Some basic properties of bounded solutions of parabolic equations with p-Laplacian diffusion

TL;DR

This paper rigorously derives fundamental properties of bounded weak solutions to initial-value problems involving p-Laplacian diffusion in parabolic equations, enhancing understanding of their behavior with bounded and integrable initial data.

Contribution

It provides a detailed and rigorous derivation of key properties of solutions to p-Laplacian parabolic equations with bounded initial data, which was previously not fully established.

Findings

Fundamental properties of solutions are rigorously derived.

Results apply to general conservative second-order parabolic equations.

Analysis covers solutions with bounded and integrable initial data.

Abstract

We provide a detailed (and fully rigorous) derivation of several fundamental properties of bounded weak solutions to initial-value problems for general conservative 2nd-order parabolic equations with p-Laplacian diffusion and (arbitrary) bounded and integrable initial data.

Equations

$$u_{t}\,+\>\mbox{div}\;\!\mbox{\boldmath$f$}(x,t,u)\,+\>\mbox{div}\,\mbox{\boldmath$g$}(t,u)\>=\;\mu(t)\;\!\;\!\mbox{div}\,(\;\!|\;\!\nabla u\,|^{\>\!p-2}\,\nabla u\;\!),$$

$$u(\cdot ,0)=u_0\in L^1({\mathbb{R}}^n)\cap L^{\infty }({\mathbb{R}}^n).$$

$$|\,\mbox{\boldmath$f$}(x,t,\mbox{u})\,|\,\leq\,\mbox{\small$F$}(t)\,|\,\mbox{u}\,|^{\>\!\kappa\,+\,1}\quad\;\;\,\forall\;x\in\mathbb{R}^{n}\!,\;t\geq 0,\;\mbox{u}\in\mathbb{R}$$

$$\|\,u(\cdot,t)\,\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R}^{n})}}\,\leq\;\mbox{\small$M$}_{\mbox{}_{\!1}}\!\;\!(\>\!\mbox{\small$T$}),\quad\;\;\,\forall\;\,0\>\!\leq\>\!t\>\!\leq\>\!\mbox{\small$T$}\!\;\!,$$

$$\|\,u(\cdot,t)\,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R}^{n})}}\leq\,\mbox{\small$M$}_{\mbox{}_{\!\infty}}\!\;\!(\>\!\mbox{\small$T$}),\quad\;\;\forall\;\,0\>\!\leq\>\!t\>\!\leq\>\!\mbox{\small$T$}\!\;\!,$$

$${\int }_{\text{mbox}{}_0}^T{\int }_{\text{mbox}{}_{{\mathbb{R}}^n}}|\nabla u(x,t){|}^{p}dxdt<\infty$$

$$\Vert u(\cdot ,t)-v(\cdot ,t){\Vert }_{\text{mbox}{}_{L^1({\mathbb{R}}^n)}}\le \Vert u(\cdot ,0)-v(\cdot ,0){\Vert }_{\text{mbox}{}_{L^1({\mathbb{R}}^n)}}$$

$$|\,\mbox{\boldmath$f$}(x,t,\mbox{u})-\mbox{\boldmath$f$}(x,t,\mbox{v})\,|\;\leq\>\mbox{\small$K$}_{\!f}(\>\!\mbox{\small$M$}\!\>\!,\;\!\mbox{\small$T$})\,|\>\mbox{u}-\mbox{v}\>|^{\>\!1\,-\,\frac{\scriptstyle 1}{\scriptstyle p}}$$

$$|\,\mbox{\boldmath$g$}(t,\mbox{u})-\mbox{\boldmath$g$}(t,\mbox{v})\,|\;\leq\>\mbox{\small$K$}_{\!\>\!g}(\>\!\mbox{\small$M$}\!\>\!,\;\!\mbox{\small$T$})\,|\>\mbox{u}-\mbox{v}\>|^{\>\!1\,-\,\frac{\scriptstyle 1}{\scriptstyle p}}$$

$$v_{\text{mbox}{}_h}(\cdot ,t):=\frac{1}{h}{\int }_t^{t+h}\tilde{v}(\cdot ,\tau )d\tau ,t\in I,$$

$${\int }_{\text{mbox}{}_0}^T{\int }_{\text{mbox}{}_{{\mathbb{R}}^n}}|\nabla u(x,t){|}^{p}dxdt<\infty$$

