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Advances in Nonlinear Analysis

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Radon measure-valued solutions of first order scalar conservation laws

Michiel Bertsch
  • Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133; and Istituto per le Applicazioni del Calcolo “M. Picone”, CNR, Roma, Italy
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/ Flavia Smarrazzo / Andrea Terracina
  • Dipartimento di Matematica “G. Castelnuovo”, Università “Sapienza” di Roma, P.le A. Moro 5, 00185 Roma, Italy
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  • Dipartimento di Matematica “G. Castelnuovo”, Università “Sapienza” di Roma, P.le A. Moro 5, 00185; and Istituto per le Applicazioni del Calcolo “M. Picone”, CNR, Roma, Italy
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Published Online: 2018-08-23 | DOI: https://doi.org/10.1515/anona-2018-0056

Abstract

We study nonnegative solutions of the Cauchy problem

{tu+x[φ(u)]=0in ×(0,T),u=u00in ×{0},

where u0 is a Radon measure and φ:[0,) is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon measures. Under some additional conditions on φ, we prove their uniqueness if the singular part of u0 is a finite superposition of Dirac masses. Regarding the behavior of φ at infinity, we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive waiting time (in the linear case φ(u)=u this happens for all times). In the latter case, we describe the evolution of the singular parts.

Keywords: First order hyperbolic conservation laws; Radon measure-valued solutions; entropy inequalities; uniqueness

MSC 2010: 35D99; 35K55; 35R25; 28A33; 28A50

1 Introduction

In this paper we consider the Cauchy problem

{tu+x[φ(u)]=0in ×(0,T)=:S,u=u0in ×{0},(P)

where T>0, u0 is a nonnegative finite Radon measure on , and φ:[0,), φ(0)=0, is a Lipschitz continuous function (see assumption H1). Therefore, φ grows at most linearly.

Problem (P) with a superlinear φ of the type φ(u)=up, p>1, was studied in [19], proving existence and uniqueness of nonnegative entropy solutions (see also [8]). By definition, in that paper the solution for positive times takes values in L1(), although the initial data u0 is a finite Radon measure. Interesting, albeit sparse results concerning (P) with φ at most linear at infinity can be found in the pioneering paper [10], in which the same definition of Radon measure-valued solutions used below (see equality (3.8)) was proposed.

When φ(u)=Cu (C), problem (P) is the Cauchy problem for the linear transport equation

{tu+Cxu=0in S,u=u0in ×{0},

whose solution is trivially the translated of u0 along the lines x=Ct+x0 (x0). In particular, the singular part us(,t) of the solution is nonzero for t>0 if and only if the same holds for t=0.

It is natural to ask what happens if φ is sublinear. To address this case we must consider solutions of problem (P) which, for t>0, possibly are finite Radon measures on as the initial data u0 . Therefore, throughout the paper we consider solutions of problem (P) as maps from [0,T] to the cone of nonnegative finite Radon measures on , which satisfy (P) in the following sense: for a suitable class of test functions ζ, we have

S[urtζ+φ(ur)xζ]𝑑x𝑑t+0Tus(,t),νζ(,t)𝑑t=-u0,ζ(,0)

(see Definition 3.3). Here the measure u(t) is defined for a.e. t(0,T), urL1(S) is the density of its absolutely continuous part, , denotes the duality map, and

νζ:=tζ+Cφxζ,Cφ:=limuφ(u)u.

Measure-valued entropy solutions are defined similarly (see Definition 3.3).

We use an approximation procedure to construct measure-valued entropy solutions of problem (P) (see Theorem 3.7). In addition, we prove that the singular part us of an entropy solution of problem (P) does not increase along the lines x=x0+Cφt (see Proposition 3.8). In particular, if Cφ=0, the map tus(,t) is nonincreasing.

Concerning the case when φ is sublinear, the following example is particularly instructive:

{tu+x[φ(u)]=0in S,u=δ0in ×{0},(1.1)

with S:=×(0,T), T>1 and

φ(u)=sgnp[(1+u)p-1](p<1p0).(1.2)

The function in (1.2) is increasing and concave, with Cφ=0, and belongs to a class for which the constructed entropy solution of problem (1.1)–(1.2) is unique (see Theorem 3.22). Hence, the following holds.

Proposition 1.1.

  • (i)

    Let p<0 . Let ξ(t) be defined by

    ξ=-(|p|tξ-1)p1-p-1(|p|tξ-1)11-p-1in (1,T),ξ(1)=0.

    Let

    A:={(x,t)S0<x|p|t, 0t1}{(x,t)Sξ(t)x|p|t, 1<tT}

    and

    us(t):=max{1-t,0}δ0,ur(x,t):=[(|p|tx-1)11-p-1]χA(x,t)((x,t)S).(1.3)

    Then u=ur+us is the unique constructed entropy solution of problem ( 1.1 )–( 1.2 ).

  • (ii)

    Let 0<p<1 . Let ξ(t) be defined by

    ξ=(|p|tξ-1)p1-p-1(|p|tξ-1)11-p-1in (0,T),ξ(0)=0.

    If B:={(x,t)Sξ(t)x|p|t, 0<tT} , then

    u(x,t)=ur(x,t):=[(|p|tx-1)11-p-1]χB(x,t)((x,t)S)(1.4)

    is the unique constructed entropy solution of problem ( 1.1 )–( 1.2 ).

Let us define the waiting time t0[0,T] for solutions u of (P):

t0:=inf{τ(0,T]us(,t)=0,ur(,t)L() for a.e. t(τ,T)}(1.5)

(by abuse of language, we call t0 “waiting time” even if t0=T). Then, by Proposition 1.1,

  • (*)

    positive waiting times occur in problem (1.1)–(1.2) if and only if p<0.

More precisely, if p<0, the singular part us(,t) persists until the waiting time t0=1 at which it disappears, whereas for 0<p<1, the singular part vanishes for all t>0, thus t0=0 – an instantaneous regularizing effect. Instantaneous regularization also occurs if p>1 (see [19] and Remark 3.24), whereas, as already remarked, in the linear case p=1, we have t0=T if u0s0.

Since φ(u)=sgnp[(1+u)p-1] (p<1,p0) is bounded if and only if p<0, and Cφ=0, statement (*) could be rephrased as follows.

Proposition 1.2.

Positive waiting times occur in problem (1.1) if and only if the map uφ(u)-Cφu, with φ as in (1.2), is bounded in [0,).

The above result is generalized to problem (P), by Theorem 3.18, for functions φ which satisfy for u large a condition implying either concavity or convexity (see assumption (H4) and Remark 3.13). The proof of Theorem 3.18 makes use of estimates of the density ur of the solution of (P), which are strongly reminiscent of the Aronson–Bénilan inequality for the porous medium equation (see Proposition 6.2). The main results on the waiting time and the regularity of solutions of (P) are collected in Section 3.3. The existence and an upper bound, in terms of φ and u0, of a waiting time was already pointed out in [10, Proposition 2.1] (see also Theorem 3.8 (ii)).

Another interesting feature of the solution of (1.1)–(1.2), with p<0, is that for t(0,1), i.e., as long as us(,t)>0, we have

limx0+ur(x,t)=.

Namely, the regular part ur(,t) diverges when approaching from the right the point x0=0, where us(,t) is concentrated. As we shall see below (see (3.24)–(3.25)), this property can be generalized to entropy solutions of a larger class of problems, characterized by the concavity/convexity property on φ mentioned before. In this class a generalized form of this property will also be used as a uniqueness criterion, provided that φ(u)-Cφu is bounded in [0,) and u0s is a finite superposition of Dirac masses (see Proposition 3.17 and Theorem 3.22). In [10] it was already observed that Kruzkov’s entropy inequalities do not guarantee the uniqueness of solutions (see also Remark 3.23 below), and the formulation of an additional uniqueness criterion was left as an open problem. This problem is addressed in a forthcoming paper, where more general compatibility conditions are given, which ensure uniqueness also for non-convex or non-concave functions φ (see [3]).

Apart from the intrinsic mathematical interest of problem (P), it is worth pointing out its connection with a class of relevant models. Ion etching is a common technique for the fabrication of semiconductor devices, also relevant in other fields of metallurgy, in which the material to be etched is bombarded with an ion beam (see [16, 25, 24]). Mathematical modelling of the process leads to the Hamilton–Jacobi equation in one space dimension

{tU+φ(xU)=0in ×(0,T),U=U0in ×{0},(HJ)

where U=U(x,t) denotes the thickness of the material and φ is bounded, non-convex and vanishing at infinity. Formal differentiation with respect to x suggests to describe the problem in terms of the unknown u:=xU, which formally solves (P) with u0=U0. In this way, discontinuous solutions of (HJ) correspond to Radon measure-valued solutions of (P) having a Dirac mass δx0 concentrated at any point x0, where U(,t) is discontinuous (t(0,T)). A rigorous justification of the above argument, relating discontinuous viscosity solutions of (HJ) to Radon measure-valued entropy solutions of (P), is to our knowledge an open problem (in this connection, see [7, 14]).

In the context of conservation laws, the term “measure-valued solution” usually refers to solutions in the sense of Young measures, after DiPerna’s seminal paper [11]. We stress that this concept of “statistical solutions” is completely different from that of Radon measure-valued solutions, introduced by Demengel and Serre [10], and discussed in the present paper. On the other hand, we do use Young measures in this paper, since they are an important ingredient in the construction of Radon measure valued solutions (see Section 3 and, in particular, Section 5).

A number of ideas used in the present paper go back to papers dealing with Radon measure-valued solutions of quasilinear parabolic problems, also of forward-backward type (in particular, see [6, 4, 5, 21, 23, 27]).

The results presented in this paper naturally lead to some open problems. Among them we mention a general statement about an instantaneous regularizing effect for fluxes with superlinear growth (singular parts should disappear instantaneously for t>0), and an appropriate generalization of our results to the case of solutions with changing signs, when additional nonuniqueness phenomena (such as N-waves, see [19]) may occur; in this regard, the general case of an initial signed Radon measure u0 in problem (P) will be considered in a forthcoming paper. Another open problem is whether new phenomena occur if φ is uniformly Lipschitz continuous on [0,) but the limit φ(s)/s as s does not exist.

The paper is organized as follows. In Section 2 we recall several known results used in the sequel and introduce some notation. In Section 3 we present the main results of the paper. In Section 4 we introduce the approximation procedure needed for the construction of solutions. Sections 57 are devoted to the proofs of existence, qualitative properties and uniqueness of solutions.

2 Preliminaries

2.1 Function spaces and Radon measures

We denote by () the Banach space of finite Radon measures on , with norm μ():=|μ|(). By +(), we denote the cone of nonnegative finite Radon measures; if μ1,μ2(), then we write μ1μ2 if μ2-μ1+(). We denote the convex set of probability measures on by 𝒫𝓁1()+(). We have τ()=τ()=1 for τ𝒫().

We denote by Cc() the space of continuous real functions with compact support in . The space of the functions of bounded variation in is denoted by BV():={uL1()u()}, where u is the distributional derivative of u. It is endowed with the norm uBV():=uL1()+u(). We say that uBVloc() if uBV(Ω) for every open bounded subset Ω.

The Lebesgue measure, either on or S:=×(0,T), is denoted by ||. Integration with respect to the Lebesgue measure on or on S will be denoted by the usual symbols dx, respectively dxdt. A Borel set E is null if |E|=0. The expression “almost everywhere”, or shortly “a.e.”, means “up to null sets”. For every measurable function f defined on and x0, we write esslimxx0f(x)=l if there is a null set E* such that f(xn)l for any sequence {xn}(E*{x0}), xnx0. We set f±:=max{±f,0} for every measurable function f on .

We denote the duality map between () and Cc() by μ,ρ:=ρ𝑑μ. By abuse of notation, we extend μ,ρ to any μ-integrable function ρ. A sequence {μn} converges strongly to μ in () if μn-μ()0 as n. A sequence {μn} of (possibly not finite) Radon measures on converges weakly* to a (possibly not finite) Radon measure μ, i.e., μn*μ, if μn,ρμ,ρ for all ρCc(). Similar definitions are used for (possibly not finite) Radon measures on Ω×(0,T), with Ω.

Every μ() has a unique decomposition μ=μac+μs, with μac() absolutely continuous and μs() singular with respect to the Lebesgue measure. We denote by μrL1() the density of μac. Every function fL1() can be identified to a finite absolutely continuous Radon measure on ; we shall denote this measure by the same symbol f used for the function.

The restriction μE of μ() to a Borel set E is defined by (μE)(A):=μ(EA) for any Borel set A. Similar notations are used for the spaces of finite Radon measures (Ω), with Ω, (S) and (S×), where S:=×(0,T).

We shall use measures u(S) which, roughly speaking, admit a parametrization with respect to the time variable.

Definition 2.1.

We denote by L(0,T;+()) the set of finite nonnegative Radon measures u+(S) such that for a.e. t(0,T), there is a measure u(,t)+() with the following properties:

  • (i)

    if ζC([0,T];Cc()), the map tu(,t),ζ(,t) belongs to L1(0,T) and

    u,ζS=0Tu(,t),ζ(,t)𝑑t,(2.1)

  • (ii)

    the map tu(,t)() belongs to L(0,T).

Accordingly, we set

uL(0,T;()):=esssupt(0,T)u(,t)()for uL(0,T;+()).

Remark 2.2.

The definition implies that for all ρCc(), the map tu(,t),ρ is measurable, thus the map u:(0,T)() is weakly* measurable (e.g., see [22, Section 6.7]). For simplicity, we prefer the notation L(0,T;()) to the more correct one Lw*(0,T;()), which is used in [22].

