proposition A.1.

Let Hypotheses 2.1 hold and let gC((a,b]×Rd) satisfy

[g]γ,:=supr(a,b)(r-a)γg(r,)<+

for some γ[0,1) and some Ia<b. Then the function z:[a,b]×RdR, defined by

z(t,x):=at(G(t,r)g(r,))(x)𝑑r,t[a,b],xd,

belongs to Cb([a,b]×Rd)C0,1+θ((a,b]×Rd) for any θ(0,1),

z(b-a)1-γ1-γ[g]γ,,xz(t,)cγ,a,b(t-a)12-γ[g]γ,(A.1)

and

xz(t,)Cθ(BR)CR[g]γ,(t-a)1-2γ-θ2(A.2)

for any t(a,b], R>0 and some positive constants cγ,a,b and CR. In particular, if γ12, then xz is bounded in (a,b]×Rd.

Finally, if [g]γ,θ,R:=supt(a,b](t-a)γg(t,)Cbθ(BR)<+ for some θ(0,1) and any R>0, then one has zCloc0,2+θ((a,b]×Rd)C1,2((a,b]×Rd). Moreover,

z(t,)Cb2(BR)c(t-a)θ2-γ[g]γ,θ,R+1,t(a,b],(A.3)

and

z(t,)Cb2+ρ(BR)c(t-a)θ-ρ2-γ[g]γ,θ,R+1,t(a,b],(A.4)

where ρ=α if θ>α, whereas ρ can be arbitrarily fixed in (0,θ) otherwise

Proof.

Throughout the proof, we will make use of [5, Proposition 2.7], where it has been shown that, for any Ia<b, R>0, η(0,1] and β[η,2+α], there exist positive constants Cβ=Cβ(a,b,R) and Cη,β=Cη,β(a,b,R) such that for any fCb(d)Clocη(d),

G(t,s)fCβ(B¯R){Cβ(t-s)-β2f,Cη,β(t-s)-β-η2fCη(B¯R+1),as<tb.(A.5)

To begin with, we observe that, for any t(a,b] and xd, the function r(G(t,r)g(r,))(x) is measurable in (a,t]. If g is bounded and uniformly continuous in d+1, this is clear. Indeed, as it has been recalled in Section 2, the function (t,s,x)(G(t,s)f)(x) is continuous in {(t,s,x)I×I×d:ts} for any fCb(d). Hence, taking (1.3) into account and adding and subtracting (G(t,r)g(r0,))(x), we can estimate

|(G(t,r)g(r,))(x)-(G(t,r0)g(r0,))(x0)|g(r,)-g(r0,)+|(G(t,r)g(r0,))(x)-(G(t,r0)g(r0,))(x0)|

for any (r,x),(r0,x0)[a,t]×d, and the last side of the previous chain of inequalities vanishes as (r,x) tends to (r0,x0).

If the function g is as in the statement of the proposition, then we can approximate it by a sequence (gn) of bounded and uniformly continuous functions in d+1 which converge to g pointwise in (a,b)×d and satisfy gn(r,)g(r,) for any r(a,b).1 Since the sequence (gn) is bounded and pointwise converges to g in (a,t]×d, by [9, Proposition 3.1 (i)] (G(t,)gn(r,))(x) converges to (G(t,)g(r,))(x) as n+ pointwise in (a,t]. Hence, the function r(G(t,r)g(r,))(x) is measurable in (a,t].

Using again (1.3), we obtain G(t,r)g(r,)g(r,)(r-a)-γ[g]γ, for any r(a,t]. It thus follows that z is bounded and the first estimate in (A.1) follows.

Proving that z is continuous in [a,b]×d is an easy task, based on estimate (1.3) and the dominated convergence theorem. Hence, the details are omitted.