$$\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}(x)\;=\;\!\;\!\Bigl{\{}\;\!e^{\!-\,\epsilon\;\!\,\sqrt{\>\!1\,+\,|\;\!x\;\!|^{2}\;\!}}-\;\!\;\!e^{\!-\,\epsilon\;\!\sqrt{\;\!1\;\!+\;\!R^{\>\!2}\>\!}}\;\!\Bigr{\}}^{\!\>\!p}\qquad\,\mbox{if }\;|\;\!x\;\!|\;\!<\>\!\mbox{\footnotesize$R$}$$

$$\mbox{\small${\displaystyle\frac{d}{d\>\!t}}$}\;\|\,u(\cdot,t)\,\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}^{\>\!2}\!\;\!+\>2\,\mu(t)\!\!\;\!\int_{\mathbb{R}^{n}}\!\!|\,\nabla u\,|^{\>\!p}\>dx\;\leq\;2\,\mbox{\small$F$}(t)\!\!\;\!\int_{\mathbb{R}^{n}}\!\!|\,u(x,t)\,|^{\>\!\kappa\>\!+\>\!1}\,|\,\nabla u\,|\>dx$$

$$\mbox{\small${\displaystyle\frac{d}{d\>\!t}}$}\;\|\,u(\cdot,t)\,\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{n})}}^{\>\!q}+\>q\;\!(q-1)\,\mu(t)\!\!\;\!\int_{\mathbb{R}^{n}}\!\!|\,u(x,t)\,|^{\>\!q\>\!-\>\!2}\,|\,\nabla u\,|^{\>\!p}\>dx$$

$$\leq\;q\;\!(q-1)\,\mbox{\small$F$}(t)\!\!\;\!\int_{\mathbb{R}^{n}}\!\!|\,u(x,t)\,|^{\>\!q\>\!-\>\!1\>\!+\>\!\kappa}\,|\,\nabla u\,|\;dx$$

$$\Vert u(\cdot ,t){\Vert }_{\text{mbox}{}_{L^q({\mathbb{R}}^n)}}^q+q(q-1){\int }_{\text{mbox}{}_{t_{\text{mbox}{}_0}}}^t\mu (\tau ){\int }_{\text{mbox}{}_{{\mathbb{R}}^n}}|u{|}^{q-2}|\nabla u{|}^{p}dxd\tau$$

$$=\;\|\,u(\cdot,t_{0})\,\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{n})}}^{\>\!q}\!\;\!+\>q\,(q-1)\!\int_{\mbox{}_{\scriptstyle t_{\mbox{}_{0}}}}^{\>\!t}\!\;\!\int_{\mbox{}_{\scriptstyle\mathbb{R}^{n}}}\!\!|\,u\,|^{\>\!q-2}\,\langle\,\mbox{\boldmath$f$}(x,\tau,u),\;\!\nabla u\,\rangle\;dx\,d\tau,$$

$$|\;\!\;\!\mbox{\boldmath$g$}(t,\mbox{u})\,|\;\!\;\!\leq\,C(\mbox{\small$T$})\;\!\;\!|\,\mbox{u}\,|^{\mbox{}^{\scriptstyle 1\,-\,\frac{\scriptstyle 1}{\scriptstyle p}}}\qquad\forall\;\;|\,\mbox{u}\,|\>\!\ll\>\!1,\;\,0<t<\mbox{\small$T$}\!\;\!.$$

$$\mbox{}\hskip 28.45274pt\|\,u(\cdot,t)\,\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R}^{n})}}\leq\;\|\,u_{0}\;\!\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R}^{n})}}\!\>\!,\qquad\;\,\forall\;\;0<t<\mbox{\small$T$}_{\!\ast}$$

$$\int_{\mbox{}_{\scriptstyle\mathbb{R}^{n}}}\!\!\!\!\;\!L_{\delta}(u(x,t))\,dx\;\>\!\leq\;\|\,u_{0}\;\!\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R}^{n})}}\>\!+\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\mbox{\footnotesize$\>\!t$}}\!\!\;\!\mu(\tau)^{\mbox{}^{\scriptstyle\!\!-\,\frac{\scriptstyle 1}{\scriptstyle p\,-\;\!1}}}\!\!\!\>\!\int_{\mbox{}_{\scriptstyle\mathbb{R}^{n}}}\!\!\!\!\;\!L_{\delta}^{\prime\prime}(u(x,\tau))\;\!\;\!|\,\mbox{\boldmath$f$}(x,\tau,u)\,|^{\mbox{}^{\scriptstyle\>\!\frac{\scriptstyle p}{\scriptstyle p\,-\;\!1}}}\;\!dx\>d\tau,$$