If uL(0,T;+()), then also uac,usL(0,T;+()) and, by (2.1),

uac,ζS=Surζ𝑑x𝑑t,us,ζS=0Tus(,t),ζ(,t)𝑑t(2.2)

for ζC([0,T];Cc()). One can easily check that for a.e. t(0,T),

uac(,t)=[u(,t)]ac,us(,t)=[u(,t)]s,ur(,t)=[u(,t)]r,(2.3)

where [u(,t)]r denotes the density of the measure [u(,t)]ac. For ρCc(), we have

[u(,t)]ac,ρ=[u(,t)]rρ𝑑x=ur(,t)ρ𝑑xfor a.e. t(0,T).

In view of (2.2)–(2.3), we shall always identify the quantities which appear on either side of equalities (2.3).

For any μ() and a, the translated measure 𝒯a(μ) is defined by

𝒯a(μ),ρ:=μ,ρ-a

for any ρCc(), where ρ-a(x):=ρ(x+a) (x). Clearly, 𝒯a(μ)() and

[𝒯a(μ)]ac=𝒯a(μac),[𝒯a(μ)]s=𝒯a(μs).

2.2 Young measures

We recall the following result [2].

Theorem 2.3.

Let ΩRN be Lebesgue measurable, let KR be closed, and let un:ΩR be a sequence of Lebesgue measurable functions such that

limn|{xΩun(x)U}|=0

for any open neighborhood U of K in R. Then there exist a subsequence {uj}{unj}{un} and a family {τx} of nonnegative measures on R, depending measurably on xΩ, such that

  • (i)

    τx():=𝑑τx1 for a.e. xΩ,

  • (ii)

    suppτxK for a.e. xΩ,

  • (iii)

    for every continuous function f: satisfying lim|ξ|f(ξ)=0 , we have

    f(uj)*f*in L(Ω),

    where

    f*(x):=τx,f=f(ξ)𝑑τx(ξ)for a.e. xΩ.(2.4)

Suppose further that {uj} satisfies the boundedness condition

limksupj|{xΩBR|uj(x)|k}|=0(2.5)

for every R>0, where BR:={xRN|x|<R}. Then

  • (iv)

    τx is a probability measure for a.e. xΩ,

  • (v)

    given any measurable subset AΩ , we have

    f(uj)f*in L1(A)(2.6)

    for all continuous functions f: such that {f(uj)} is sequentially weakly compact in L1(A).

Below we shall always refer to the family {τx} of probability measures given by the previous theorem as the disintegration of the Young measure τ (or briefly Young measure) associated to the sequence {uj}. We denote the set of Young measures on Ω× by 𝒴(Ω;); in particular, 𝒴(S;) denotes the set of Young measures on S×, with S:=×(0,T).

Remark 2.4.

  • (i)

    The argument used in the proof of Theorem 2.3 shows that, under hypothesis (2.5), the convergence in (2.6) holds true for Carathéodory functions f:A× if {f(,uj)} is sequentially weakly relatively compact in L1(A).

  • (ii)

    Condition (2.5) is very weak. It is equivalent to the statement that for any R>0, there is a continuous nondecreasing function gR:[0,) such that

    limξgR(ξ)=,supjΩBRgR(|uj(x)|)𝑑x<.

    Therefore, Theorem 2.3 applies to bounded sequences {uj} in L1(Ω) (in which case gR(ξ)=ξ).

If ΩN is bounded and {uj} is a bounded but not uniformly integrable sequence in L1(Ω), it is possible to extract a uniformly integrable subsequence “by removing sets of small measure”. This is the content of the following “Biting lemma” (e.g., see [17, 28] and references therein).

Theorem 2.5.

Let {un} be a bounded sequence in L1(Ω), where ΩRN is a bounded open set. Moreover, let {uj}{un} and {τx} be the subsequence and the Young measure given in Theorem 2.3, respectively. Then there exist a subsequence {uk}{ujk}{uj} and a decreasing sequence of measurable sets EkΩ of Lebesgue measure |Ek|0 such that the sequence {ukχΩEk} is uniformly integrable and

ukχΩEkZ:=ξ𝑑τ(ξ)in L1(Ω),

where ZL1(Ω) is called the barycenter of the disintegration {τx}.

3 Main results

Throughout the paper we assume that u0+(). Concerning φ, we always suppose that

  • (H1)

    φC([0,)), φ(0)=0, φL(0,), and limuφ(u)u=:Cφ exists.

Hence, there exists M>0 such that

|φ(u)|M,|φ(u)|Mufor a.e. u>0.(3.1)

3.1 Definition of solution

In the following definitions, we denote by

νζ:=tζ+Cφxζ(3.2)

the derivative of any ζC1(S) along the vector τ¯(Cφ,1).

Definition 3.1.

By a solution of problem (P) in the sense of Young measures, we mean a pair (u,τ) such that

  • (i)

    uL(0,T;+()), τ𝒴(S;),

  • (ii)

    suppτ(x,t)[0,) for a.e. (x,t)S, and

    ur(x,t)=[0,)ξ𝑑τ(x,t)(ξ),(3.3)

    where τ(x,t)𝒫𝓁1() is the disintegration of τ,

  • (iii)

    for all ζC1([0,T];Cc1()), with ζ(,T)=0 in , we have

    S[urtζ+φ*xζ]𝑑x𝑑t+0Tus(,t),νζ(,t)𝑑t=-u0,ζ(,0),(3.4)

    where νζ is defined by (3.2) and

    φ*(x,t):=[0,)φ(ξ)𝑑τ(x,t)(ξ)for a.e. (x,t)S.(3.5)

By an entropy solution of problem (P) in the sense of Young measures, we mean a solution such that

S[E*tζ+F*xζ]𝑑x𝑑t+CE0Tus(,t),tζ(,t)𝑑t+CF0Tus(,t),xζ(,t)𝑑t-E(u0r)ζ(x,0)𝑑x-CEu0s,ζ(,0)(3.6)

for all ζ as above, ζ0, and for every pair (E,F), E,F:[0,), such that

  • (C1)

    E is convex, E,FL(0,), F=Eφ in (0,), and limuE(u)u=:CE, limuF(u)u=:CF exist.

In (3.6), for a.e. (x,t)S, we set

E*(x,t):=[0,)E(ξ)𝑑τ(x,t)(ξ),F*(x,t):=[0,)F(ξ)𝑑τ(x,t)(ξ).

Entropy subsolutions (respectively supersolutions) of problem (P) in the sense of Young measures are defined by requiring that inequality (3.6) be satisfied for all ζ and (E,F) as above, with E nondecreasing (respectively nonincreasing).

Observe that choosing E(u)=±u in the entropy inequality (3.6) plainly gives the weak formulation (3.4).

Remark 3.2.

  • (i)

    By (3.1), (3.3) and (3.5),

    |φ*(x,t)|M[0,)ξ𝑑τ(x,t)(ξ)=Mur(x,t)for a.e. (x,t)S.(3.7)

    Since urL(0,T;L1()), by (3.7), we have that φ*L(0,T;L1()).

  • (ii)

    By (C1), the functions E, F have at most linear growth. Arguing as in (i), it follows that E* and F* belong to L(0,T;Lloc1()) and L(0,T;L1()), respectively, if E(0)=F(0)=0.

Definition 3.3.

A measure uL(0,T;+()) is called a solution of problem (P) if for all ζC1([0,T];Cc1()), ζ(,T)=0 in , we have

S[urtζ+φ(ur)xζ]𝑑x𝑑t+0Tus(,t),νζ(,t)𝑑t=-u0,ζ(,0),(3.8)

where νζ is defined by (3.2). A solution of problem (P) is called an entropy solution if for all ζ0 as above and for all (E,F) as in (C1), it satisfies the entropy inequality

S[E(ur)tζ+F(ur)xζ]𝑑x𝑑t+CE0Tus(,t),tζ(,t)𝑑t+CF0Tus(,t),xζ(,t)𝑑t-E(u0r)ζ(x,0)𝑑x-CEu0s,ζ(,0).(3.9)

Entropy subsolutions (respectively supersolutions) of problem (P) are defined by requiring (3.9) to be satisfied for all ζ and (E,F) as before, with E nondecreasing (respectively nonincreasing).

A solution of problem (P) is also a solution in the sense of Young measures. Moreover, it follows from (3.1) that φ(ur)L(0,T;L1()). Similar remarks hold for entropy solutions, subsolutions and supersolutions.

Remark 3.4.

  • (i)

    If Cφ=0, equality (3.8) reads

    S[utζ+φ(ur)xζ]𝑑x𝑑t=-u0,ζ(,0),

    whence tu=-x[φ(ur)] in 𝒟(S).

  • (ii)

    For the Kružkov entropies E(u)=|u-k|, F(u)=sgn(u-k)[φ(u)-φ(k)] (k[0,)), we have CE=1, CF=Cφ. Then inequality (3.9), for all k[0,), reads

    S{|ur-k|tζ+sgn(ur-k)[φ(ur)-φ(k)]xζ}𝑑x𝑑t+0Tus(,t),νζ(,t)𝑑t-|u0r-k|ζ(x,0)dx-u0s,ζ(,0).(3.10)

The following proposition states that for any solution of (P) in the sense of Young measures, the map tu(t), possibly redefined in a null set, is continuous up to t=0 with respect to the weak* topology of +(). In particular, it explains in which sense the initial condition is satisfied.

Proposition 3.5.

Let (H1) be satisfied, let (u,τ) be a solution of problem (P) in the sense of Young measures, and let ρCc(R). Then

esslimt0+u(,t),ρ=u0,ρ,(3.11)esslimtt0u(,t),ρ=u(,t0),ρfor a.e. t0(0,T).(3.12)

The map tu(t) has a representative, defined for all t[0,T], such that

limtt0u(,t),ρ=u(,t0),ρfor all t0[0,T].(3.13)

3.2 Existence and monotonicity

The existence of solutions is proven by an approximation procedure. If u0+(), then there exist u0nL1()L() such that

u0n0in ,u0nL1()u0(),(3.14)u0n*u0,u0nu0ra.e. in ,u0n-u0rLloc1(suppu0s)0(3.15)

(e.g., see [23, Lemma 4.1]). Consider the approximating problem

{tun+x[φ(un)]=0in S,un=u0nin ×{0}(n).(Pn)

Let us recall the definition of entropy solution of problem (Pn) (e.g., see [9]).

Definition 3.6.

A function unL(0,T;L1())L(S) is called an entropy solution of problem (Pn) if for every ζC1([0,T];Cc1()), with ζ(,T)=0 in and ζ0, and for any couple (E,F), with E convex and F=Eφ, we have

S[E(un)tζ+F(un)xζ]𝑑x𝑑t-E(u0n)ζ(x,0)𝑑x.(3.16)

Entropy solutions are weak solutions if ζC1([0,T];Cc1()), ζ(,T)=0 in and

S[untζ+φ(un)xζ]𝑑x𝑑t+u0nζ(x,0)𝑑x=0.(3.17)

By studying the limiting points of the sequence {un}, we shall prove the following result.

Theorem 3.7.

  • (i)

    Let (H1) be satisfied. Then problem ( (P) ) has a solution u , which is obtained as a limiting point of the sequence {un} of entropy solutions to problems ( (Pn) ). In addition, u is an entropy solution of problem ( (P) ) in the sense of Young measures.

  • (ii)

    Let (H1) and the following assumption be satisfied:

    • (i)(C1)

      φC1([0,)) , and for every u¯>0 , there exist a,b0, a+b>0 , such that φ is strictly monotone in [u¯-a,u¯+b].

    Then u is an entropy solution of problem ( (P) ).

Hypothesis (C1) fails if for example φ is affine in an interval (a,b)(0,). In that case, Proposition 5.9 (iii), which characterizes the limiting Young measure, gives some additional information.

The following proposition shows that the singular part of an entropy subsolution of (P) does not increase along the lines x=Cφt+x0.

Proposition 3.8.

Let (H1) be satisfied.

  • (i)

    Let u be an entropy subsolution of problem ( (P) ) in the sense of Young measures. Then

    us(,t2)𝒯Cφ(t2-t1)(us(,t1))in +(),for a.e. 0t1t2T.(3.18)

    In particular,

    us(,t)𝒯Cφt(u0s)in +(),for a.e. t(0,T),(3.19)

    whence us(,t)()u0s() for a.e. t(0,T).

  • (ii)

    Let u be a solution of problem ( (P) ). Then there is conservation of mass, i.e.,

    u(,t)()=u0()for a.e. t(0,T).

The linear case φ(u)=u shows that equality may hold in (3.18). Moreover, if Cφ=0, it follows from (3.18) that the map tus(,t) is nonincreasing.

3.3 Waiting time and regularity

It is convenient to distinguish two cases: Cφ=0 (sublinear growth at infinity) and Cφ0 (linear growth at infinity), with Cφ defined by (H1).

3.3.1 Sublinear growth

Beside (H1), we will use the following assumption:

  • (H2)

    φC([0,)), Cφ=0, there exist H-1, K such that φ′′(u)[Hφ(u)+K]-[φ(u)]2<0 for all u[0,).

By (H2) the map uφ′′(u)[Hφ(u)+K] is strictly negative and continuous in [0,), hence two cases are possible: either (a) Hφ+K>0, φ′′<0, or (b) Hφ+K<0, φ′′>0 in [0,). In case (a), we have φ>0 in [0,), since φ′′<0 and limuφ(u)=Cφ=0. Similarly, in case (b), we have plainly φ<0 in [0,). In particular, in both cases (H2) implies (C1). Moreover, if also (H1) holds, thus φ(0)=0, we have Hφ+K>0 in [0,) if and only if K>0.

Remark 3.9.

The following examples show that all values of H-1 may occur in (H2):

φ(u)=sgnp[(1+u)p-1](p<1,p0)H=p1-p(-1,0)(0,),K=|H|,φ(u)=1-e-αu(α>0)H=-1,K=1,φ(u)=log(1+u) or φ(u)=1-1log(e+u)H=0,K=1.

The following property of constructed entropy solutions plays an important role as a uniqueness criterion (see its generalized form given by Proposition 3.17 and Theorem 3.22 below).

Proposition 3.10.

