Pattern formation in a flux limited reaction–diffusion equation of porous media type

Abstract

A non-linear PDE featuring flux limitation effects together with those of the porous media equation (non-linear Fokker–Planck) is presented in this paper. We analyze the balance of such diverse effects through the study of the existence and qualitative behavior of some admissible patterns, namely traveling wave solutions, to this singular reaction–diffusion equation. We show the existence and qualitative behavior of different types of traveling waves: classical profiles for wave speeds high enough, and discontinuous waves that are reminiscent of hyperbolic shock waves when the wave speed lowers below a certain threshold. Some of these solutions are of particular relevance as they provide models by which the whole solution (and not just the bulk of it, as it is the case with classical traveling waves) spreads through the medium with finite speed.

Appendix: Entropy solutions

Our purpose in this Appendix is to give the necessary background in order to introduce the notion of entropy solutions to (1.2), to state some existence and uniqueness results for them, and to give sense to the properties stated in Sect. 1.2.

Equation (1.2) belongs to the more general class of flux limited diffusion equations, which has been extensively studied in [2, 4, 5, 21, 22]. As shown in those papers, the notion of entropy solution is the right one in order to prove existence and uniqueness results and to describe the qualitative features of solutions. In particular, and closely related to this work, the so-called relativistic heat equation [which corresponds to $m=1$ in (1.2)] coupled with a Fisher–Kolmogorov type reaction term has been studied in [3, 19]. Existence and uniqueness results for that model were proved in [3], the construction of traveling waves being the object of [19].

Thus, our first purpose is to give a brief review of the concept of entropy solution for flux limited diffusion equations. Although we are only concerned with the case $N=1$, we state the results in the more general context where $N\ge 1$ since this may be useful for future reference. For a more detailed treatment we refer to [4, 21]. We consider parabolic equations of the form

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \frac{\partial u}{\partial t} = \text{ div }\, \, \mathbf{a}(u, \nabla u)+F(u), &{} \quad \hbox {in}\quad Q_T=]0,T[\times \mathbb {R}^N\\ \displaystyle \\ \displaystyle u(0,x) = u_{0}(x), &{} \quad \text{ in }\quad x \in \mathbb {R}^N \end{array} \right. \end{aligned}$$

(4.1)

where F(u) is a Lipschitz continuous function such that $F(0)=0$ and $\mathbf{a}(z, \zeta ) = \nabla _{\zeta }f(z, \zeta )$ is associated to a Lagrangian f satisfying a set of technical assumptions. Let us give a brief account of them, referring to [4, 21] for a thorough presentation. Thus, we assume that

(H) f is continuous on $[0,\infty [ \times \mathbb {R}^N$ and is a convex differentiable function of $\zeta$ such that $\nabla _{\zeta }f(z, \zeta ) \in C([0,\infty [ \times \mathbb {R}^N)$. Further, we require f to satisfy the coercivity and linear growth conditions

$$\begin{aligned} C_0(z) \vert \zeta \vert - D_0(z) \le f(z, \zeta ) \le M_0(z)(\vert \zeta \vert + 1), \end{aligned}$$

(4.2)

for any $(z,\zeta )\in [0,\infty [\times \mathbb {R}^N$, and some positive and continuous functions $C_0, D_0,$ $M_0 \in C([0,\infty [)$ with $C_0(z) > 0$ for any $z\ne 0$. Notice that $\vert \zeta \vert$ denotes the Euclidian norm of $\zeta \in \mathbb {R}^N$. We assume that

$$\begin{aligned} C_0(z) \ge c_0 z^{m}, \quad \hbox { for some }c_0 > 0,\quad {m} \ge 1, \quad z\in [0,\infty [. \end{aligned}$$

Let $\mathbf{a}(z, \zeta ) = \nabla _{\zeta }f(z, \zeta )$, $(z,\zeta )\in [0,\infty [\times \mathbb {R}^N$. We assume that there is a vector field $\mathbf{b}(z,\zeta )$ and a constant $M > 0$ such that

$$\begin{aligned} \mathbf{a}(z,\zeta ) = z^{m} \mathbf{b}(z,\zeta ) \quad \hbox { with}\quad \vert \mathbf{b}(z, \zeta ) \vert \le M, \ \ \ \forall \ (z, \zeta ) \in [0,\infty [ \times \mathbb {R}^N. \end{aligned}$$