$$\int_{\mbox{}_{\scriptstyle\!B_{\mbox{}_{\!\;\!R}}}}\!\!\!\!\;\!|\,u(x,t)\,|\;\zeta_{\mbox{}_{R}}\!\;\!(x)\;\!\;\!dx\;\>\!\leq\;\>\!\|\,u_{0}\;\!\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R}^{n})}}\;\!+\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\mbox{\footnotesize$\>\!t$}}\!\!\!\;\!\;\!\mu(\tau)\!\;\!\int_{\mbox{}_{\scriptstyle\!B_{\mbox{}_{\!\;\!R}}}}\!\!\!\!\;\!|\,\nabla u(x,\tau)\,|^{\mbox{}^{\scriptstyle\>\!p\;\!-\>\!1}}\;\!|\,\nabla\zeta_{\mbox{}_{R}}\!\;\!(x)\,|\;dx\;\!\;\!d\tau$$

$$\int_{\mbox{}_{\scriptstyle\mathbb{R}^{n}}}\!\!\!\>\!u(x,t)\>dx\;=\;\!\int_{\mbox{}_{\scriptstyle\mathbb{R}^{n}}}\!\!\!\>\!u_{0}(x)\,dx,\qquad\;\,\forall\;\,0\>\!<\>\!t\>\!<\>\!\mbox{\small$T$}_{\!\ast}.$$

$$\int_{\mbox{}_{\scriptstyle\!R/2\,<\,|\,x\,|\,<\,R\;\!+\;\!2\;\!S}}|\,u(x,t)\,|\,\zeta_{\mbox{}_{\scriptstyle R,\>\!S}}(x)\;\!\;\!dx\;\leq\;\!\int_{\mbox{}_{\scriptstyle\!\;\!|\,x\,|\,>\,R/2}}|\,u_{0}(x)\,|\;\!\;\!dx\;+\;\mbox{\footnotesize$I$}(\mbox{\footnotesize$R$},\,\!\mbox{\footnotesize$S$})\;\!\;\!+\,\mbox{\footnotesize$J$}_{\mbox{}_{\scriptstyle\!\>\!1}}\!\;\!(\mbox{\footnotesize$R$})\;\!\;\!+\,\mbox{\footnotesize$J$}_{\mbox{}_{\scriptstyle\!\,\!2}}\!\;\!(\mbox{\footnotesize$R$},\,\!\mbox{\footnotesize$S$})\;\!\;\!+\,\mbox{\footnotesize$H$}_{\mbox{}_{\scriptstyle\!\>\!1}}\!\,\!(\mbox{\footnotesize$R$})\;\!\;\!+\,\mbox{\footnotesize$H$}_{\mbox{}_{\scriptstyle\!2}}\!\;\!(\mbox{\footnotesize$R$},\,\!\mbox{\footnotesize$S$})$$

$$\mbox{\footnotesize$I$}(\mbox{\footnotesize$R$},\,\!\mbox{\footnotesize$S$})\;=\;\!\int_{\mbox{}_{\scriptstyle 0}}^{\>\!T}\!\!\!\mbox{\footnotesize$F$}(\tau)\!\!\;\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!R/2\,<\,|\,x\,|\,<\,R\;\!+\;\!2\;\!S}}|\,u(x,\tau)\,|^{\>\!\kappa\;\!+\;\!1}\;\!\;\!|\,\nabla\zeta_{\mbox{}_{\scriptstyle R,\>\!S}}(x)\,|\;dx\,d\tau,$$

$$\mbox{\footnotesize$J$}_{\mbox{}_{\scriptstyle\!\>\!1}}\!\>\!(\mbox{\footnotesize$R$})\;=\;\!\int_{\mbox{}_{\scriptstyle 0}}^{\>\!T}\!\!\!\mu(\tau)\!\!\;\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!R/2\,<\,|\,x\,|\,<\,R}}|\,\nabla u\,|^{\>\!p\;\!-\;\!1}\;\!\;\!|\,\nabla\zeta_{\mbox{}_{\scriptstyle R,\>\!S}}(x)\,|\;dx\,d\tau,$$