Let (H1)(H2) be satisfied, and let φ be bounded in [0,). Then every entropy solution u of problem (P) given by Theorem 3.7 satisfies, for a.e. t(0,T) and all x0suppus(,t),

esslimxx0+ur(x,t)=if φ>0 in [0,),(3.20)esslimxx0-ur(x,t)=if φ<0 in [0,).

Theorem 3.11.

  • (i)

    Let (H1) be satisfied, let u0s({x0})>0 for some x0 and let u be a solution of problem ( (P) ). If φ is bounded in (0,) (in particular, Cφ=0 ), then the waiting time t0 defined by ( 1.5 ) satisfies

    t0min{T,u0s({x0})φL(0,)}>0.(3.21)

  • (ii)

    Let (H1) (H2) be satisfied, and let u be the entropy solution of problem ( (P) ) given by Theorem 3.7.

    • (ii)(a)

      If φ is bounded in (0,) and, moreover, H>-1, |K|<limu|φ(u)|=:γ , then

      t0min{T,(H+1)u0()γ-|K|}.(3.22)

    • (ii)(b)

      If φ is unbounded in (0,) , then t0=0.

Remark 3.12.

Concerning estimates (3.21) and (3.22), it is worth considering the case in which u0=δ0 and φ(u)=1-(1+u)p, p<0. By explicit calculations, in Proposition 1.1, we show that in this case the waiting time defined in (1.5) is t0=1. Hence, in this case, estimates (3.21)–(3.22) are sharp, since

δ0({0})φL(0,)=1and(H+1)δ0()γ-|K|=(p/(1-p)+1)δ0()1+p/(1-p)=1.

Remark 3.13.

In part (ii) of Theorem 3.11, it is enough to require condition (H2) for large values of u. More precisely (see Remark 6.10), Theorem 3.11 (ii) remains valid if instead of (H2), for some k>0, the following holds:

  • (H3)

    the function φk:[0,), φk(u):=φ(u+k)-φ(k), satisfies (H2).

In this connection, observe that the conditions H>-1 and |K|<limu|φ(u)| exclude the function φ(u)=1-e-u. The same conditions also exclude the function φ(u)=1-1log(e+u), where K=1=γ. However, in this case, we can use hypothesis (H3) for k>0, which is satisfied with H=0 and K=log-2(e+k)<γk=log-1(e+k).

Let us finally mention the following regularization result.

Proposition 3.14.

Let (H1)(H2) be satisfied, and let φ be bounded in [0,) (in particular, Cφ=0). Then, for a.e. t(0,T), suppus(t) is a null set.

Remark 3.15.

It suffices to prove Proposition 3.10, Theorem 3.11 and Proposition 3.14 by assuming φ′′<0 in (H2) (hence, K>0, by (H2) and the assumption φ(0)=0). Otherwise, it can be easily seen that if uL(0,T;+()) is a solution of problem (P), the map u~ defined by setting

u~,ζS:=0Tu(,t),ζ(-,t)dt

for every ζC([0,T];Cc()) is a solution of the problem

{tu~+x[φ~(u~)]=0in S,u~=u~0in ×{0}.(3.23)

Here u~0,ρ:=u0,ρ(-) for all ρCc(), and the function φ~:=-φ satisfies (H2) with K~:=-K. The same holds for entropy solutions.

3.3.2 Linear growth

Let φ satisfy the following assumption:

  • (H4)

    φC([0,)) and there exist H-1, K such that

    φ′′(u){H[φ(u)-Cφu]+K}-[φ(u)-Cφ]2<0for all u[0,)

(observe that (H4) reduces to (H2) if Cφ=0). If (H4) holds, the function φ~:=φ(u)-Cφu satisfies (H2) since Cφ~=0.

Remark 3.16.

It is easily seen that if u is a solution (respectively an entropy solution) of problem (P), then vL(0,T;+(Ω)), defined by

v(,t)=𝒯-h(u(,t))in ()

for any h, is a solution (respectively an entropy solution) of (P) with u0 replaced by v0:=𝒯-h(u0). Similarly, u~(,t):=𝒯-Cφt(u(,t)) is a solution (respectively an entropy solution) of problem (3.23), with u~0=u0 and φ~(u)=φ(u)-Cφu.

By Remark 3.16, the above results for the case Cφ=0 can be generalized as follows.

Proposition 3.17.

Let (H1) and (H4) be satisfied, and let uφ(u)-Cφu be bounded in (0,). Then every entropy solution u of problem (P) given by Theorem 3.7 satisfies, for a.e. t(0,T) and all x0suppus(,t),

esslimxx0+ur(x+Cφt,t)=if φ>Cφ in [0,),(3.24)esslimxx0-ur(x+Cφt,t)=if φ<Cφ in [0,).(3.25)

Theorem 3.18.

  • (i)

    Let (H1) be satisfied, let u0s({x0})>0 for some x0 , and let u be a solution of problem ( (P) ). If uφ(u)-Cφu is bounded in (0,) , then

    t0min{T,u0s({x0})φ-CφuL(0,)}>0.

  • (ii)

    Let (H1) and (H4) be satisfied, and let u be the entropy solution of problem ( (P) ) given by Theorem 3.7.

    • (ii)(a)

      Let uφ(u)-Cφu be bounded in (0,) . If H>-1 and |K|<limu|φ(u)-Cφu|=:γ~ , then

      t0min{T,(H+1)u0()γ~-|K|}.

    • (ii)(b)

      Let uφ(u)-Cφu be unbounded in (0,) . Then t0=0.

Again, Theorem 3.18 (ii) remains valid if, for some k>0, the function φk defined in Remark 3.13 satisfies (H4).

Proposition 3.19.

Let (H1) and (H4) be satisfied, and let uφ(u)-Cφu be bounded in (0,). Then for a.e. t(0,T), suppus(t) is a null set.

3.4 Uniqueness

In connection with equality (3.11), observe that if u0s0 and the waiting time t0 is equal to 0, then the map tu(,t) is not continuous at t=0 in the strong topology of () (otherwise we would have limt0+us(,t)()=0=u0s(), a contradiction). Instead, continuity along the lines x=x0+Cφt may occur if the waiting time t0 is positive.

Proposition 3.20.

Let (H1) be satisfied. Let uφ(u)-Cφu be bounded in (0,), and let u0 satisfy

u0s=l=1Nclδxl,with cl[0,)l=1,,N for some N.(3.26)

  • (i)

    If condition (C1) holds, then every entropy solution u of problem ( (P) ) given by Theorem 3.7 (ii) satisfies

    esslimt0+𝒯-Cφt(u(,t))-u0()=0.(3.27)

  • (ii)

    All entropy solutions u of problem ( (P) ) satisfy 𝒯-Cφt(u(,t))C((0,T];()).

Let us mention that the above statement (ii) holds for any u0+() if φ satisfies (H1) and (H4) (see Proposition 6.2).

The following uniqueness result will be proven in Section 7.

Theorem 3.21.

Let (H1) be satisfied and let uφ(u)-Cφu be bounded and monotonic in (0,). Let u0 satisfy (3.26). Then there exists at most one entropy solution u of problem (P) which satisfies either (3.24) or (3.25), and the condition

esslimt0+ur(,t)-u0rL1()=0.(3.28)

By Propositions 3.17, 3.20 and Theorem 3.21, we have the following existence and uniqueness result (observe that (H4) implies C1).

Theorem 3.22.

Let (H1) and (H4) be satisfied, and let uφ(u)-Cφu be bounded in (0,). Let u0 satisfy (3.26). Then there exists a unique entropy solution of problem (P) which satisfies (3.24)–(3.25).

Remark 3.23.

Conditions (3.24)–(3.25) in Theorem 3.22 cannot be omitted. In fact, there exist entropy solutions of problem (P) which do not satisfy either (3.24) or (3.25), depending on φ. Therefore, by Proposition 3.17, they are different from those given by Theorem 3.7, thus uniqueness fails.

For example, let u0s0 and u0rL1()L(). Let uL(0,T;+()) be defined by

u(,t):=ur(,t)+𝒯Cφt(u0s)for a.e. t(0,T),

where urC([0,T];L1())L(S) is the unique entropy solution of problem (P) with u0 replaced by u0r. Since u(,0)=ur(,0)+u0s=u0r+u0s=u0, one easily checks that (3.8)–(3.9) are satisfied, thus u is an entropy solution of (P). On the other hand, urL(S), so ur(,t)L() for a.e. t(0,T), and (3.24)–(3.25) fails.

Remark 3.24.

If uφ(u)-Cφu is unbounded and satisfies assumptions (H1) and (H4), by [19, Theorem 1.1] and Theorem 3.18, for every u0+() there exists a unique entropy solution of problem (P) with waiting time t0 equal to 0. In fact, every entropy solution u given by Theorem 3.18 is a solution according to [19]. This follows if we show that

u=urL(×(τ,T))for every τ>0(3.29)

and esslimt0u(,t)=u0 narrowly in (), i.e. esslimt0u(,t),ρ=u0,ρ for all bounded ρC(). The latter follows from (3.11) and Proposition 3.8 (ii) (see [17, Proposition 2, p. 38]).

To prove (3.29) we fix τ>0. By (1.5) we may assume that ur(,τ)L() and u(,t)=ur(,t) for all tτ. By standard approximation arguments, we may substitute in the entropy inequality (3.9) E(u)=[s-kτ]+, with kτ=ur(,τ)L(), and ζ(x,t)χ[τ,t](t). Hence, [ur(,t)-kτ]+𝑑x0 for a.e. tτ and (3.29) follows.

4 Approximating problems

In this section we consider problem (Pn). Let u0nL1()L() satisfy (3.14) and let {u0nε}Cc(), u0nε0 be any sequence such that

u0nεL1()u0nL1()u0(),u0nεL()u0nL(),(4.1)u0nεu0nin L1(),u0nε*u0nin L().(4.2)

Let ηCc() be a standard mollifier, let ηε(u):=1εη(uε) for ε>0, and set

φε(u):=(ηε*φ¯)(u)-(ηε*φ¯)(0)=ηε(u-v)φ¯(v)𝑑v-ηε(-v)φ¯(v)𝑑v,u

(here φ¯(u)=φ(u) for u0 and φ¯(u)=0 for u<0). The regularized problem associated with (Pn) is

{tunε+x[φε(unε)]=εx2unεin S,unε=u0nεin ×{0}(4.3)

(where ε>0, n), has a unique strong solution unεC([0,T];H2())L(S), tunεL2(S) (e.g., see [20]). Some properties of the family {unε} are collected in the following lemmata. Up to minor changes, the proof is standard (e.g., see [9]), thus is omitted.

Lemma 4.1.

Let unε be the solution of problem (4.3). Then, for every nN and ε>0,

unε0in S,unεL(S)u0nL(),(4.4)unε(x,t)dx=u0nε(x)dx(t(0,T)),supt(0,T)unε(,t)L1()u0nL1()u0(),(4.5)supt(0,T)unε(+h,t)-unε(,t)L1()u0nε(+h)-u0nεL1()for any h.(4.6)

Lemma 4.2.

Let φ satisfy (3.1). Then there exists C>0, which only depends on u0M(R), such that for all nN, ε(0,1) and p(0,1),

εS(1+unε)p-2(xunε)2𝑑x𝑑tCp(1-p).(4.7)

Proof.

Let UC2([0,)), U0 in (0,), and set

ΘU,ε(u):=0uU(s)φε(s)ds+θU(θU).(4.8)

Multiplying the first equation in (4.3) by U(unε) gives

t[U(unε)]+x[ΘU,ε(unε)]=εx2[U(unε)]-εU′′(unε)(xunε)2in S.(4.9)

Hence, for all ζC1([0,T];Cc2()),

εSU′′(unε)(xunε)2ζ𝑑x𝑑t+U(unε(x,T))ζ(x,T)𝑑x=U(u0nε)ζ(x,0)𝑑x+S{U(unε)tζ+ΘU,ε(unε)xζ+εU(unε)x2ζ}𝑑x𝑑t.(4.10)

By (3.1) and the definition of the function φε, for all u0,

|ΘU,ε(u)|0uU(s)|φε(s)|𝑑s+|θU|M[U(u)-U(0)]+|θU|.(4.11)

Choose θU=0, U(u)=(1+u)p-1, with p(0,1), and

ζ=ρk:=χ{|x|k}+ρ(-k)χ{kx<k+1}+ρ(+k)χ{-(k+1)<x-k}(k),

with any ρCc2((-1,1)) such that ρ(0)=1, 0ρ1, and the derivatives ρ, ρ′′ vanish at 0. Then 0U(u)u for u0 and, by (4.5), (4.10) and (4.11),

εp(1-p)S(1+unε)p-2(xunε)2ρk𝑑x𝑑tu0nε(x)𝑑x+S{Munε|ρk|+εunε|ρk′′|}𝑑x𝑑t{1+(M+1)TρC2([-1,1])}u0()=:C

for all ε(0,1) and k. Passing to the limit as k, we obtain (4.7). ∎

Lemma 4.3.

Let φ satisfy (3.1) and let UC2([0,)) be such that

|U′′(u)|K(1+u)p-2for all u[0,), for some K0 and p(0,1).(4.12)

Then there exists Cp>0 such that for all nN and ε>0,

εS|U′′(unε)|(xunε)2dxdtCp.(4.13)

If, moreover, UL(0,), then the family {Un,ρε}, where

Un,ρε(t):=U(unε)(x,t)ρ(x)dx(t(0,T))(4.14)

and ρCc2(R), is bounded in BV(0,T).

Proof.

Inequality (4.13) follows immediately from (4.7) and (4.12). To prove that {Un,ρε} is bounded in BV(0,T), observe that, by (4.9),

(Un,ρε)(t)=[ΘU,ε(unε)ρ+εU(unε)ρ′′-εU′′(unε)(xunε)2ρ](x,t)𝑑x.(4.15)

Since UL(0,), there exists N>0 such that |U(u)|N(1+u) for u0. Hence, |U(unε)|N(1+unε) and, by (4.8), (3.1) and the definition of φε, we have

|ΘU,ε(unε)|φεUL((0,))|unε|+|θU|=:M~unε+|θU|.