(4.3)

We consider the function $h : [0,\infty [ \times \mathbb {R}^N \rightarrow \mathbb {R}$ defined by

$$\begin{aligned} h(z, \zeta ):= \mathbf{a}(z, \zeta ) \cdot \zeta . \end{aligned}$$

(4.4)

From the convexity of f in $\zeta$, (4.2) and (4.3), it follows that

$$\begin{aligned} C_0(z) \vert \zeta \vert - D_1(z) \le h(z, \zeta ) \le M z^{ m}\vert \zeta \vert , \end{aligned}$$

for any $(z,\zeta )\in [0,\infty [\times \mathbb {R}^N$, where $D_1(z) = D_0(z)+f(z,0)$. We also assume that the recession functions $f^0$, $h^0$ exist, being the recession function $g^0$ of a function g defined as

$$\begin{aligned} g^0(x, z, \zeta ) = \lim _{t \rightarrow 0^+} tg \left( x, z, \frac{\zeta }{t} \right) . \end{aligned}$$

It is convex and homogeneous of degree 1 in $\zeta$.

Other technical assumptions on f, h are required:

• $f^0(z,\zeta )=h^0(z,\zeta )$ for all $z \in \mathbb {R}$ and $\zeta \in \mathbb {R}^N$

• $h(z,\zeta )=h(z,-\zeta )$ for all $z \in \mathbb {R}$ and $\zeta \in \mathbb {R}^N$

• $\frac{\partial \mathbf{a}_i}{\partial \zeta _i}(z,\zeta ) \in C(\mathbb {R}\times \mathbb {R}^N)$ for any $i=1,\ldots , N$

• $\mathbf{a}(z,\zeta )\cdot \eta \le h^0(z,\eta )$ for all $z \in \mathbb {R}$ and $\zeta , \eta \in \mathbb {R}^N$

• $h^0(z,\zeta )$ can be written as $h^0(z,\zeta )=\varphi (z) \psi ^0(\zeta )$, where $\varphi$ is a Lipschitz continuous function such that $\varphi (z)>0$ for any $z \ne 0$ and $\psi ^0$ is a convex function which is homogeneous of degree one

• For any $R>0$ there exists a constant $C>0$ such that

  $$\begin{aligned} |(\mathbf{a}(z,\zeta )-\mathbf{a}(\hat{z},\zeta ))\cdot (\zeta - \hat{\zeta })|\le C |z-\hat{z}| \Vert \zeta -\hat{\zeta }\Vert \end{aligned}$$

  for any $(z,\zeta ), (z,\zeta ) \in \mathbb {R}\times \mathbb {R}^N, \ |z|,|\hat{z}| \le R$.

When we say that assumption (H) holds, we refer to the complete set of assumptions above.

For the generalized relativistic heat equation (1.2) the function

$$\begin{aligned} f(z,\zeta ) = \frac{c^2}{\nu } z^m \sqrt{z^2 + \frac{\nu ^2}{c^2} \vert \zeta \vert ^2} \end{aligned}$$

(4.5)

satisfies all the assumptions that allow to work in the context of entropy solutions (see [2, 4]). In this case

$$\begin{aligned} \mathbf{a}(z,\zeta ) = \nu \frac{ z^m \zeta }{\sqrt{z^2 + \frac{\nu ^2}{c^2} \vert \zeta \vert ^2}} \quad \hbox { and } \quad h(z,\zeta ) = \mathbf{a}(z,\zeta )\cdot \zeta = \nu \frac{z^m \vert \zeta \vert ^2}{\sqrt{z^2 + \frac{\nu ^2}{c^2} \vert \zeta \vert ^2}}. \end{aligned}$$

Due to the linear growth condition on the Lagrangian, the natural energy space to study the solutions of (4.1) is the space of functions of bounded variation, or BV functions. In Sect. 4.1 we recall some basic basic facts about them.