$$\mbox{\footnotesize$J$}_{\mbox{}_{\scriptstyle\!\>\!2}}\!\;\!(\mbox{\footnotesize$R$},\,\!\mbox{\footnotesize$S$})\;=\;\!\int_{\mbox{}_{\scriptstyle 0}}^{\>\!T}\!\!\!\mu(\tau)\!\!\;\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!R\;\!+\;\!S\,<\,|\,x\,|\,<\,R\;\!+\;\!2\;\!S}}|\,\nabla u\,|^{\>\!p\;\!-\;\!1}\;\!\;\!|\,\nabla\zeta_{\mbox{}_{\scriptstyle R,\>\!S}}(x)\,|\;dx\,d\tau,$$

$$\mbox{\footnotesize$H$}_{\mbox{}_{\scriptstyle\!\>\!1}}\!\>\!(\mbox{\footnotesize$R$})\;=\;\!\int_{\mbox{}_{\scriptstyle 0}}^{\>\!T}\!\!\;\!\>\!\int_{\mbox{}_{\scriptstyle\!\>\!R/2\,<\,|\,x\,|\,<\,R}}|\,\mbox{\boldmath$g$}(\tau,u)\,|\;|\,\nabla\zeta_{\mbox{}_{\scriptstyle R,\>\!S}}(x)\,|\;dx\,d\tau,$$

$$\mbox{\footnotesize$H$}_{\mbox{}_{\scriptstyle\!\>\!2}}\!\;\!(\mbox{\footnotesize$R$},\,\!\mbox{\footnotesize$S$})\;=\;\!\int_{\mbox{}_{\scriptstyle 0}}^{\>\!T}\!\!\;\!\>\!\int_{\mbox{}_{\scriptstyle\!\>\!R\;\!+\;\!S\,<\,|\,x\,|\,<\,R\;\!+\;\!2\;\!S}}|\,\mbox{\boldmath$g$}(\tau,u)\,|\;|\,\nabla\zeta_{\mbox{}_{\scriptstyle R,\>\!S}}(x)\,|\;dx\,d\tau.$$

$$\mbox{\footnotesize$J$}_{\mbox{}_{\scriptstyle\!\>\!1}}\!\;\!(\mbox{\footnotesize$R$})\;\>\!\leq\;\;\!\epsilon\!\>\!\int_{\mbox{}_{\scriptstyle 0}}^{\>\!T}\!\!\!\mu(\tau)\!\!\;\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!R/2\,<\,|\,x\,|\,<\,R}}|\,\nabla\zeta_{\mbox{}_{\scriptstyle R,\>\!S}}(x)\,|^{\>\!p}\;dx\,d\tau\;\>\!+\;\>\!\epsilon^{\scriptstyle\!\>\!-\,\frac{\scriptstyle 1}{\scriptstyle\;\!p\;\!-\;\!1\>\!}}\!\!\>\!\int_{\mbox{}_{\scriptstyle 0}}^{\>\!T}\!\!\!\mu(\tau)\!\!\;\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!R/2\,<\,|\,x\,|\,<\,R}}|\,\nabla u\,|^{\>\!p}\;\!\;\!dx\,d\tau,$$

$$\mbox{}\hskip 28.45274pt\int_{\mbox{}_{\scriptstyle\!\>\!|\,x\,|\,>\,R}}|\,u(x,t)\,|\;\!\;\!dx\;\leq\;\;\!\epsilon\,+\;\!\;\!\mbox{\footnotesize$K$}_{\mbox{}_{\scriptstyle\!\>\!n}}\;\!\epsilon\;\!\;\!\Bigl{\{}\;\!1+\!\int_{\mbox{}_{\scriptstyle 0}}^{\>\!T}\!\!\!\mu(\tau)\;\!\;\!d\tau\Bigr{\}}\;\;\qquad\forall\;\,0\>\!<\>\!t\>\!\leq\>\!\mbox{\footnotesize$T$}\!\;\!.$$