Then it follows from (4.15) that

|(Un,ρε)|(t)ρC2()suppρ{(M~+εN)unε(x,t)+εN+|θU|}𝑑x+ερL()[|U′′(unε)|(xunε)2](x,t)𝑑x,

and, by (4.5) and (4.13), there exists a constant Cp,ρ>0 such that

0T|(Un,ρε)|(t)dtρC2(){(M~+N)Tu0()+Cp,ρ}.(4.16)

On the other hand, by (4.5) and since |U(unε)|N(1+unε), we have

0T|Un,ρε|(t)dtNTρL()(u0()+|suppρ|),(4.17)

whence the result follows. ∎

From the above lemmata, we get the following convergence results.

Lemma 4.4.

  • (i)

    If φC([0,)) , there exist a subsequence {unεm}{unε} and unL(S)L(0,T;L1()) such that, as εm0,

    unεm*unin L(S),unεmun  𝑎𝑛𝑑  φεm(unεm)φ(un)a.e. in S,(4.18)unεmunin L1((-L,L)×(0,T)),for all L>0.(4.19)

    Moreover, un0 a.e. in S, unL(S)u0nL() and

    supt(0,T)un(,t)L1()u0nL1()u0().(4.20)

  • (ii)

    Let φ satisfy ( 3.1 ), let ρCc2() , and let UC2([0,)) , with UL(0,) , satisfy ( 4.12 ). Let Un,ρεm be defined by ( 4.14 ) and set

    Un,ρ(t):=U(un)(x,t)ρ(x)dx(t(0,T)).(4.21)

    Then

    Un,ρεmUn,ρin L1(0,T) and a.e. in (0,T).(4.22)

Proof.

By (4.4), unεm*un in L(S), where unL(S), unL(S)u0nL() and un0 a.e. in S. The a.e.-convergence of unεm and part (ii) follow from (4.19), and since φε converges uniformly to the continuous function φ on compact subsets of , we also obtain the a.e.-convergence of φεm(unεm).

It remains to prove (4.19) and (4.20). We claim that for a.e. t(0,T),

unεm(,t)*un(,t)in L() as εm0.(4.23)

Set In,ρεm(t):=unεm(x,t)ρ(x)𝑑x for t(0,T)) and let ρCc2(). By Lemma 4.3, with U(u)=u, the sequence {In,ρεm} is bounded in BV(0,T) and has a subsequence (not relabeled) {In,ρεm} such that

In,ρεmIn,ρin L1(0,T) as εm0(4.24)

for some In,ρBV(0,T). Since unεm*un in L(S),

limm0TIn,ρεm(t)dt=Sun(x,t)ρ(x)dxdt=0T(un(x,t)ρ(x)dx)dt,

whence In,ρ=un(x,t)ρ(x)𝑑x for a.e. t(0,T)), and the convergence in (4.24) is satisfied along the whole sequence {In,ρεm}. Hence, for all ρCc2(), there exists a null set N(0,T) such that

limεm0unεm(x,t)ρ(x)𝑑x=un(x,t)ρ(x)𝑑xfor all t(0,T)N.

Since Cc2() is dense in L1() and L1() is separable, the choice of the set N can be made independent of ρ. Hence, we have proven (4.23).

By (4.2), (4.5), (4.6), and the Fréchet–Kolmogorov theorem, {unεm(,t)} is relatively compact in L1((-L,L)) for all t(0,T) and L>0. Hence, by (4.23),

unεm(,t)un(,t)in L1((-L,L)) as εm0,for L>0 and a.e. t(0,T),(4.25)

and (4.20) follows from (4.5). Finally, (4.19) follows from (4.5), (4.25) and the dominated convergence theorem. ∎

Proposition 4.5.

Let φC([0,)). For all nN, problem (Pn) has an entropy solution un, which is unique if φ is locally Lipschitz continuous. For a.e. t(0,T), we have

un(+h,t)-un(,t)L1()u0n(+h)-u0nL1()for any h,(4.26)un(x,t)𝑑x=u0n(x)𝑑x.(4.27)

Moreover, given ρCc2(R) and UC2([0,)), with UL(0,), satisfying (4.12), the sequence {Un,ρ} defined by (4.21) is bounded in BV(0,T).

Proof.

Let ζ and E be as in Definition 3.6, and Fε=Eφε. Then

S{E(unε)(tζ+εx2ζ)+Fε(unε)xζ}𝑑x𝑑t+E(u0nε)ζ(x,0)𝑑x0,(4.28)

where unεm is defined by Lemma 4.4. By (4.4), it is not restrictive to assume that E(u)=|u-k| and Fε(u)=sgn(u-k)[φε(u)-φε(k)] (k[0,)). By (4.4),

φεm(unεm)L(S)sup|v|u0nL()|φεm(v)|sup|v|u0nL()+1|φ(v)|.

Since φεm(unεm)φ(un) a.e. in S (see (4.18)), it follows from (4.19) and the dominated convergence theorem that

SFεm(unεm)xζdxdtSF(un)xζdxdtas εm0.

The remaining terms in (4.28) (with ε=εm) are dealt with similarly. Letting εm0, we obtain (3.16), so un is an entropy solution of problem (Pn). Its uniqueness follows from Kružkov’s theorem [26].

Inequality (4.26) follows from (4.6) and (4.25). Concerning (4.27), it follows from (3.17) that for all ρCc1() and a.e. t(0,T),

un(x,t)ρ(x)𝑑x-u0n(x)ρ(x)𝑑x=0tφ(un)(x,s)ρ(x)𝑑x𝑑s.(4.29)

Let {ρk}Cc1() be such that ρk(x)=1 for x[-k,k], ρk(x)=0 if |x|k+1, and ρkL()2. Setting ρ=ρk in (4.29) and letting k, we get

|0tφ(un)(x,s)ρk(x)dxds|2M0t{xk|x|k+1}|un(x,s)|dxds0,

since unL1(S). On the other hand, by the monotone convergence theorem,

un(x,t)ρk(x)𝑑xun(x,t)𝑑x,u0n(x)ρk(x)𝑑xu0n(x)𝑑x,

and (4.27) follows from (4.29).

Finally, let us show that {Un,ρ} is bounded in BV(0,T). By (4.17) and (4.22),

0T|Un,ρ(t)|dt=limεm00T|Un,ρεm(t)|dtNρL()(Tu0()+|suppρ|),

and, by (4.16) and the lower semicontinuity of the total variation in L1(0,T) ([15, Theorem 1, Section 5.2.1]), we get

Un,ρ(0,T)ρC2(){(M~+N)Tu0()+Cp,ρ},

with Cp,ρ>0 as in (4.16). This completes the proof. ∎

5 Existence and monotonicity: Proofs

We proceed with the proof of Theorem 3.7.

Proposition 5.1.

Let (H1) hold and let un be the entropy solution of problem (Pn). Then there exist a sequence {unj} and uL(0,T;M+(R)) such that

unj*uin (S).(5.1)

For all L>0, there exists a decreasing sequence {Ej}(-L,L)×(0,T) of Lebesgue measurable sets, with |Ej|0 as j, such that

unjχ((-L,L)×(0,T))Ejub:=[0,)ξ𝑑τ(ξ)in L1((-L,L)×(0,T)),(5.2)

where τY(S;R) is the Young measure associated with {unj}, and

unjχEj*μ:=u-ubin ((-L,L)×(0,T)).(5.3)

Proof.

By (4.20), there exist u+(S) and a sequence {unj} such that unj*u in (S). Arguing as in [27, Proposition 4.2], we obtain that uL(0,T;+()).

Since by (4.20) the sequence {unj} is bounded in L1(S), by Theorem 2.3 there exist a subsequence of {unj} (not relabeled) and a Young measure τ𝒴(S;) such that

  • (i)

    for every measurable set AS, (2.4)–(2.6) are valid for any fC() such that the sequence {f(unj)} is sequentially weakly relatively compact in L1(A),

  • (ii)

    suppτ(x,t)[0,) for a.e. (x,t)S (here τ(x,t) is the disintegration of τ).

Then the result follows by Theorem 2.5 and a standard diagonal procedure. ∎

Remark 5.2.

The function ub in (5.2) is defined for a.e. in (x,t)S, since τ is globally defined in S. In addition, by (4.20) and the arbitrariness of L in Proposition 5.1, a routine proof shows that ubL(0,T;L1()) and ub0 a.e. in S. Therefore, the Radon measure μ0 (see (5.3)) is defined on S, μL(0,T;+()), and

μ=u-ubu=ub+μin (S).(5.4)

Proposition 5.3.

Let (H1) hold, let μ be as in (5.4) and let UC([0,)). If

limuU(u)u=:CU[0,),(5.5)

then, for all L>0,

U(unj)*U*+CUμin ((-L,L)×(0,T)),(5.6)

where U*L(0,T;Lloc1(R)) is defined by

U*(x,t):=[0,)U(ξ)𝑑τ(x,t)(ξ)for a.e. (x,t)S.

Remark 5.4.

If UC([0,)) satisfies (5.5), there exists N>0 such that

|U(u)|N(1+u)for u0.(5.7)

Moreover, U*L(0,T;L1()) if |U(u)|Nu, since ubL(0,T;L1()) and

|U*(x,t)|[0,)|U(ξ)|dτ(x,t)(ξ)N[0,)ξdτ(x,t)(ξ)=Nub(x,t)for a.e. (x,t)S.

Proof of Proposition 5.3.

For all ε>0, there exist mε>0 such that

-εu<U(u)-CUu<εuif u>mε.(5.8)

For any m, m>mε, let l1m,l2mC([0,)) be such that 0l1m1, 0l2m1, l1m+l2m=1 in [0,), suppl1m[0,m+1] and suppl2m[m,). Then, by (5.8),

|U(unj)-[U(unj)l1m(unj)+CUunjl2m(unj)]|<εunjl2m(unj)for j.(5.9)

Since supS[|U(unj)|l1m(unj)]supu[0,m+1]|U(u)|<, it follows that {U(unj)l1m(unj)} is uniformly integrable in (-L,L)×(0,T). Hence, by Theorem 2.3, for all L>0,

U(unj)l1m(unj)U1m*:=[0,)U(ξ)l1m(ξ)𝑑τ(ξ)(5.10)

in L1((-L,L)×(0,T)). Here U1m* belongs to L(0,T;Lloc1()), since, by (5.7),

|U1m*|[0,)|U(ξ)|l1m(ξ)dτ(ξ)N[0,)(1+ξ)dτ(ξ)N(1+ub).(5.11)

Similarly, by (5.1), (5.2), (5.4) and (5.10), with U(u)=u,

unjl2m(unj)=unj-unjl1m(unj)*u-[0,)ξl1m(ξ)𝑑τ(ξ)=ub-[0,)ξl1m(ξ)𝑑τ(ξ)+μ=[0,)ξ[1-l1m(ξ)]𝑑τ(ξ)+μ=[0,)ξl2m(ξ)𝑑τ(ξ)+μ=:l2m*+μin ((-L,L)×(0,T)).(5.12)

From (5.9)–(5.12), for any ζCc((-L,L)×(0,T)), ζ0, and m as above, we get

(-L,L)×(0,T)[U1m*+(CU-ε)l2m*]ζ𝑑x𝑑t+(CU-ε)μ,ζ(-L,L)×(0,T)lim infnj(-L,L)×(0,T)U(unj)ζ𝑑x𝑑tlim supnj(-L,L)×(0,T)U(unj)ζ𝑑x𝑑t(-L,L)×(0,T)[U1m*+(CU+ε)l2m*]ζ𝑑x𝑑t+(CU+ε)μ,ζ(-L,L)×(0,T).(5.13)

Since U1m*L(0,T;Lloc1()),

0l2m*[m,)ξ𝑑τ(ξ)ubL(0,T;L1())

and

limεm0l2m*(x,t)=0,limεm0U1m*(x,t)=U*(x,t)for a.e. (x,t)S,

by letting m in (5.13), we get plainly

(-L,L)×(0,T)U*ζ𝑑x𝑑t+(CU-ε)μ,ζ(-L,L)×(0,T)lim infnj(-L,L)×(0,T)U(unj)ζ𝑑x𝑑tlim supnj(-L,L)×(0,T)U(unj)ζ𝑑x𝑑t(-L,L)×(0,T)U*ζ𝑑x𝑑t+(CU+ε)μ,ζ(-L,L)×(0,T),

whence

0lim supnj(-L,L)×(0,T)U(unj)ζ𝑑x𝑑t-lim infnj(-L,L)×(0,T)U(unj)ζ𝑑x𝑑t2εμ,ζ(-L,L)×(0,T).

From the above inequalities, the conclusion follows. ∎

Proposition 5.5.

Let (H1) hold. Let μ, U and U* be as in Proposition 5.3. Then

0T|U(unj)(x,t)ρ(x)dx-U*(x,t)ρ(x)dx-CUμ(,t),ρ|dt0(5.14)

as j for ρCc(R). Moreover, for all L>0, there exist a null set N(0,T) and a subsequence of {unj} (not relabeled), such that for all t(0,T)N,

U(unj)(,t)*U*(,t)+CUμ(,t)in ((-L,L)).(5.15)

Remark 5.6.

Choosing U(u)=u in (5.15), we obtain that

unj(,t)*u(,t)in ((-L,L)) for a.e. t(0,T) and L>0.(5.16)

If UC([0,)) satisfies (5.5), U*L(0,T;Lloc1()) and {U(un)} is bounded in L(0,T;Lloc1()) (see (4.20) and (5.7)). Since every ζC(2)L(2) can be uniformly approximated in bounded sets by finite sums i=1pfi,p(x)gi,p(t), with fi,p, gi,p bounded and continuous functions (1ip; e.g., see [12, Théorème D.1.1]), it follows from (5.14) that, as j, for all ζC([0,T];Cc()),

0T|[U(unj)ζ](x,t)dx-[U*ζ](x,t)dx-CUμ(,t),ζ(,t)|dt0.(5.17)

Proof of Proposition 5.5.