The notion of entropy solutions is based on a set of Kruzkov’s type inequalities and it requires to define a functional calculus for functions whose truncations are in BV. We briefly review in Sect. 4.2 this functional calculus which is based on the works [27, 28], which prove lower semicontinuity results for functionals on BV. After this, in Sect. 4.3 we state without proof an existence and uniqueness result for entropy solutions of (4.1). The proof can be obtained by a suitable adaptation of the techniques in [3]. Since the traveling wave solutions we construct are functions in $L^\infty (\mathbb {R}^N)^+$, we give a uniqueness result for solutions in that space (see Sect. 4.3.3). A similar result was proved in [3] for the case $m=1$.

This section gives the necessary background for the characterization of entropy conditions given in Sect. 1.2.

1.1 Functions of bounded variation and some generalizations

Denote by ${\mathcal {L}}^N$ and ${\mathcal {H}}^{N-1}$ the N-dimensional Lebesgue measure and the $(N-1)$-dimensional Hausdorff measure in $\mathbb {R}^N$, respectively. Given an open set $\Omega$ in $\mathbb {R}^N$ we denote by ${\mathcal {D}}(\Omega )$ the space of infinitely differentiable functions with compact support in $\Omega$. The space of continuous functions with compact support in $\mathbb {R}^N$ will be denoted by $C_c(\mathbb {R}^N)$.

Recall that if $\Omega$ is an open subset of $\mathbb {R}^N$, a function $u \in L^1(\Omega )$ whose gradient Du in the sense of distributions is a vector valued Radon measure with finite total variation in $\Omega$ is called a function of bounded variation. The class of such functions will be denoted by $BV(\Omega )$. For $u \in BV(\Omega )$, the vector measure Du decomposes into its absolutely continuous and singular parts $Du = D^a u + D^s u$. Then $D^a u = \nabla u \ {\mathcal {L}}^N$, where $\nabla u$ is the Radon–Nikodym derivative of the measure Du with respect to the Lebesgue measure ${\mathcal {L}}^N$. We also split $D^su$ in two parts: the jump part $D^j u$ and the Cantor part $D^c u$. It is well known (see for instance [1]) that

where $u^+(x),u^-(x)$ denote the upper and lower approximate limits of u at x, $J_u$ denotes the set of approximate jump points of u (i.e. points $x\in \Omega$ for which $u^+(x)\ne u^-(x)$), and $\nu _u(x) = \frac{Du}{\vert D u \vert }(x)$, being $\frac{Du}{\vert D u \vert }$ the Radon–Nikodym derivative of Du with respect to its total variation $\vert D u \vert$. For further information concerning functions of bounded variation we refer to [1].

We need to consider the following truncation functions. For $a < b$, let $T_{a,b}(r) := \max (\min (b,r),a)$. We denote

$$\begin{aligned} \mathcal {T}_r:= \{ T_{a,b} \ : \ 0 < a < b \}. \ \ \ \end{aligned}$$

Given any function w and $a,b\in \mathbb {R}$ we shall use the notation $\{w\ge a\} = \{x\in \mathbb {R}^N: w(x)\ge a\}$, $\{a \le w\le b\} = \{x\in \mathbb {R}^N: a \le w(x)\le b\}$, and similarly for the sets $\{w > a\}$, $\{w \le a\}$, $\{w < a\}$, etc.

We need to consider the following function space

$$\begin{aligned} TBV_\mathrm{r}^+(\mathbb {R}^N):= \left\{ w \in L^1(\mathbb {R}^N)^+ \ : \ \ T_{a,b}(w) - a \in BV(\mathbb {R}^N), \ \ \forall \ T_{a,b} \in \mathcal {T}_r \right\} . \end{aligned}$$

Notice that $TBV_\mathrm{r}^+(\mathbb {R}^N)$ is closely related to the space $GBV(\mathbb {R}^N)$ of generalized functions of bounded variation introduced by Di Giorgi and Ambrosio (see [1]) Using the chain rule for BV-functions (see for instance [1]), one can give a sense to $\nabla u$ for a function $u \in TBV^+(\mathbb {R}^N)$ as the unique function v which satisfies

We refer to Lemma 2.1 of [9] or [1] for details.

1.2 Functionals defined on BV

In order to define the notion of entropy solutions of (4.1) and give a characterization of them, we need a functional calculus defined on functions whose truncations are in BV.