Full text

Some basic properties of bounded solutions

of parabolic equations with $p$-Laplacian diffusion

J. Q. Chagas,$\mbox{}^{\!\>\!1}$ P. L. Guidolin$\mbox{}^{\;\!2}$ and J. P. Zingano$\mbox{}^{\;\!3}$

$\mbox{}^{1}$Departamento de Matemática e Estatística

Universidade Estadual de Ponta Grossa

Ponta Grossa, PR 84030-900, Brazil

$\mbox{}^{2}$Instituto Federal de Educação, Ciência e Tecnologia

Farroupilha, RS 95180-000, Brazil

$\mbox{}^{3}$Departamento de Matemática Pura e Aplicada

Universidade Federal do Rio Grande do Sul

Porto Alegre, RS 91509-900, Brazil

Abstract

We provide a detailed derivation of several fundamental properties of bounded weak solutions to initial value problems for general conservative 2nd-order parabolic equations with $p$-Laplacian diffusion and arbitrary initial data $\;\!u_{0}\!\!\;\!\;\!\in L^{1}(\mathbb{R}^{n})\cap\!\;\!\;\!L^{\infty}(\mathbb{R}^{n})$.

1. Introduction

In this work, we provide a detailed derivation of several fundamental properties of (bounded, weak) solutions of the initial value problem for evolution $p$-Laplacian equations of the type

[TABLE]

[TABLE]

Here, $p>2$ is constant, $\mu\in C^{0}(\>\![\;\!0,\infty)\,\!)$ is assumed to be positive everywhere, and \mbox{\boldmathf}\!\>\!=(\>\!f_{\mbox{}_{1}}\!\>\!,f_{\mbox{}_{2}}\!\;\!,\!\;\!...\>\!,f_{\mbox{}_{\scriptstyle n}}), \mbox{\boldmathg}\!\>\!=(\;\!g_{\mbox{}_{1}}\!\!\;\!\;\!,g_{\mbox{}_{2}}\!\;\!,\!\;\!...\>\!,g_{\mbox{}_{\scriptstyle n}}) are given continuous fields such that \mbox{\boldmathg}(t,0)={\bf 0} for all $\>\!t\geq 0\>\!$ and with  $f$  satisfying the growth condition

[TABLE]

for some \mbox{\smallF}\in C^{0}(\!\;\!\;\![\;\!0,\infty)\,\!) and some constant $\>\!\kappa\geq 0$, where $\>\!|\!\;\!\;\!\cdot\!\;\!\;\!|$ denotes the absolute value (in case of scalars) or the Euclidean norm (in case of vectors), as in (1.1$a$). By a (bounded) solution of (1.1) in some time interval [\;\!0,\;\!\mbox{\smallT}_{\!\ast}) we mean any function u(\cdot,t)\in C^{0}([\;\!0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!L^{1}_{\mbox{\scriptsize loc}}(\mathbb{R}^{n}))\!\;\!\;\!\cap\!\;\!\;\!L^{p}_{\mbox{\scriptsize loc}}((\!\;\!\;\!0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!W^{1,\,p}_{\mbox{\scriptsize loc}}(\mathbb{R}^{n})) satisfying the equation (1.1$a$) in {\cal D}^{\;\!\prime}(\;\!\mathbb{R}^{n}\!\times\!\!\;\!\;\!(\;\!0,\;\!\mbox{\smallT}_{\!\ast})\,\!) with ${\displaystyle\>\!u(\cdot,0)=\>\!u_{0}\,\!}$ and {\displaystyle\;\!u(\cdot,t)\in L^{\infty}_{\mbox{\scriptsize loc}}([\;\!0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!L^{1}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n}))} — that is, for every \;\!0<\mbox{\smallT}\!\>\!<\mbox{\smallT}_{\!\ast} given, we have

[TABLE]

[TABLE]

for suitable bounds {\displaystyle\mbox{\smallM}_{\mbox{}_{\!1}}\!\;\!(\>\!\mbox{\smallT}),\;\!\mbox{\smallM}_{\mbox{}_{\!\infty}}\!\;\!(\>\!\mbox{\smallT})} depending on $T$ ​​ ​ ​(and the solution $u$ considered). For the local (in time) existence of such solutions, see e.g. [10, 11, 14, 15, 16], while, for global existence, [4, 10] can be consulted. Our main objective in this work is to provide a complete, rigorous derivation of important fundamental properties possessed by the solutions, following the lines of [2, 3, 4, 6, 9, 10, 14]. Thus, for example, in Section 2 we show that