(i) Let us first prove (5.14) for UC2([0,)), with UL(0,), satisfying (4.12) and (5.5). Let ρCc(), hCc(0,T), and fix any L>0 such that suppρ(-L,L). Then, by (5.6),

0TUnj,ρ(t)h(t)𝑑t0TUρ*(t)h(t)𝑑t+CU0Th(t)μ(,t),ρ𝑑t,(5.18)

where Unj,ρ is defined by (4.21) and Uρ*(t):=U*(x,t)ρ(x)𝑑x. Since, by Proposition 4.5, {Unj,ρ} is bounded in BV(0,T) if ρCc2(), there exists a subsequence which converges in L1(0,T). Combined with (5.18), this yields that Unj,ρUρ*+CUμ(,),ρ in 𝒟(0,T) and in L1(0,T) for all ρCc2(). Since the sequence {U(unj)} is bounded in L(0,T;L1((-L,L))) and U*L(0,T;L1((-L,L))), the condition ρCc2() may be relaxed to ρCc(), and we have found (5.14).

(ii) Next we prove (5.14) for all UC([0,))L((0,)) (in this case CU=0). To this end, let Uk(u):=(Uχ[0,k]*θk)(u) for any u0, where θk0 is a sequence of standard mollifiers (k). Then {Uk}Cc2([0,)), UkU uniformly on compact subsets of [0,) and UkL()UL(). By part (i) and (4.20), for all ρCc() and k, M>0,

lim supj0T𝑑t|U(unj)ρ(x)𝑑x-U*(x,t)ρ(x)𝑑x|lim supj{0unjM}|U(unj)-Uk(unj)||ρ|dxdt+lim supj{unj>M}|U(unj)-Uk(unj)||ρ|dxdt   +S|U*-Uk*||ρ|dxdtρ|suppρ|TU-UkL(0,M)+ρ{2TMu0()UL()   +suppρ×(0,T)dxdt[0,)|Uk(ξ)-U(ξ)|dτ(x,t)(ξ)}2ρ|suppρ|TU-UkL(0,M)+2ρUL(){Tu0()M   +suppρ×(0,T)dxdt{ξ>M}dτ(x,t)(ξ)},

where we have used Chebychev’s inequality and the inequality

{0ξM}|Uk(ξ)-U(ξ)|dτ(x,t)(ξ)+{ξ>M}|Uk(ξ)-U(ξ)|dτ(x,t)(ξ)Uk-UL(0,M)+2UL(){ξ>M}dτ(x,t)(ξ).

Letting k, since UkU uniformly on compact sets in [0,), we obtain

lim supj0T𝑑t|U(unj)ρ(x)𝑑x-U*(x,t)ρ(x)𝑑x|2ρC(¯)UL(){Tu0()M+suppρ×(0,T)𝑑x𝑑t{ξ>M}𝑑τ(x,t)(ξ)}.(5.19)

Since τ(x,t) is a probability measure, we have {ξ>M}𝑑τ(x,t)(ξ)0 as M for a.e. (x,t)S, thus, by the dominated convergence theorem,

suppρ×(0,T)𝑑x𝑑t{ξ>M}𝑑τ(x,t)(ξ)0as M.

Then, letting M in (5.19), we obtain (5.14).

(iii) Now let UC([0,)) be any function satisfying (5.5). Arguing as in the proof of Proposition 5.3, let l1m,l2mC2([0,)) (m) satisfy l1m,l2m0 and l1m+l2m=1 in [0,), suppl1m[0,m+1], and suppl2m[m,). Then

U(unj)=U(unj)l1m(unj)+U(unj)l2,m(unj)(5.20)

and, by (5.8), for all ε>0 and m>mε,

(CU-ε)unjl2m(unj)U(unj)l2m(unj)(CU+ε)unjl2m(unj).(5.21)

Since Ul1mL()UC([0,m+1])<, the function Ul1m belongs to C([0,))L(). Then, by part (ii),

0T|[U(unj)l1m(unj)](x,t)ρ(x)dx-U*1m(x,t)ρ(x)dx|dt0(5.22)

as j, where ρCc() and U1m* is defined by (5.10). By (5.21) and (4.20),

0T|[U(unj)l2m(unj)-CUunjl2m(unj)](x,t)ρ(x)dx|dtεS|unj||ρ(x)|dxεTu0()ρ.

Then we obtain that

0T|[U(unj)l2m(unj)-CUl2m*](x,t)ρ(x)dx-CUμ(,t),ρ|dtεTu0()ρ+CU0T|[unjl2m(unj)-l2m*](x,t)ρ(x)dx-μ(,t),ρ|dt,

with l2m* defined as in (5.12). The map uul2m(u) belongs to C2([0,)), has bounded derivative and satisfies (4.12) and (5.5), with CU=1. Then, by part (i), (5.20) and (5.22),

lim supj0T|[U(unj)-U1m*-CUl2m*](x,t)ρ(x)dx-CUμ(,t),ρ|dtεTu0()ρif m>mε.(5.23)

To complete the proof of (5.14), we show that

limmS|U*-U1m*-CUl2m*|(x,t)|ρ(x)|dxdt=0.(5.24)

By (5.21),

|U*-U1m*-CUl2m*|(x,t)[0,)|U(ξ)-U(ξ)l1m(ξ)-CUξl2m(ξ)|dτ(x,t)(ξ)=[0,)|U(ξ)l2m(ξ)-CUξl2m(ξ)|dτ(x,t)(ξ)ε[m,)ξ𝑑τ(x,t)(ξ)εub(x,t)

for a.e. (x,t)S. Since ubL(0,T;L1()) and [m,)ξ𝑑τ(x,t)(ξ)0 as m for a.e. (x,t)S, (5.24) follows from the dominated convergence theorem.

Letting m in (5.23), it follows from (5.24) that

lim supj0T|[U(unj)-U*](x,t)ρ(x)dx-CUμ(,t),ρ|dtlim supm(lim supj0T|[U(unj)-U1m*-CUl2m*]ρdx-CUμ(,t),ρ|dt)εTu0()ρ,

and (5.14) follows from the arbitrariness of ε.

Finally, (5.15) follows from (5.14), the separability of Cc() and a diagonal argument; we leave the details to the reader. ∎

Proposition 5.7.

Let (H1) hold. Then (5.4) is the Lebesgue decomposition of u, i.e.,

ub=ura.e. in S,μ=usin (S).(5.25)

Proof.

Let U be a convex function with U(0)=0 and UL(0,). By (3.16),

U(unj)(x,t¯)ζ(x,t¯)𝑑x-U(u0nj)(x)ζ(x,0)𝑑x×(0,t¯){U(unj)tζ+ΘU(unj)xζ}𝑑x𝑑t(5.26)

for all ζC1([0,T];Cc1()) and a.e. t¯(0,t), where

ΘU(u):=0uU(s)φ(s)ds+θU(θU).(5.27)

Let Um(u)=(u-m)χ[m,)(u) and θUm=0 (m). Since Um(u)/uCUm=1 and ΘUm(u)/uCφ as u (with Cφ as in H1), it follows from (5.17) that

0t¯|[Um(unj)tζ](x,t)dx-[Um*tζ](x,t)dx-μ(,t),tζ(,t)|dt0

and

0t¯|[ΘUm(unj)xζ](x,t)dx-[ΘUm*xζ](x,t)dx-Cφμ(,t),xζ(,t)|dt0

as j, where

Um*(x,t):=[0,)Um(ξ)𝑑τ(x,t)(ξ),ΘUm*(x,t):=[0,)ΘUm(ξ)𝑑τ(x,t)(ξ)

belong to L(0,T;Lloc1()). In particular, setting νζ:=tζ+Cφxζ, we have that

×(0,t¯){Um(unj)tζ+ΘUm(unj)xζ}𝑑x𝑑t×(0,t¯){Um*tζ+ΘUm*xζ}𝑑x𝑑t+0t¯μ(,t),νζ(,t)𝑑t.(5.28)

By (5.15) and a diagonal argument, there exist a null set N(0,T) and a subsequence, denoted again by {unj}, such that for all t¯(0,T)N and m,

limnjUm(unj)(x,t¯)ζ(x,t¯)𝑑x=Um*(x,t¯)ζ(x,t¯)𝑑x+μ(,t¯),ζ(x,t¯).(5.29)

Since {Um(u0nj)-u0nj} is bounded in L() and converges a.e. to Um(u0r)-u0r, it follows from (3.15) that

limnjUm(u0nj)(x)ζ(x,0)𝑑x=Um(u0r)(x)ζ(x,0)𝑑x+u0s,ζ(,0).(5.30)

Setting U=Um in (5.26) and letting j, we obtain from (5.28)–(5.30) that

Um*(x,t¯)ζ(x,t¯)𝑑x+μ(,t¯),ζ(,t¯)×(0,t¯){Um*tζ+ΘUm*xζ}𝑑x𝑑t+0t¯μ(,t),νζ(,t)𝑑t+Um(u0r)(x)ζ(x,0)𝑑x+u0s,ζ(,0)(5.31)

for all t¯(0,T)N and m. Since for all u0 (see (3.1)),

0Um(u)uχ[m,)(u),|ΘUm(u)|=|φ(u)-φ(m)|χ[m,u)(u)Muχ[m,)(u),

we have that |Um*|ub, |ΘUm*|Mub, Um*0 and ΘUm*(x,t)0 (as m) a.e. in S. Thus, by the dominated convergence theorem and (5.31), for all t¯(0,T)N,

μ(,t¯),ζ(,t¯)0t¯μ(,t),νζ(,t)𝑑t+u0s,ζ(,0).(5.32)

Let ρCc1() and ζ(x,t)=ρ(x-Cφt), so ζν0. By (5.32), μ(,t¯),ρ(-Cφt¯)u0s,ρ. Hence, μ(,t¯) is singular with respect to the Lebesgue measure and, since μ(,t¯)=[μ(,t¯)]s=μs(,t¯) for a.e. t¯(0,T) (see (2.3)), (5.25) follows from the uniqueness of the Lebesgue decomposition. ∎

The following result is based on the concept of compensated compactness (e.g., see [13]).

Proposition 5.8.

Let (H1) hold. Then φ(ur)=[0,)φ(ξ)𝑑τ(ξ) a.e. in S.

Proof.

Let U,VC2([0,))L((0,)) satisfy (4.12), and assume that ΘU,ΘV, defined by (5.27), belong to L((0,)). By (4.13), we have

εU′′(unε)(xunε)2L1(S)CpandεV′′(unε)(xunε)2L1(S)Cp

for all ε(0,1) and n, and up to a subsequence,

εU′′(unε)(xunε)2*λn,εV′′(unε)(xunε)2*μnin (S) as ε0,(5.33)

for some λn,μn(S). By the lower semicontinuity of the norm,

λn(S)Cp,μn(S)Cpfor n.(5.34)

Let ζCc2(S). Then (see (4.9))

εSU′′(unε)(xunε)2ζ𝑑x𝑑t=S{U(unε)tζ+ΘU,ε(unε)xζ+εU(unε)x2ζ}𝑑x𝑑t,(5.35)

where ΘU,ε(u)=0uU(s)φε(s)𝑑s+θU, θU. By (3.1) and (4.4), for all n,

|ΘU,ε(unε)|0u0n|U(s)φε(s)|ds+|θU|M0u0n|U(s)|ds+|θU|γn,U

for some γn,U0, so for fixed n, the family {ΘU,ε(unε)}ε is uniformly bounded in L(S). Similar results hold for V and ΘV,ε(u)=0uV(s)φε(s)𝑑s+θV, and letting ε0 in (5.35) along some subsequence {εm} (see the proof of Proposition 4.5), it follows from by (5.33) that for all n and ζCc1(S),

S{U(un)tζ+ΘU(un)xζ}𝑑x𝑑t=λn,ζS,S{V(un)tζ+ΘV(un)xζ}𝑑x𝑑t=μn,ζS,(5.36)

where un is the entropy solution of the approximating problem (Pn) (see (4.18)).

Let AS be a bounded open set and let Yn,Zn:A2 be defined by

Yn:=(ΘU(un),U(un)),Zn:=(V(un),-ΘV(un)).

By (5.36),

divYn=-λn,curlZn=-μnin 𝒟(A).(5.37)

Since U,ΘU,V,ΘV are bounded in (0,), the sequences U(un), ΘU(un), V(un) and ΘV(un) are bounded in L1(A) and uniformly integrable, and, by Theorem 2.3,

U(un)U*:=[0,)U(ξ)𝑑τ(,)(ξ),ΘU(un)ΘU*:=[0,)ΘU(ξ)𝑑τ(,)(ξ),V(un)V*:=[0,)V(ξ)𝑑τ(,)(ξ),ΘV(un)ΘV*:=[0,)ΘV(ξ)𝑑τ(,)(ξ)

in L1(A), where τ(,) denotes the disintegration of the Young measure τ associated with {un}. Since the sequences U(un), ΘU(un), V(un) and ΘV(un) are bounded in L(A)L2(A), they also converge weakly in L2(A), so

YnY*:=(ΘU*,U*),ZnZ*:=(V*,-ΘV*)in [L2(A)]2.