Let $\Omega$ be an open subset of $\mathbb {R}^N$. Let $g: \Omega \times \mathbb {R}\times \mathbb {R}^N \rightarrow [0, \infty [$ be a Borel function such that

$$\begin{aligned} C(x) \vert \zeta \vert - D(x) \le g(x, z, \zeta ) \le M'(x) + M \vert \zeta \vert \end{aligned}$$

for any $(x, z, \zeta ) \in \Omega \times \mathbb {R}\times \mathbb {R}^N$, $\vert z\vert \le R$, and any $R>0$, where M is a positive constant and $C,D,M' \ge 0$ are bounded Borel functions which may depend on R. Assume that $C,D,M' \in L^1(\Omega )$.

Following Dal Maso [27] we consider the functional:

$$\begin{aligned} {\mathcal {R}}_g(u):= & {} \displaystyle \int _{\Omega } g(x,u(x), \nabla u(x)) \, dx + \int _{\Omega } g^0 \left( x, \tilde{u}(x),\frac{Du}{\vert D u \vert }(x) \right) \, \vert D^c u \vert \nonumber \\&+ \displaystyle \int _{J_u} \left( \int _{u_-(x)}^{u_+(x)} g^0(x, s, \nu _u(x)) \, ds \right) \, d {\mathcal {H}}^{N-1}(x), \end{aligned}$$

for $u \in BV(\Omega ) \cap L^\infty (\Omega )$, being $\tilde{u}$ is the approximated limit of u [1].

In case that $\Omega$ is a bounded set, and under standard continuity and coercivity assumptions, Dal Maso proved in [27] that ${\mathcal {R}}_g(u)$ is $L^1$-lower semi-continuous for $u \in BV(\Omega )$. More recently, De Cicco, Fusco, and Verde [28] have obtained a very general result about the $L^1$-lower semi-continuity of ${\mathcal {R}}_g$ in $BV(\mathbb {R}^N)$.

Assume that $g:\mathbb {R}\times \mathbb {R}^N \rightarrow [0, \infty [$ is a Borel function such that

$$\begin{aligned} C \vert \zeta \vert - D \le g(z, \zeta ) \le M(1+ \vert \zeta \vert ) \qquad \forall (z,\zeta )\in \mathbb {R}^N, \, \vert z \vert \le R, \end{aligned}$$

(4.6)

for any $R > 0$ and for some constants $C,D,M \ge 0$ which may depend on R. Observe that both functions f, h defined in (4.5), (4.4) satisfy (4.6).

Assume that

for any $u\in L^1(\mathbb {R}^N)^+$. Let $u \in TBV_\mathrm{r}^+(\mathbb {R}^N) \cap L^\infty (\mathbb {R}^N)$ and $T = T_{a,b}\in {\mathcal {T}}_r$. For each $\phi \in C_c(\mathbb {R}^N)$, $\phi \ge 0$, we define the Radon measure g(u, DT(u)) by

$$\begin{aligned} \langle g(u, DT(u)), \phi \rangle&: = {\mathcal {R}}_{\phi g}(T_{a,b}(u))+ \displaystyle \int _{\{u \le a\}} \phi (x) \left( g(u(x), 0) - g(a, 0)\right) \, dx \nonumber \\&\displaystyle \quad + \int _{\{u \ge b\}} \phi (x) \left( g(u(x), 0) - g(b, 0) \right) \, dx. \end{aligned}$$

(4.7)

If $\phi \in C_c(\mathbb {R}^N)$, we write $\phi = \phi ^+ - \phi ^-$ with $\phi ^+= \max (\phi ,0)$, $\phi ^- = - \min (\phi ,0)$, and we define $\langle g(u, DT(u)), \phi \rangle : = \langle g(u, DT(u)), \phi ^+ \rangle - \langle g(u, DT(u)), \phi ^- \rangle$.

Recall that, if $g(z,\zeta )$ is continuous in $(z,\zeta )$, convex in $\zeta$ for any $z\in \mathbb {R}$, and $\phi \in C^1(\mathbb {R}^N)^+$ has compact support, then $\langle g(u, DT(u)), \phi \rangle$ is lower semi-continuous in $TBV^+(\mathbb {R}^N)$ with respect to $L^1(\mathbb {R}^N)$-convergence [28]. This property is used to prove existence of solutions of (4.1).