Fix θ(0,1). The first estimate in (A.5) with β=1+θ and the assumptions on g allow to differentiate z with respect to xj (j=1,,d), under the integral sign, and obtain that Djz(t,) is locally θ-Hölder continuous in d, uniformly with respect to t(a,b), and

Djz(t,)Cθ(BR)CR[g]γ,(t-a)1-2γ-θ2,t(a,b].(A.6)

To conclude that Djz is continuous in (a,b]×d, it suffices to prove that, for any xd, the function Djz(,x) is continuous in (a,b]. For this purpose, we apply an interpolation argument. We fix R>0 such that xB¯R. Applying the well-known interpolation estimate

fC1(B¯R)KfC(B¯R)θ/(1+θ)fC1+θ(BR)1/(1+θ)

with f=z(t,)-z(t0,) and t,t0(a,b], from the continuity of z in [a,b]×d and the local boundedness in (a,b] of the function tf(t,)C1+θ(BR), we conclude that the function Djz(,x) is continuous in (a,b]. Hence, zCloc0,1+θ((a,b]×d). Estimate (A.2) follows from (A.6). Further, estimate (2.4) and the assumption on g imply that

|Djz(t,x)|C0[g]γ,at(r-a)-γ(1+(t-r)-12)𝑑r=Cγ,a,b(t-a)12-γ[g]γ,

for any (t,x)(a,b]×d, whence the second estimate in (A.1) follows at once.

Let us now assume that supt(a,b)(t-a)γg(t,)Cbθ(BR)<+ for any R>0. Arguing as above and taking the second estimate in (A.5) with β=2 (resp. β=2+α) into account, we can show that z(t,)Cloc2(d) (resp. z(t,)Cloc2+α(d)) for any t(a,b] and (A.3) (resp. (A.4)) holds true. Applying the interpolation inequality

φC2(B¯R)Cφθ/(2+θ)φC2+θ(B¯R)2/(2+θ)

with φ=z(t,)-z(t0,) we deduce that the second-order spatial derivatives of z are continuous in (a,b]×BR and, hence, in (a,b]×d due to the arbitrariness of R>0.

Finally, to prove the differentiability of z, we introduce the sequence (zn), where

zn(t,x)=at-1n(G(t,r)g(r,))(x)𝑑r,t[a+1/n,b],xd,n.

As it is immediately seen, zn converges to z, locally uniformly in (a,b]×d and each function zn is differentiable in [a+1/n,b]×d with respect to t and

Dtzn(t,x)=at-1n(𝒜(t)G(t,r)g(r,))(x)𝑑r+(G(t,t-1n)g(t-1n,))(x)

for such values of (t,x). Since 𝒜(t)G(t,r)g(r,)Cb(BR)CR[g]γ,(t-r)θ/2-γ(r-a)-γ for any r(a,t), and g(t-1n,) converges to g(t,) locally uniformly in d, by [9, Proposition 3.6] and the dominated convergence theorem, we conclude that Dtzn converges locally uniformly in (a,b]×d to 𝒜z+g. Thus, we conclude that z is continuously differentiable in (a,b]×d and, therein, Dtz=𝒜z+g. ∎

Lemma A.2.

Let J be an interval and let gC(J×Rd) be such that g(t,) is bounded in Rd for any tJ. Then the function tg(t,) is measurable in J.

Proof.

To begin with, we observe that for any n the function tg(t,)C(B¯n) is continuous in J. This is a straightforward consequence of the uniform continuity of g in J0×Bn for any bounded interval J0 compactly embedded into J. To complete the proof, it suffices to show that zn(t):=g(t,)C(B¯n) converges to g(t,) for any tJ. Clearly, for any fixed tJ, the sequence (zn(t)) is increasing and is bounded from above by g(t,). To prove that (zn(t)) converges to g(t,), we fix a sequence (xn)d such that |g(t,xn)| tends to g(t,) as n+. For any n, let kn be such that xnBkn. Without loss of generality, we can assume that the sequence (kn) is increasing. Then zkn(t)=g(t,)C(B¯kn)|g(t,xn)| for any n. Hence, the sequence (zkn(t)) converges to g(t,) and this is enough to conclude that the whole sequence (zn(t)) converges to g(t,) as n+. ∎

Finally, we prove some interior Lp-estimates.

Proposition A.3.

Let ΩRd be a bounded open set and let uC1,2((s,T)×Ω) solve the equation Dtu=Au in (s,T)×Ω. Then, for any x0Ω and any radius R1>0 such that BR1(x0)Ω, there exists a positive constant c~=c~(R1,x0,s,T) such that

(t-s)u(t,)W2,p(BR1(x0))+t-su(t,)W1,p(BR1(x0))c~supr(s,T)u(r,)Lp(Ω).