By a similar argument,

YnZn:=ΘU(un)V(un)-ΘV(un)U(un)[0,)[ΘU(ξ)V(ξ)-ΘV(ξ)U(ξ)]𝑑τ(,)(ξ)in L2(A).(5.38)

By (5.34) and (5.37), {divYn} and {curlZn} are precompact in W-1,2(A) (see [13, Chapter 1, Corollary 1]) and, by the div-curl lemma,

YnZnY*Z*=ΘU*V*-ΘV*U*in 𝒟(A).(5.39)

By (5.38) and (5.39),

[0,)[ΘU(ξ)-ΘU*]V(ξ)𝑑τ(ξ)=[0,)[U(ξ)-U*]ΘV(ξ)𝑑τ(ξ)a.e. in A.(5.40)

For every U as above with U>0 in (0,), by a standard approximation argument, we may choose V(u)=|U*-U(u)|, so ΘV(u)=sgn(U(u)-U*)[ΘU(u)-ΘU(U-1(U*))] and, by (5.40),

[ΘU*-ΘU(U-1(U*))][0,)|U*-U(ξ)|dτ(ξ)=0.(5.41)

Let UkC2([0,))L((0,)) satisfy (4.12) and

Uk(0)=0,0<UkUk+11in [0,),Uk(u)1for u0 as k.(5.42)

By (3.1),

|ΘUk(u)|0uUk(s)|φ(s)|𝑑s+|θUk|MUk(u)+|θUk|,

thus ΘUk is bounded in (0,)) for every k. We claim that, as k,

Uk*:=[0,)Uk(ξ)𝑑τ(ξ)ura.e. in A,(5.43)ΘUk*-ΘUk(Uk-1(Uk*))[0,)φ(ξ)𝑑τ(ξ)-φ(ur)a.e. in A,(5.44)

where ΘUk*:=[0,)ΘUk(ξ)𝑑τ(ξ) (recall that φL1([0,);dτ(x,t)), see Remark 3.2). By (5.43) and the dominated convergence theorem, for a.e. (x,t)A,

[0,)|Uk*(x,t)-Uk(ξ)|dτ(x,t)(ξ)[0,)|ur(x,t)-ξ|dτ(x,t)(ξ)as k,

since 0Uk(ξ)ξ for all k and I(ξ):=ξ belongs to L1([0,),dτ(x,t)) (recall that, by (5.25) and the definition of ub in (5.2), ur(x,t)=[0,)ξ𝑑τ(x,t)(ξ)< for a.e. (x,t)S). Letting k in (5.41), with U=Uk, we obtain that for a.e. (x,t)A,

[[0,)φ(ξ)dτ(x,t)(ξ)-φ(ur)(x,t)][0,)|ur(x,t)-ξ|dτ(x,t)(ξ)=0,

and Proposition 5.8 follows from the arbitrariness of A.

It remains to prove (5.43) and (5.44). By (5.42) and the monotone convergence theorem, Uk(ξ)ξ for any ξ[0,), and (5.43) follows (recall that I(ξ)=ξL1([0,),dτ)). Concerning (5.44), we observe that

ΘUk*-ΘUk(Uk-1(Uk*))=[0,)(0ξUk(s)φ(s)ds)dτ(ξ)-0Uk-1(Uk*)Uk(s)φ(s)ds.(5.45)

Since Uk(ξ)1 and |Uk(ξ)φ(ξ)|M for ξ0 (see (5.42) and (3.1)), it follows from the dominated convergence theorem that

[0,)(0ξUk(s)φ(s)ds)dτ(x,t)(ξ)[0,)φ(ξ)dτ(x,t)(ξ).(5.46)

On the other hand,

0Uk-1(Uk*(x,t))Uk(s)φ(s)𝑑s-φ(ur)(x,t)=0ur(x,t)[Uk(s)-1]φ(s)𝑑s+ur(x,t)Uk-1(Uk*(x,t))Uk(s)φ(s)𝑑s.(5.47)

Arguing as before, one can show that the first term in the right-hand side of (5.47) vanishes as k. As for the second term, we observe, by (5.42) and (5.43), that

|ur(x,t)Uk-1(Uk*(x,t))Uk(s)φ(s)𝑑s|M|ur(x,t)-Uk-1(Uk*(x,t))|M(|ur(x,t)-Uk-1(ur(x,t))|+supsIδ(ur(x,t))1U1(s)|ur(x,t)-Uk*(x,t)|)

for some δ>0 and k sufficiently large, where Iδ(q)(q-δ,q+δ). Hence,

0Uk-1(Uk*(x,t))Uk(s)φ(s)𝑑sφ(ur)(x,t)for a.e. (x,t)A,(5.48)

and we obtain (5.44) from (5.45), (5.46) and (5.48). ∎

To prove the second part of Theorem 3.7, we need the following result which characterizes the disintegration of the Young measure τ.

Proposition 5.9.

Let (H1) hold and φC1([0,)) satisfy for all u¯>0 either (C1) or the following:

  • (C3)

    there exist a>0, b(0,] such that φ is constant in Ia,b=[u¯-a,u¯+b] and, if b< , then φ is strictly monotone in [u¯+b,u¯+b~] and [u¯-a~,u¯-a] for some b~>b and a~(a,u¯).

Then, for a.e. (x,t)S, the following hold:

  • (i)

    If ur(x,t)=0 , then τ(x,t)=δ0.

  • (ii)

    If φ is strictly monotone in Ia,b=[ur(x,t)-a,ur(x,t)+b] , with a,b0, a+b>0 , then

    τ(x,t)=δur(x,t).(5.49)

  • (iii)

    If φ is constant in the above interval Ia,b for some a>0, b>0 , then

    suppτ(x,t)I(x,t)for a.e. (x,t)S,(5.50)

    where I(x,t)Ia,b is the maximal interval where φ()φ(ur(x,t)).

Proof.

Let (x,t)S be fixed. If ur(x,t)=0, it follows from (5.25) and the definition of ub in (5.2) that [0,)ξ𝑑τ(x,t)(ξ)=0, which implies part (i): τ(x,t)=δ0.

So let ur(x,t)>0. Let l1:=ur(x,t), l2>l1 and

Vk(u):=k(u-l1)χ(l1,l1+1k)(u)+χ[l1+1k,l2)(u)+k(l2+1k-u)χ[l2,l2+1k)(u)

for u0 and sufficiently large k. Then Vk(u)χ(l1,l2](u) as k, and

ΘVk(u)=0uVk(s)φ(s)dsφ(l1)χ(l1,l2](u)+[φ(l1)-φ(l2)]χ(l2,)(u)(u0).

By standard approximation arguments, (5.40) is satisfied with U=Uk and V=Vk, where {Uk} is the sequence in the proof of Proposition 5.8 (see (5.42)), i.e.,

[0,)[ΘUk(ξ)-ΘUk*(x,t)]Vk(ξ)𝑑τ(x,t)(ξ)=[0,)[Uk(ξ)-Uk*(x,t)]ΘVk(ξ)𝑑τ(x,t)(ξ).

Letting k and arguing as in the proof of Proposition 5.8, we obtain that

Uk(ξ)-Uk*(x,t)ξ-[0,)ξ𝑑τ(x,t)(ξ)=ξ-ur(x,t)=ξ-l1,ΘUk*(x,t)-ΘUk(ξ)[0,)φ(ξ)𝑑τ(x,t)(ξ)-φ(ξ)=φ(ur)(x,t)-φ(ξ)=φ(l1)-φ(ξ)

for all ξ0 (see (5.25) and Proposition 5.8). This implies that

[0,)[ΘUk(ξ)-ΘUk*(x,t)]Vk(ξ)𝑑τ(x,t)(ξ)(l1,l2][φ(ξ)-φ(l1)]𝑑τ(x,t)(ξ),[0,)[Uk(ξ)-Uk*(x,t)]ΘVk(ξ)𝑑τ(x,t)(ξ)(l1,l2]φ(l1)(ξ-l1)𝑑τ(x,t)(ξ)+[φ(l1)-φ(l2)](l2,)(ξ-l1)𝑑τ(x,t)(ξ),

whence

(l1,l2][φ(ξ)-φ(l1)-φ(l1)(ξ-l1)]𝑑τ(x,t)(ξ)=[φ(l1)-φ(l2)](l2,)(ξ-l1)𝑑τ(x,t)(ξ).(5.51)

Similarly, let l0(0,l1) and set

V~k(u):=k(u-l0)χ[l0,l0+1k](u)+χ(l0+1k,l1-1k)(u)+k(l1-u)χ[l1-1k,l1](u).

Then V~k(u)χ(l0,l1)(u) and

ΘV~k(u)=l1uV~k(s)φ(s)dsφ(l1)χ(l0,l1)(u)+[φ(l1)-φ(l0)]χ[0,l0](u)(u0).

Letting k in (5.40), with U=Uk as above and V=V~k, we obtain that

(l0,l1)[φ(ξ)-φ(l1)-φ(l1)(ξ-l1)]𝑑τ(x,t)(ξ)=[φ(l1)-φ(l0)][0,l0](ξ-l1)𝑑τ(x,t)(ξ).(5.52)

By (C1) and (C3), we can distinguish two cases. (a) If φ is strictly convex or strictly concave in [l1,l2], it follows from (5.51) that

(l1,l2]|φ(ξ)-φ(l1)-φ(l1)(ξ-l1)|dτ(x,t)(ξ)+|φ(l1)-φ(l2)|(l2,)|l1-ξ|dτ(x,t)(ξ)=0,

where

χ(l1,l2](ξ)|φ(ξ)-φ(l1)-φ(l1)(ξ-l1)|>0and|φ(l1)-φ(l2)|>0.

This implies that suppτ(x,t)[0,l1]. Since τ(x,t) is a probability measure and l1:=ur(x,t),

ur(x,t)=[0,ur(x,t)]ξ𝑑τ(x,t)(ξ)=[0,ur(x,t)][ξ-ur(x,t)]𝑑τ(x,t)(ξ)+ur(x,t)

(see (5.2) and (5.25)), thus

[0,ur(x,t)]|ξ-ur(x,t)|dτ(x,t)(ξ)τ(x,t)([0,ur(x,t)))=0.

Hence, suppτ(x,t)={ur(x,t)} and (5.49) follows since τ(x,t) is a probability measure.

Similarly, if φ is strictly convex or strictly concave in (l0,l1), it follows from (5.52) that τ(x,t)([0,l1))=0 (we omit the details). Thus, suppτ(x,t)[l1,), and arguing as above we obtain (5.49). (b) If φ is affine in [l1-c,l1+c] for some c>0, let I=[l¯0,l¯2] be the maximal interval containing l1, where φ(ξ)=φ(l1). If I=[0,), (5.50) is satisfied. If l¯2<, by (C3) and the maximality of I, φ is strictly convex (or concave) in [l¯2,l¯2+b] for some b>0 (and affine in [l1,l¯2]). By (5.51), with l2(l¯2,l¯2+b), we obtain that

(l¯2,l2]|φ(ξ)-φ(l1)-φ(l1)(ξ-l1)|dτ(x,t)(ξ)+|φ(l1)-φ(l2)|(l2,)|l1-ξ|dτ(x,t)(ξ)=0,

where

χ(l¯2,l2](ξ)|φ(ξ)-φ(l1)-φ(l1)(ξ-l1)|>0and|φ(l1)-φ(l2)|>0.

It follows that τ(x,t)((l¯2,))=0, whence suppτ(x,t)[0,l¯2]. Similarly, if l¯0>0, by (C3) and the maximality of I, φ is strictly convex (or concave) in [l¯0-a,l¯0] for some a>0 (and affine in [l¯0,l1]). Arguing as before, we obtain from (5.52), with l0(l¯0-a,l¯0), that suppτ(x,t)[l¯0,) (we omit the details). Summing up, we obtain (5.50): suppτ(x,t)[0,l¯2][l¯0,)=I. ∎

Remark 5.10.

If (C1) is satisfied for all u¯>0, it follows from (5.49) and standard properties of narrow convergence of Young measures (see [28]) that unjur in measure, where {unj} is the subsequence in Proposition 5.1. Therefore, up to a subsequence, unjur a.e. in S. Hence, if φ is bounded, it follows from the dominated convergence theorem that φ(unj)φ(ur) in L1((-L,L)×(0,T)) for all L>0.

Now we can prove Theorem 3.7.

Proof of Theorem 3.7.

Let ζC1([0,T];Cc1()), with ζ(,T)=0 in , and let L>0 be such that suppζ(-L,L)×[0,T]. By (5.17), with U(u)=u and U(u)=φ(u),

SunjtζdxdtSurtζdxdt+0Tus(,t),tζ(,t)𝑑t,Sφ(unj)xζdxdtSφ*xζdxdt+Cφ0Tus(,t),xζ(,t)𝑑t

(see also (5.25)). Letting j in (3.17), with n=nj, we obtain (3.4). Inequality (3.6) is proven similarly, since by arguing as in Proposition 5.3, we get

E(u0nj)*E(u0r)+CEu0sin ()

(in this regard, see also (3.15)). Thus, the function uL(0,T;+()) given by Proposition 5.1 is an entropy solution of problem (P) in the sense of Young measures. By Proposition 5.8, it is also a solution in the sense of Definition 3.3. This proves the first part of the theorem. The second part is an immediate consequence of Proposition 5.9; in fact, (3.9) follows from (3.6) and (5.49). ∎

Let us end this section by proving Proposition 3.8.

Proof of Proposition 3.8.

For every ζ~C1([0,T];Cc1()), ζ~(,T)=0, we set E(u)=Um(u)=(u-m)χ{u>m}(u) and F(u)=Fm(u)=0uUm(ξ)φ(ξ)𝑑ξ=(φ(u)-φ(m))χ{u>m}(u) in the entropy inequalities (3.6) (m). Then we get

S{Um*tζ~+Fm*xζ~}𝑑x𝑑t+0Tus(,t),tζ~(,t)𝑑t+Cφ0Tus(,t),xζ~(,t)𝑑t-Um(u0r)ζ~(x,0)𝑑x-u0s,ζ~(,0),

where, for a.e. (x,t)S,

Um*(x,t):=[0,)Um(ξ)𝑑τ(x,t)(ξ),Fm*(x,t):=[0,)Fm(ξ)𝑑τ(x,t)(ξ).