We can now define the required functional calculus. We follow [21] and note that it represents an extension of the functional calculus in [2, 4] that uses a more restrictive class of test functions.

Let us denote by ${\mathcal {P}}$ the set of Lipschitz continuous functions $p : [0, +\infty [ \rightarrow \mathbb {R}$ satisfying $p^{\prime }(s) = 0$ for s large enough. We write ${\mathcal {P}}^+:= \{ p \in {\mathcal {P}} \ : \ p \ge 0 \}$.

Let $S \in C([0,\infty [)$ and $p \in {\mathcal {P}} \cap C^1([0,\infty [)$. We denote

$$\begin{aligned} f_{S:p}(z,\zeta ) = S(z)p'(z) f(z,\zeta ), \qquad h_{S:p}(z,\zeta ) = S(z)p'(z)h(z,\zeta ). \end{aligned}$$

If $Sp'\ge 0$, then the function $f_{S:p}(z,\zeta )$ satisfies the assumptions implying the lower semicontinuity of the associated energy functional [28].

Assume that $p(r)=p(T_{a,b}(r))$, $0 < a < b$. We assume that $u \in TBV_\mathrm{r}^+(\mathbb {R}^N)$ and

Since $h(z, 0) = 0$, the last assumption clearly holds for h.

Finally, we define $f_{S:p}(u,DT_{a,b}(u)),$ $h_{S:p}(u,DT_{a,b}(u))$ as the Radon measures given by (4.7) with $g(z,\zeta ) = f_{S:p}(z,\zeta )$ and $g(z,\zeta ) = h_{S:p}(z,\zeta )$, respectively.

1.3 Existence and uniqueness of entropy solutions

1.3.1 The class of test functions

Let us introduce the class of test functions required to define entropy sub- and super-solutions. If $u\in TBV_\mathrm{r}^+(\mathbb {R}^N)$, we define $\mathcal {TSUB}$ (resp. $\mathcal {TSUPER}$ ) as the class of functions $S,T \in \mathcal {P}$ such that

$$\begin{aligned} S \ge 0 , S'\ge 0 \quad \text{ and }\quad T\ge 0,T'\ge 0, \end{aligned}$$

$$\begin{aligned} ( \text{ resp. }\, S \le 0, S'\ge 0 \quad \text{ and }\quad T\ge 0,T'\le 0) \end{aligned}$$

and $p(r) = \tilde{p}(T_{a,b}(r))$ for some $0 < a < b$, where $\tilde{p}$ is differentiable in a neighborhood of [a, b] and p represents either S or T.

Although the proof of uniqueness and the development of the theory requires only the use of test functions $S,T\in \mathcal {T}^+$ and this was the family used in [4], the analysis of the entropy conditions is facilitated by the use of more general test functions in $\mathcal {TSUB}$ and $\mathcal {TSUPER}$.

1.3.2 Entropy solutions in $L^1\cap L^\infty$

Let $L^1_{w}(0,T,BV(\mathbb {R}^N))$ be the space of weakly$^*$ measurable functions $w:[0,T] \rightarrow BV(\mathbb {R}^N)$ (i.e., $t \in [0,T] \rightarrow \langle w(t),\phi \rangle$ is measurable for every $\phi$ in the predual of $BV(\mathbb {R}^N)$) such that $\int _0^T \Vert w(t)\Vert _{BV} \, dt< \infty$. Observe that, since $BV(\mathbb {R}^N)$ has a separable predual (see [1]), it follows easily that the map $t \in [0,T]\rightarrow \Vert w(t) \Vert _{BV}$ is measurable. By $L^1_{loc, w}(0, T, BV(\mathbb {R}^N))$ we denote the space of weakly$^*$ measurable functions $w:[0,T] \rightarrow BV(\mathbb {R}^N)$ such that the map $t \in [0,T]\rightarrow \Vert w(t) \Vert _{BV}$ is in $L^1_{loc}(]0, T[)$.