[TABLE]

for every {\displaystyle\>\!0<\mbox{\smallT}<\mbox{\smallT}_{\!\ast}\!\;\!}, so that {\displaystyle\>\!u(\cdot,t)\in L^{p}_{\mbox{\scriptsize loc}}(\>\![\;\!0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!W^{1,\,p}(\mathbb{R}^{n})\,\!)}, along with the monotonicity of ${\displaystyle\;\!\|\,u(\cdot,t)\,\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R}^{n})}}}$ and other basic results. In Section 3, solutions are shown to contract in $L^{1}(\mathbb{R}^{n})$, so that we have

[TABLE]

for any given solution pair $\;\!u,\;\!v$, and any $t>0$ for which both solutions are defined, provided that the flux functions \mbox{\boldmathf}\!,\;\!\mbox{\boldmathg} in the equation (1.1$a$) above satisfy additional conditions, which include

[TABLE]

[TABLE]

for all $\;\!x\in\mathbb{R}^{n}\!\;\!$, \;\!0\leq t\leq\mbox{\smallT}\!\;\!, \;\!|\,\mbox{u}\,|\leq\mbox{\smallM}\!\>\!, \;\!|\,\mbox{v}\,|\leq\mbox{\smallM}\!\>\!, for each given \mbox{\smallM}>0, \mbox{\smallT}>0, where the Lipschitz constants {\displaystyle\mbox{\smallK}_{\!f}(\>\!\mbox{\smallM}\!\>\!,\;\!\mbox{\smallT}),\;\!\mbox{\smallK}_{\!\>\!g}(\>\!\mbox{\smallM}\!\>\!,\;\!\mbox{\smallT})} may depend upon the values of  $M$ , $T$ (see Section 3 for further details). Also, under such extra assumptions, the solutions are shown to obey a familiar comparison principle, as expected for 2nd-order parabolic problems. From this, it follows in particular that solutions are uniquely defined by their initial data, which is not necessarily the situation in Section 2.

2. Some fundamental basic properties

We begin by recalling an important regularization technique [6, 13, 14]: given an interval $I\subseteq\mathbb{R}$ (arbitrary), $\>\!h>0$ (small), and some function $\>\!v(\cdot,t)\in L^{r}(\>\!I\!\;\!,\!\;\!\;\!L^{q}_{\mbox{\scriptsize loc}}(\!\;\!\;\!\mathbb{R}^{n}))$, where $\>\!q,\>\!r\in[\;\!1,\infty\>\!]$, let $v_{h}(\cdot,t)\in C^{0}(\>\!I\!\>\!,\>\!L^{q}_{\mbox{\scriptsize loc}}(\>\!\mathbb{R}^{n}))$ be the Steklov average

[TABLE]

where $\!\;\!\;\!\tilde{v}(\cdot,\tau)=v(\cdot,\tau)$ if $\tau\!\in I\!\;\!$, $\!\;\!\;\!\tilde{v}(\cdot,\tau)=0\!\;\!\;\!$ if $\tau\!\notin I\!\;\!$. For {\displaystyle\;\!u(\cdot,t)\in C^{0}([\;\!0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!L^{1}_{\mbox{\scriptsize loc}}(\mathbb{R}^{n}))} {\displaystyle\cap\;\!L^{p}_{\mbox{\scriptsize loc}}((\>\!0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!W^{1,\,p}_{\mbox{\scriptsize loc}}(\mathbb{R}^{n}))} solution of (1.1), we then obtain (see [6], Ch. II ​; [14], Ch. 1) that, for any ball \;\!\mbox{\smallB}_{\mbox{}_{\!R}}\!\>\!=\{\;\!x\in\mathbb{R}^{n}\!\!\;\!:\;\!|\;\!x\;\!|<\!\;\!\mbox{\smallR}\;\!\}\>\!:

          {\displaystyle\int_{\mbox{}_{\scriptstyle\!\>\!B_{\mbox{}_{\!\>\!R}}}}\Bigl{\{}\,u_{\mbox{}_{\scriptstyle h,\,t}}\!\>\!(x,t)\,\phi(x)\;\!+\;\!\langle\;\!\bigl{[}\,\mu(t)\,|\;\!\nabla u\,|^{\>\!p-2}\,\nabla u\,\bigr{]}_{\!\;\!h},\nabla\phi\;\!\;\!\rangle\,\Bigr{\}}\;\!\;\!dx\;=}