As in the proof of Proposition 5.7, we have S{Um*tζ~+Fm*xζ~}𝑑x𝑑t0 and Um(u0r)ζ~(x,0)𝑑x0 as m, whence

0Tus(,t),tζ~(,t)𝑑t+Cφ0Tus(,t),xζ~(,t)𝑑t-u0s,ζ~(,0).(5.53)

Let ζC([0,T];Cc()). By Definition 2.1 (for L(0,T;())), the map tus(,t),ζ(,t) belongs to L(0,T). Hence,

limh01ht¯t¯+hus(,t),ζ(,t)𝑑t=us(,t¯),ζ(,t¯)for every t¯(0,T)N,(5.54)

for some null set N(0,T) (by separability arguments, we have that N is independent of ζ; see the proof of [23, Lemma 3.1]). Let t1,t2(0,T)N, 0<t1<t2<T. By standard approximation arguments, we can choose ζ~(x,t)=gh(t)ζ(x,t) in (5.53), where

gh(t):=1h(t-t1)χ{t1tt1+h}(t)+χ{t1+h<t<t2}(t)+1h(t2+h-t)χ{t2tt2+h}(t)(5.55)

and h(0,min{t2-t1,T-t2}). Letting h0 in (5.53), we obtain that

us(,t2),ζ(,t2)t1t2us(,t),ζν(,t)𝑑t+us(,t1),ζ(,t1).(5.56)

Similarly, let fh(t):=χ{0t<t2}(t)+1h(t2+h-t)χ{t2tt2+h}(t). Setting ζ~(x,t)=fh(t)ζ(x,t) in (5.53) and letting h0+, we obtain that

us(,t),ζ(,t)0tus(,τ),ζν(,τ)𝑑τ+u0s,ζ(,0).(5.57)

Arguing as in the last part of the proof of Proposition 5.7, we obtain (3.18) and (3.19) from, respectively, (5.56) and (5.57) (we omit the details).

(ii) It follows from (3.8) that for a.e. τ(0,T) and m,

u(,τ),ρm-u0,ρm=0τ{Ωmφ(ur)(x,t)ρmdx+Cφus(,t)Ωm,ρm}dt,(5.58)

where {ρm}Cc1() is such that ρm=1 in [-m,m], suppρm[-m-1,m+1], 0ρm1 and |ρm|2 in , and Ωm:=[-m-1,-m][m,m+1]. Since usL(0,T;+()) and φ(ur)L(0,T;L1()), a routine proof shows that

limm0τφ(ur)(x,t)ρm(x)𝑑x𝑑t=limm0τus(,t)Ωm,ρm𝑑t=0.

Since ρm(x)1 for all x, we also get that u(,τ),ρmu(,τ)() and u0,ρmu0() as m. Letting m in (5.58), we obtain claim (ii). ∎

6 Regularity: Proofs

The first regularity result which we prove is Proposition 3.5. Hence, we need the following lemma.

Lemma 6.1.

Let (H1) be satisfied. Let (u,τ) be a Young measure solution of problem (P). Then there exists a null set F*(0,T) such that for every t0,t1(0,T)F*, t0<t1, and any ρCc1(R), we have

u(,t1),ρ-u0,ρ=0t1{φ*(x,t)ρ(x)dx+Cφus(,t),ρ}dt,(6.1)u(,t1),ρ-u(,t0),ρ=t0t1{φ*(x,t)ρ(x)dx+Cφus(,t),ρ}dt.(6.2)

Proof.

Since uL(0,T;+()), there exists a null set F0(0,T) such that the spatial disintegration u(,t)+() is defined for every t(0,T)F0. Arguing as in the proof of [23, Lemma 3.1], we can show that there exists a null set F*(0,T), F0F*, such that for every ρCc() and t(0,T)F*,

limq{2qt-1qt+1q|u(,s),ρ-u(,t),ρ|ds}=0.(6.3)

The proof of (6.1) is based on (3.4) and (6.3). Let ρCc1() and t1(0,T)F*. By standard regularization arguments, we can set ζ=ρ(x)kq(t) in (3.4), with q1T-t1+1 (q) and

kq(t):=min{1,q(t1+1q-t)+}χ(0,t1]in (0,T) as q

to get

qt1t1+1qu(,t),ρdt-u0,ρ=0T{φ*(x,t)ρ(x)dx+Cφus(,t),ρ}kq(t)dt.

Letting q, we obtain (6.1) from (3.7) and (6.3). Subtracting from (6.1) the same inequality with t1 replaced by t0, we obtain (6.2). ∎

Proof of Proposition 3.5.

Let F*(0,T) be the null set given by Lemma 6.1. Let {τn}(0,T)F*, with τn0+ as n. Since, by (3.7), uL(0,T;+()) and φ*L(0,T;L1()), it follows from (6.1) that u(,τn),ρu0,ρ for all ρCc1(). Since, by Definition 2.1 (ii), supnu(,τn)()C, there exist μ0+() and a subsequence {τnk} such that u(,τnk)*μ0 in () as k. By standard density arguments, this implies that μ0=u0. Hence, u(,τn)*u0 along the whole sequence {τn}, and (3.11) follows from (6.1) and the arbitrariness of {τn}.

Similarly, it follows from (6.2) that u(,τn),ρu(,t0),ρ for all ρCc1() as τnt0 if t0,τn(0,T)F*, and we obtain (3.12).

To prove (3.13), we observe that, given t0[0,T] and two sequences τn1 and τn2 contained in (0,T)F* and converging to t0, we have u(,τn1)-u(,τn2),ρ0 for all ρCc(). Hence, if t0F*, the continuous extension of u(,t) from (0,T)F* with respect to the weak* topology is well-defined. ∎

Let us now prove the results of Section 3.3. As explained there, replacing x by x-Cφt we may assume, without loss of generality, that Cφ=0; namely, it suffices to prove Proposition 3.10, Theorem 3.11 and Proposition 3.14. Moreover, replacing x by -x and φ by -φ, it suffices to do so by assuming that (H2) is satisfied with φ′′<0, φ>0 in (0,) (see Remark 3.15). Therefore, we make use of the following assumption:

  • (H5)

    φC([0,)), Cφ=0, φ′′(u)<0, and there exist H-1, K>0 such that

    φ′′(u)[Hφ(u)+K]-[φ(u)]2<0for all u[0,).

(Recall that in this case φ>0 and Hφ(u)+K>0 in [0,).)

First we prove some estimates of the constructed entropy solutions. As already said, these estimates are analogous to the Aronson–Bénilan inequality for the convex case up, p>1 (see [1]).

Proposition 6.2.

Let (H1) and (H5) be satisfied, and let u be an entropy solution of problem (P) given by Theorem 3.7. Then, for a.e. 0<t1<t2T,

φ(ur)(,t2)+KH(t2t1)H[φ(ur)(,t1)+KH]a.e. in  if H0,(6.4)φ(ur)(,t2)-Klog(t2)φ(ur)(,t1)-Klogt1a.e. in  if H=0.(6.5)

Moreover, if

  • (C4)

    there exists L>0 such that

    Hφ(u)+KL(1+u)φ(u)for u0,

then tuM(Ω×(τ,T)), t[φ(ur)]M(Ω×(τ,T)), and uC((0,T];M(Ω)) for every bounded open set ΩR and τ>0.

Remark 6.3.

If φ(u)=sgnp[(1+u)p-1] (p<1,p0), (6.4) becomes

ur(,t2)(t2t1)11-p[1+ur(,t1)]-1a.e. in , for a.e. 0<t1t2T

(see Remark 3.9). Similarly, if φ(u)=log(1+u), (6.5) becomes

ur(,t2)(t2t1)[1+ur(,t1)]-1a.e. in , for a.e. 0<t1t2T.

Let (H5) hold. To prove Proposition 6.2, we use a different regularization of (Pn), that is,

{tynε+x[φ(ynε)]=εx2[φ(ynε)]in S,ynε=u0nεin ×{0},(6.6)

where {u0nε} satisfies (4.1)–(4.2). The existence, uniqueness and regularity results recalled in Section 4 for problem (4.3), as well as the a priori estimates in Lemma 4.1 and the convergence results in Lemma 4.4 (i), continue to hold for solutions of (6.6) (see [18]). In particular, there exist a sequence {ynεm} and ynL(S)L(0,T;L1()) such that ynεm*yn in L(S) and for all L>0,

ynεmynin L1((-L,L)×(0,T)) as εm0.(6.7)

From (6.6), for every E convex, F=Eφ, and ζ as in Definition 3.6, we get

S{E(ynεm)tζ+F(ynεm)xζ}𝑑x𝑑t+E(u0nε)ζ(x,0)𝑑xεmSF(ynεm)xynεmxζdxdt.

Arguing as in the proof of Proposition 4.5 and letting εm0, we obtain that

S[E(yn)tζ+F(yn)xζ]𝑑x𝑑t-E(u0n)ζ(x,0)𝑑x.

So yn satisfies (3.16) and, by Kružkov’s uniqueness theorem, yn=un. Hence, we have shown the following lemma.

Lemma 6.4.

Let (H1) and (H5) be satisfied, and let un be the unique entropy solution of problem (Pn) given by Proposition 4.5. Then there exists a subsequence {ynεm} of solutions of (6.6) such that ynεm*un in L(S) and satisfies (6.7).

Lemma 6.5.

Let (H1) and (H5) be satisfied. Then

t[Hφ(ynε)(,t)+KtH]{0in  if H>0,0in  if H<0,(6.8)t[φ(ynε)(,t)-Klogt]0in  if H=0,(6.9)

for all t(0,T), ε>0 and nN. Moreover, if (C4) is satisfied, then

ttynεL(1+ynε)in S.(6.10)

Proof.

For convenience, we set Aεx2-x, thus tynε=A[φ(ynε)] in S. Let

znε:=ttynε-g(ynε),where g(ynε):=Hφ(ynε)+Kφ(ynε)(n).

It follows from (H5) and a straightforward calculation that

tznε=A[φ(ynε)znε]+[H+1-g(ynε)]0znε+g(ynε)tA[φ(ynε)znε]+[H+1-g(ynε)]znεt

in S. Since znε=-g(u0nε)0 in ×{0}, it follows from the comparison principle for parabolic equations that znε0 in S for all n. Hence, ttynε(,t)g(ynε)(,t) in for all t(0,T), which implies (6.8), (6.9) and, if (C4) is satisfied, (6.10). ∎

Proof of Proposition 6.2.

Let {ynεm} be as in the proof of Lemma 6.4. By (6.8)–(6.9),

φ(ynεm)(,t2)+KH(t2t1)H[φ(ynεm)(,t1)+KH]in  if H0,φ(ynε)(x,t2)-Klog(t2)φ(ynε)(x,t1)-Klogt1in  if H=0,

for all 0<t1t2T and n. Hence, by Lemma 6.4,

φ(un)(,t2)+KH(t2t1)H[φ(un)(,t1)+KH]a.e. in  if H0,(6.11)φ(un)(,t2)-Klog(t2)φ(un)(,t1)-Klogt1a.e. in  if H=0,(6.12)

for a.e. 0<t1t2T. Since φ is strictly decreasing in [0,) (recall that φ is concave by assumption H5), possibly extracting another subsequence (denoted again by {nj}), φ(unj)φ(ur) a.e. in S (see Remark 5.10). Letting j in (6.11)–(6.12) (with n=nj), we obtain (6.4)–(6.5).

Let Ω=(-L,L). If (C4) is satisfied, it follows from (6.10) and (4.5) that

tΩ[tynε]+(x,t)𝑑xL|Ω|+u0()for all t(0,T].(6.13)

Since |tynε|=2[tynε]+-tynε a.e. in S, there exists CΩ>0 such that

τTΩ|tynε|(x,t)𝑑x𝑑t2(T-τ)L|Ω|+u0()τ+Ω{ynε(x,τ)-ynε(x,T)}CΩτ

for all τ>0, ε>0 and n, and, by (3.1),

τTΩ|tynε|dxdtCΩτ,τTΩ|t[φ(ynε)]|dxdtMCΩτ.(6.14)

Let {εm} and {nj} be as in Lemma 6.4 and (5.1). Then

limnjlimεm0ynjεm,tζΩ×(τ,T)=u,tζΩ×(τ,T)for all ζCc1(Ω×(τ,T)),

whence, by (6.14) and the lower semicontinuity of the total variation,

tu(Ω×(τ,T))CΩτ.

Similarly, by (5.6), (5.25) and Proposition 5.8,

limnjlimεm0φ(ynjεm),tζΩ×(τ,T)=τTΩφ(ur)tζdxdtfor all ζCc1(Ω×(τ,T)),

and, by (6.14) and the lower semicontinuity of the total variation,

t[φ(ur)](Ω×(τ,T))MCΩτ.

It remains to prove that uC((0,T];(Ω)). Observe that for all t1,t2(0,T], 0<τ<t1<t2, and ρCc2(), 0ρ1 in , ρ=1 in Ω,

Ω|ynε(x,t2)-ynε(x,t1)|dx|ynε(x,t2)-ynε(x,t1)|ρ(x)dxt1t2|tynε|ρdxdt=t1t2(2[tynε]+-tynε)ρ𝑑x𝑑t=2t1t2[tynε]+ρ𝑑x𝑑t-t1t2{φ(ynε)ρ(x)+εφ(ynε)ρ′′(x)}𝑑x𝑑t2L|suppρ|+u0()τ(t2-t1)-t1t2{φ(ynε)ρ+εφ(ynε)ρ′′}𝑑x𝑑t,

where we have used (6.13). We let ε=εm0 and use (3.1) and (4.20) to obtain

Ω|un(x,t2)-un(x,t1)|dx2L|suppρ|+u0()τ(t2-t1)-t1t2φ(un)ρ(x)dxdt(2L|suppρ|+u0()τ+Mu0()ρL())(t2-t1)=:C~τ(t2-t1).

By (5.16) and the lower semicontinuity of the total variation,

u(,t2)-u(,t1)(Ω)C~τ|t1-t2|for a.e. 0<τ<t1<t2T.

So we may define u(,t) for all t[τ,T] such that uC([τ,T];(Ω)). Since τ>0 is arbitrary, the proof is complete. ∎

To prove Proposition 3.10, we need the following lemma.