Proof.

Throughout the proof, we denote by c a positive constant, independent of n and u, which may vary from line to line.

Let us fix 0<R1<R2 such that BR2(x0)¯Ω and a sequence of cut-off functions (ϑn)Cc(Ω) such that 𝟙Brn(x0)ϑn𝟙Brn+1(x0) and ϑnCbk(Ω)2knc for any n{0} and k=0,1,2,3, where

rn:=2R1-R2+(2-2-n)(R2-R1).

Since the function un:=ϑnu solves the equation Dtun=𝒜un+gn in (s,T)×Brn+1(x0), where

gn=-u𝒜ϑn-Qxu,ϑn,

we can write

un(t,x)=(Gn+1𝒟(t,s)ϑnu(s,))(x)+st(Gn+1𝒟(t,σ)gn(σ,))(x)𝑑σ,(A.7)

where Gn+1𝒟(t,s) is the evolution operator associated to the realization of the operator 𝒜 in Lp(Brn+1(x0)) with homogeneous Dirichlet boundary conditions. It is well known that

G𝒟(t,r)ψW2,p(Brn+1(x0))c(t-r)-1+α2ψWα,p(Brn+1(x0))

for any α(0,1), ψWα,p(Brn+1(x0)) and sr<tT. Since gn(σ,)Wα,p(Brn+1(x0)) for any σ(s,t), from (A.7) we obtain

(t-s)u(t,)W2,p(Brn(x0))cu(s,)Lp(Brn+1(x0))+cst(t-σ)-1+α2gn(σ,)Wα,p(Brn+1(x0))𝑑σ.

Now, for any n we set ζn:=supt(s,T)(t-s)u(t,)W2,p(Brn(x0)) and estimate the function under the integral sign. At first, we note that

gn(σ,)Wα,p(Brn+1(x0))cϑnCb2+α(Brn+1(x0))u(σ,)W1+α,p(Brn+1(x0)).

By interpolation and using Young’s inequalities we obtain, for any σ(s,t),

u(σ,)W1,p(Brn+1(x0))c(σ-s)-12u(σ,)Lp(Brn+1(x0))12ζn+1(σ-s)-12(cε-1u(σ,)Lp(Ω)+εζn+1)

and

xu(σ,)Wα,p(Brn+1(x0))(σ-s)-1+α2(cε-1+α1-αu(σ,)Lp(Ω)+εζn+1).

Collecting the above estimates together, we get

ζn8ncεζn+1+csupr(s,T)u(r,)Lp(Ω)(1+8nε-(1+α)/(1-α)).

Now we fix 0<η<64-1/(1+α) and ε=8-nc-1η. Multiplying both the sides of the previous inequality by ηn and summing up from 0 to N yields

ζ0-ηN+1ζN+1csupr(s,T)u(r,)Lp(Ω).(A.8)

Since {ζn}n is bounded, taking the limit as N+ on the left-hand side of (A.8) we conclude that

(t-s)u(t,)W2,p(BR1(x0))csupr(s,T)u(r,)Lp(Ω)

for any t(s,T). An interpolation argument gives u(t,)W1,p(BR1(x0))c(t-s)-1/2supr(s,T)u(r,)Lp(Ω) for any t(s,T), and this completes the proof. ∎

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Footnotes

  • 1

    This can be done, for instance, setting gn(t,x)=ϑn(t)(g¯(t,)ρn)(x) for any (t,x)d+1, n, where g¯:(a,+)×d equals g in (a,b)×d and g¯(t,)=g(b,) for any t>b, (ϑn)C() is a sequence of smooth functions such that 𝟙[a+2/n,+)ϑn𝟙[a+1/n,+) for any n and “” denotes convolution with respect to the spatial variables. 

About the article

Received: 2016-07-24

Revised: 2016-12-21

Accepted: 2017-01-08

Published Online: 2017-02-04


The authors are members of GNAMPA of the Italian Istituto Nazionale di Alta Matematica. This work has been supported by the INdAM-GNAMPA Project 2016 “Equazioni e sistemi di equazioni ellittiche e paraboliche associate ad operatori con coefficienti illimitati e discontinui”.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 225–252, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0166.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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