Definition 4.1

Assume that $u_0 \in (L^1(\mathbb {R}^N)\cap L^\infty (\mathbb {R}^N))^+$. A measurable function $u: ]0,T[\times \mathbb {R}^N \rightarrow \mathbb {R}$ is an entropy sub-solution (resp. super-solution) of (4.1) in $Q_T = ]0,T[\times \mathbb {R}^N$ if $u \in C([0, T]; L^1(\mathbb {R}^N))$, $T_{a,b}(u(\cdot )) - a \in L^1_{loc, w}(0, T, BV(\mathbb {R}^N))$ for all $0 < a < b$, and

• (i) $u(0) \le u_0$ (resp. $u(0) \ge u_0$), and

• (ii) the following inequality is satisfied

  $$\begin{aligned}&\displaystyle \int _0^T\int _{\mathbb {R}^N} \phi h_{S:T}(u,DT_{a,b}(u)) \, dt + \int _0^T\int _{\mathbb {R}^N} \phi h_{T:S}(u,DS_{c,d}(u)) \, dt \nonumber \\&\quad \le \displaystyle \int _0^T\int _{\mathbb {R}^N} \Big \{ J_{TS}(u(t)) \phi ^{\prime }(t) - \mathbf{a}(u(t), \nabla u(t)) \cdot \nabla \phi \ T(u(t)) S(u(t))\Big \} dxdt \nonumber \\&\qquad +\displaystyle \int _0^T\int _{\mathbb {R}^N}\phi (t)T(u(t))S(u(t))F(u(t)) \, dxdt, \end{aligned}$$

  (4.8)

  for truncation functions $(S , \, T) \in \mathcal {TSUB}$ (resp. $(S , \, T) \in \mathcal {TSUPER}$) with $T=\tilde{T}\circ T_{a,b}$, $S=\tilde{S}\circ S_{c,d}$, $0 < a < b$, $0 < c < d$, and any smooth function $\phi$ of compact support, in particular those of the form $\phi (t,x) = \phi _1(t)\rho (x)$, $\phi _1\in {\mathcal {D}}(]0,T[)$, $\rho \in {\mathcal {D}}(\mathbb {R}^N)$.

We say that $u: ]0,T[\times \mathbb {R}^N \rightarrow \mathbb {R}$ is an entropy solution of (4.1) if it is an entropy sub- and super-solution.

Notice that if u is an entropy sub-solution (resp. super-solution), then $u_t \le \mathrm{div} \, \mathbf{a}(u(t), \nabla u(t)) + F(u(t))$ (resp. $\ge$) in $\mathcal {D}^\prime (Q_T)$. We notice also that u is an entropy solution if $u_t = \mathrm{div} \, \mathbf{a}(u(t), \nabla u(t)) + F(u(t))$ in $\mathcal {D}^\prime (Q_T)$, $u(0)=u_0$ and the inequalities (4.8) hold for truncations $(S , \, T) \in \mathcal {TSUB}$ and any test functions as in (ii) [21].

We have the following existence and uniqueness result, which is an extension of those in [3].

Theorem 4.2

Let the set of assumptions (H) be satisfied and let F be Lipschitz continuous with $F(0) = 0$. Then, for any initial datum $0 \le u_0 \in L^{\infty }(\mathbb {R}^N) \cap L^{1}(\mathbb {R}^N)$ there exists a unique entropy solution u of (4.1) in $Q_T$ for every $T > 0$ such that $u(0) = u_0$, satisfying $u \in C([0, T]; L^1(\mathbb {R}^N))$ and $F(u(t)) \in L^1(\mathbb {R}^N)$ for almost all $0 \le t \le T$. Moreover, if u(t), $\overline{u}(t)$ are entropy solutions corresponding to initial data $u_0$, $\overline{u}_0 \in \left( L^{\infty }(\mathbb {R}^N) \cap L^{1}(\mathbb {R}^N)\right) ^+$, respectively, then

$$\begin{aligned} \Vert u(t) - \overline{u}(t) \Vert _1 \le e^{t\Vert F \Vert _{Lip}} \, \Vert u_0 - \overline{u}_0 \Vert _1 \ \ \ \ \ \ \mathrm{for \ all} \ \ t \ge 0. \end{aligned}$$

1.3.3 Entropy solutions in $L^\infty$

In order to cover the case of bounded traveling waves, we extend the notion of entropy solutions to functions in $L^{\infty }(\mathbb {R}^N)^+$. We follow the presentation in [3].