Lemma 6.6.

Let (H1) be satisfied, and let u be the solution of problem (P) given by Theorem 3.7. Let {unj} be as in the proof of Theorem 3.7. Then, for a.e. t(0,T) and all x0suppus(,t), there exist a sequence {x0k}R and a subsequence {unk} of {unj} such that x0kx0 and unk(x0k,t) as k.

Proof.

Let x0suppus(,t). We may assume that the convergence in (5.16) is satisfied for this t. Since x0suppus(,t), there is no neighborhood Iδ(x0) such that the sequence {unj(,t)} lies in a bounded subset of L(Iδ(x0)). Otherwise, up to a subsequence, unj(,t)*ft in L(Iδ(x0)) for some ftL(Iδ(x0)), ft0. However, this would imply that us(,t)=0 in Iδ(x0), a contradiction.

Setting δ=1/k, we obtain that supnjunj(,t)L(I1/k(x0))= for all k. Hence, for all k, there exists x0kI1/k(x0) such that unk(x0k,t)k. ∎

Proof of Proposition 3.10.

As pointed out above, it suffices to prove equality (3.20) by assuming (H5). Let {unj} be as in the proof of Lemma 6.6. By Lemma 6.4, for every nj, there exists εm0 such that

ynjεm(,t)unj(,t)in Lloc1() as εm0 for a.e. t(0,T).(6.15)

By the proof of Lemma 6.5, for all t(0,T),

εmx2[φ(ynjεm)(,t)]-x[φ(ynjεm)(,t)]=t(ynjεm)g(ynjεm)(,t)tin ,(6.16)

where g(u)=Hφ(u)+Kφ(u)>0. For every x¯<x¯, let ρCc1((x¯,x¯)), ρ0. Multiplying (6.16) by ρ/g(ynjεm(,t)), integrating by parts and setting Ψ(y):=0yφ(u)g(u)𝑑u, we find that

x¯x¯Ψ(ynjεm)(x,t)[εmρ′′(x)+ρ(x)]𝑑x1tx¯x¯ρ(x)𝑑x-εmx¯x¯φ(ynjεm)g(ynjεm)[(ynjεm)x]2[g(ynjεm)]2(x,t)ρ(x)𝑑x1tx¯x¯ρ(x)𝑑x

(observe that by (H5) we have g(u)H+10 and Ψ is bounded). Hence, by (6.15),

x¯x¯Ψ(unj)(x,t)ρ(x)𝑑x1tx¯x¯ρ(x)𝑑x.(6.17)

Let x0suppus(,t), and let {x0k}, {unk} be as in Lemma 6.6, for a.e. t(0,T). Let x¯>x0 be fixed. Since x0kx0, there exists k¯ such that x¯>x0k for all k>k¯. Consider any sequence {ρm}Cc1((x0k,x¯)), 0ρm1, ρmχ(x0k,x¯) in . Without loss of generality, we may assume that both x0k and x¯ are Lebesgue points of unk(,t) for all k. Setting ρ=ρm and x¯=x0k in (6.17), and letting m, we find that

Ψ(unk)(x0k,t)Ψ(unk)(x¯,t)+1t(x¯-x0k)for all nk.

Since Ψ is continuous, by Lemma 6.6 and Remark 5.10 (recall that φ satisfies (C1), since φ is strictly concave by assumption H5), letting nk gives

Ψ(ur)(x¯,t)+1t(x¯-x0)Ψ() for a.e. x¯>x0,

whence, by the invertibility of Ψ,

ur(x¯,t)Ψ-1(Ψ()-1t(x¯-x0))for a.e. x¯>x0.(6.18)

Letting x¯x0+ in the previous inequality, we obtain (3.20). ∎

To prove Theorem 3.11, we need the following result.

Proposition 6.7.

Let (H1) be satisfied. Let Cφ=0, and let u be a solution of problem (P). Then, for a.e. 0t1t2T,

  • (i)

    the map xΦ(x,t1,t2):=t1t2φ(ur)(x,t)𝑑t belongs to BV(),

  • (ii)

    for all x0,x1, x0x1,

    u(,t2)([x0,x1])-u(,t1)([x0,x1])=Φ(x0-,t1,t2)-Φ(x1+,t1,t2),(6.19)u(,t2)([x0,x1])-u0([x0,x1])=Φ(x0-,0,t2)-Φ(x1+,0,t2).(6.20)

Remark 6.8.

It is easily seen that for Cφ0, equalities (6.19)–(6.20) are replaced by

u(,t2)([x0,x1])-𝒯Cφ(t2-t1)(u(,t1))([x0,x1])=Φ(x0-,t1,t2)-Φ(x1+,t1,t2),(6.21)u(,t2)([x0,x1])-𝒯Cφt2(u0)([x0,x1])=Φ(x0-,0,t2)-Φ(x1+,0,t2),

where now

Φ(x,t1,t2):=t1t2[φ(ur)-Cφur](x+Cφ(t-t1),t)𝑑t.(6.22)

Proof of Proposition 6.7.

(i) By (3.1), |t1t2φ(ur)(x,t)𝑑t|Mt1t2ur(x,t)𝑑tL1(). We argue as in the proof of Proposition 3.8 (see (5.54)). There exists a null set N(0,T) such that

limh01ht¯t¯+hu(,t),ρ𝑑t=u(,t¯),ρfor all ρCc() and t¯(0,T)N.(6.23)

Let t1,t2(0,T)N, 0<t1<t2<T, ρCc1(), and ζ(x,t)=gh(t)ρ(x), with gh as in (5.55). Since Cφ=0, we obtain from (3.8) that

1ht1t1+hu(,t),ρ𝑑t-1ht2t2+hu(,t),ρ𝑑t+0Tgh(t)ρ(x)φ(ur)(x,t)𝑑x𝑑t=0.

Letting h0, it follows from (6.23) that

u(,t2),ρ-u(,t1),ρ=Φ(x,t1,t2)ρ(x)𝑑x.(6.24)

Hence, the distributional derivative Φx(x,t1,t2) belongs to ().

(ii) We set, for m and x,

ρm(x):=m(x-x0+1m)χ[x0-1m,x0]+χ(x0,x1)(x)+m(-x+x1+1m)χ[x1,x1+1m].

By standard regularization arguments, we can choose ρ=ρm in (6.24) to obtain

u(,t2),ρm-u(,t1),ρm=mx0-1mx0Φ(x,t1,t2)𝑑x-mx1x1+1mΦ(x,t1,t2)𝑑x.(6.25)

By the dominated convergence theorem, u(,ti),ρmu(,ti)([x0,x1]) as m (i=1,2), whereas, by part (i),

mx0-1mx0Φ(x,t1,t2)𝑑xΦ(x0-,t1,t2),mx1x1+1mΦ(x,t1,t2)𝑑xΦ(x1+,t1,t2).

Hence, (6.19) follows from (6.25). The proof of (6.20) is similar. ∎

Remark 6.9.

Observe that, by (3.18) and (6.21) with x0=x1=x, all entropy solutions of problem (P) satisfy, for a.e. 0t1t2T,

Φ(x-,t1,t2)Φ(x+,t1,t2)for all x,

with Φ defined by (6.22).

Now we are ready to prove Theorem 3.11 and Proposition 3.14. As pointed out at the beginning of this section, in doing so it is not restrictive to assume that (H5) holds.

Proof of Theorem 3.11.

(i) By (6.20), for a.e. 0tT,

us(t)({x0})=u0s({x0})+Φ(x0-,0,t)-Φ(x0+,0,t)u0s({x0})-φL(0,)t,

whence us(t)({x0})>0 if t(0,u0s({x0})φL(0,)). Hence, (3.21) follows.

(ii) Let un be the entropy solution of problem (Pn) given by Proposition 4.5. We argue as in the proof of Proposition 6.7. For all n, the map xΦn(x,t1,t2):=t1t2φ(un)(x,t)𝑑t belongs to BV() and, for a.e. 0t1t2T and a.e. x0x1,

x0x1un(x,t2)𝑑x-x0x1un(x,t1)𝑑x=Φn(x0-,t1,t2)-Φn(x1+,t1,t2).

Letting x1, it follows from (4.27) and (3.14) that

t1t2φ(un)(x,t)𝑑tu0()for n and a.e. x.(6.26)

Let {ynεm} be the subsequence used in the proof of Lemma 6.4. By (6.8) and (6.9), for every 0<t1tT and x,

t1tφ(ynεm)(x,s)𝑑s=1Ht1tHφ(ynεm)(x,s)+KsHsH𝑑s-KH(t-t1)Hφ(ynεm)(x,t)+KHtHtH+1-t1H+1H+1-KH(t-t1)if H0,t1tφ(ynεm)(x,s)𝑑s=t1t[φ(ynεm)(x,s)-Klogs]𝑑s+Kt1tlogsds[φ(ynεm)(x,t)-Klogt](t-t1)+K[tlogt-t]-K[t1logt1-t1]if H=0.

Letting εm0, by (6.26), we obtain, for a.e. t(t1,T) and a.e. x,

u0()Φn(x,t1,t){Hφ(un)(x,t)+KHtHtH+1-t1H+1H+1-KH(t-t1)if H0,[φ(un)(x,t)-K](t-t1)+Kt1logtt1if H=0.

Letting t10+, we find in both cases that

φ(un)(x,t)(H+1)u0()t+Kfor a.e. t(t1,T) and a.e. x(6.27)

(recall that we have assumed H>-1 if φ is bounded; otherwise, if φ is unbounded, we have H0, since φ>0 and Hφ+K>0 in [0,) by H5). If limuφ(u)=:γ<, K<γ and H>-1, the sequence {un(,t)} lies in a bounded subset of L() (thus, by (5.16) us(,t)=0 and ur(,t)L()) for a.e. t(0,T) such that

(H+1)u0()t+K<γt>(H+1)u0()γ-K.

This proves claim (ii) (a).

If γ=, we have H0, since Hφ+K>0 in [0,) (see H5). Then, by (6.27), the sequence {un(,t)} lies in a bounded subset of L() for a.e. t(0,T), hence, by (5.16) as n, we obtain that t0=0. Thus, claim (ii) (b) follows. This completes the proof. ∎

Remark 6.10.

As we claimed in Remark 3.13, in Theorem 3.11 (ii), we may relax hypothesis (H2) to (H3), with k>0. To prove this, for every u0+(Ω), let {u0n} be any sequence as in (3.14)–(3.15), and let un be the entropy solution of problem (Pn). Set v0n:=Gk(u0n), where Gk(u):=(u-k)+ for every u0, and let vn be the entropy solution of the following problem:

{tvn+x[φk(vn)]=0in S,vn=v0nin ×{0}

(φk(u)=φ(u+k)-φ(k)). A standard calculation shows that Gk(un) is an entropy subsolution of the above problem, whence

Gk(un)vna.e. in S.(6.28)

Following the proof of Theorem 3.7, the sequence {vn} converges to an entropy solution v of problem (P) with initial datum v0=u0s+Gk(u0r). Moreover, by assumption (H3), φk satisfies (H2) and we may apply Theorem 3.11 (ii) to v. Therefore, the conclusion follows from (6.28).

Proof of Proposition 3.14.

By the proof of Proposition 3.10, inequality (6.18) is satisfied for a.e. t(0,T) and all x0suppus(,t). We fix such t. Let x1suppus(,t) and set 1:=(x1-ε,x1+ε) with ε>0. By (6.18),

1ur(x,t)dxx1x1+εΨ-1(Ψ()-1t(x-x1))dx=0εΨ-1(Ψ()-yt)dy=:Bε.

If suppus(,t)1, let x2suppus(,t)1 and set 2:=(x2-ε,x2+ε). Since (x1,x1+ε)(x2,x2+ε)=, we have that

12ur(x,t)𝑑xx1x1+εur(x,t)𝑑x+x2x2+εur(x,t)𝑑x2Bε.

We continue this construction recursively as long as suppus(,t)1n-1, with n-1:=(xn-1-ε,xn-1+ε): there exists xnsuppus(,t){1n-1} such that, setting n:=(xn-ε,xn+ε),

nBε1nur(x,t)𝑑xu0().

Hence, this construction stops at some n=nε, and nεBεu0(). Therefore,

suppus(,t)1nε,|supps(,t)||1nε|2nεε2εBεu0().

Since Bε/ε as ε0, the claim follows. ∎

7 Uniqueness: Proofs

Again, without loss of generality, we may assume that Cφ=0 in the following proofs (see Remark 3.16).

Proof of Proposition 3.20.

(i) The first step of the proof consists in showing that

esslimt0+ur(,t)-u0rL1()=0.(7.1)

Let {unε} be the sequence of solutions to problems (4.3) considered in Section 4, and let {xl} (l=1,,N) be as in (3.26). We set Il:=(xl,xl+1), Ql:=Il×(0,τ) (l=1,,N-1), I-:=(-,x1), I+:=(xN,), and Q±:=I±×(0,τ).

Let 1lN-1 and ρCc2(Il), ρ0. Let h0>0 be such that x+hIl if xsuppρ and |h|<h0. Let δ>0. Setting vnε(x,t):=unε(x+h,t) and z:=(unε-vnε)(ρ+δ), we apply the L1-contraction property to the parabolic equation

tz+x[(R+2ερρ+δ)z]-εx2z=(Rρρ+δ+ερ′′ρ+δ)z=(φε(unε)-φε(vnε))ρ+ε[unε-vnε]ρ′′,

where

R:={φε(unε)-φε(vnε)unε-vnεif unεvnε,φε(unε)otherwise.

Hence,

Il|z(x,τ)|dxIl|z(x,0)|dx+0τIl|φε(unε(x,t))-φε(unε(x+h,t))||ρ(x)|dxdt+ε0τIl|unε(x,t)-unε(x+h,t)||ρ′′(x)|dxdtfor τ(0,T).

First we let δ0 and then ε=