Definition 4.3

Given $0 \le u_0 \in L^{\infty }(\mathbb {R}^N)$, we say that a measurable function $u: ]0,T[\times \mathbb {R}^N \rightarrow \mathbb {R}$ is an entropy sub-solution (respectively, entropy super-solution) of the Cauchy problem (4.1) in $Q_T = ]0,T[\times \mathbb {R}^N$ if $u \in C([0, T]; L_{loc}^1(\mathbb {R}^N))$, $u(0)\le u_0$ (resp. $u(0) \ge u_0$), $F(u(t)) \in L_{loc}^1(\mathbb {R}^N)$ for almost every $0 \le t \le T$, $T_{a,b}(u(\cdot )) - a \in L^1_{loc, w}(0, T, BV_\mathrm{loc}(\mathbb {R}^N))$ for all $0 < a < b$, $\mathbf{a}(u( \cdot ), \nabla u(\cdot )) \in L^{\infty }(Q_T)$, and the inequalities (4.8) are satisfied for truncations $(S , \, T) \in \mathcal {TSUB}$ (resp. $(S , \, T) \in \mathcal {TSUPER}$) with $T=\tilde{T}\circ T_{a,b}$, $S=\tilde{S}\circ S_{c,d}$, $0 < a < b$, $0 < c < d$, and any smooth function $\phi$ of compact support, in particular those of the form $\phi (t,x) = \phi _1(t)\rho (x)$, $\phi _1\in {\mathcal {D}}(]0,T[)$, $\rho \in {\mathcal {D}}(\mathbb {R}^N)$.

We say that $u: ]0,T[\times \mathbb {R}^N \rightarrow \mathbb {R}$ is an entropy solution of (4.1) if u is an entropy sub-solution and super-solution.

Definition 4.4

Let u be a sub- or a super-solution of (4.1) in $Q_T$. We say that u has a null flux at infinity if

$$\begin{aligned} \lim _{R \rightarrow + \infty } \int _0^T \int _{ \mathbb {R}^N} \vert \mathbf{a}(u(t),\nabla u(t)) \vert \, \vert \nabla \psi _R (x) \vert \, dx dt = 0 \end{aligned}$$

for all $\psi _R \in {\mathcal {D}}(\mathbb {R}^N)$ such that $0 \le \psi _R \le 1$, $\psi _R \equiv 1$ on $B_R$, $\text{ supp }(\psi _R) \subset B_{R+2}$ and $\Vert \nabla \psi _R \Vert _{\infty } \le 1$.

We have uniqueness of entropy solutions for initial data in $L^{\infty }(\mathbb {R}^N)$ when they have null flux at infinity.

Theorem 4.5

Let the set of assumptions (H) be satisfied and let F be Lipschitz continuous with $F(0) = 0$.

• (i) Let u(t), $\overline{u}(t)$ be two entropy solutions of (4.1) with initial data $u_0, \overline{u}_0 \in L^{\infty }(\mathbb {R}^N)^+$, respectively. Assume that u(t) and $\overline{u}(t)$ have null flux at infinity. Then

  $$\begin{aligned} \Vert u(t) - \overline{u}(t) \Vert _1 \le e^{t\Vert F \Vert _{Lip}} \, \Vert u_0 - \overline{u}_0 \Vert _1, \ \ \ \ \ \mathrm{for \ all} \ \ t \ge 0. \end{aligned}$$

• (ii) Assume that $u_0\in (L^1(\mathbb {R}^N)\cap L^\infty (\mathbb {R}^N))^+$, $\overline{u}_0 \in L^{\infty }(\mathbb {R}^N)^+$. Let u(t) be the entropy solution of (4.1) with initial datum $u_0$. Let $\overline{u}(t)$ be an entropy super-solution of (4.1) with initial datum $\overline{u}_0 \in L^{\infty }(\mathbb {R}^N)^+$ having a null flux at infinity. Assume in addition that $\overline{u}(t) \in BV_\mathrm{loc}(\mathbb {R}^N)$ for almost every $0 < t < T$. Then

  $$\begin{aligned} \Vert (u(t) - \overline{u}(t))^+ \Vert _1 \le e^{t\Vert F \Vert _{Lip}} \, \Vert (u_0 - \overline{u}_0)^+ \Vert _1, \ \ \ \ \ \mathrm{for \ all} \ \ t \ge 0. \end{aligned}$$