(2.2)

                             {\displaystyle=\int_{\mbox{}_{\scriptstyle\!\>\!B_{\mbox{}_{\!\>\!R}}}}\Bigl{\{}\;\!\langle\,\bigl{[}\;\!\mbox{\boldmathf}(x,t,u)\,\bigr{]}_{\!\;\!h},\>\!\nabla\phi\;\!\;\!\rangle\,+\;\!\langle\,\bigl{[}\;\!\mbox{\boldmathg}(t,u)\,\bigr{]}_{\!\;\!h},\>\!\nabla\phi\;\!\;\!\rangle\;\!\Bigr{\}}\;\!\;\!dx}

for all \;\!0<t<\mbox{\smallT}_{\!\ast}\!\;\!-h,  ​and any \>\!\phi\in W^{1,\,p}_{0}(\mbox{\smallB}_{\mbox{}_{\!\;\!R}})\cap L^{\infty}(\mbox{\smallB}_{\mbox{}_{\!\;\!R}}), where \;\!u_{\mbox{}_{\scriptstyle h,\,t}}\!\;\!(\cdot,t)=\!\;\!\;\!\mbox{\footnotesize{\displaystyle\frac{\partial}{\partial>!t}}}\,u_{\mbox{}_{\scriptstyle h}}\!(\cdot,t) $=\!\;\![\;\!u(\cdot,t+h)-u(\cdot,t)\>\!]\!\;\!\;\!/\!\;\!\;\!h$ is the strong pointwise derivative of $u_{\mbox{}_{\scriptstyle h}}\!(\cdot,t)$ in \mbox{\smallL}^{\!\;\!1}(\mbox{\footnotesizeB}_{\mbox{}_{\!R}}\!\;\!), and where $\langle\,\cdot\;\!,\>\!\cdot\,\rangle$ denotes the standard inner product of a pair of $n$-dimensional vectors. As in [6, 13, 14], the expression (2.2) is a very useful starting point for the derivation of a number of important solution properties, as illustrated by the following results.

Proposition 2.1. *Let {\displaystyle\;\!u(\cdot,t)\in C^{0}(\,\![\,0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!L^{1}_{\mbox{\scriptsize\em loc}}(\mathbb{R}^{n})\,\!)\!\;\!\;\!\cap\!\;\!\;\!L^{p}_{\mbox{\scriptsize\em loc}}(\,\!(\!\;\!\;\!0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!W^{1,\,p}_{\mbox{\scriptsize\em loc}}(\mathbb{R}^{n})\,\!)\;\cap}

{\displaystyle L^{\infty}_{\mbox{\scriptsize\em loc}}(\>\![\;\!0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!L^{1}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n}))} be any given solution to the problem $\;\!(1.1)$, $(1.2)$,

where $\,\kappa\geq 0$. Then *

[TABLE]

*for every \;\!0<\mbox{\smallT}\!\;\!<\mbox{\smallT}_{\!\ast}, so that {\displaystyle\;\!u(\cdot,t)\in L^{p}_{\mbox{\scriptsize\em loc}}(\>\![\;\!0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!W^{1,\,p}(\mathbb{R}^{n})\,\!)}.

**Proof. Let 0<t_{0}<\mbox{\footnotesizeT}. ​Given \mbox{\footnotesizeR}>0, $\epsilon>0$, let $\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}\!\in C^{2}(\mathbb{R}^{n})$ be the cut-off function

**

[TABLE]

and ${\displaystyle\;\!\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}(x)\;\!=\;\!0\;\!}$ if \;\!|\;\!x\;\!|\geq\mbox{\footnotesizeR}. Taking ${\displaystyle\;\!\phi(x)\>\!=\;\!2\,u_{\mbox{}_{\scriptstyle h}}\!(x,t)\,\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}\!\;\!(x)}$ in (2.2) above, integrating the resulting equation in (\!\;\!\!\;\!\;\!t_{0},\!\;\!\;\!\mbox{\footnotesizeT}), and letting $h\;\!{\scriptstyle\searrow}\;\!0$, we get, letting (as always) {\displaystyle\!\;\!\mbox{\footnotesizeB}_{\mbox{}_{\!\>\!R}}\!\!\;\!\;\!} denote the ball {\displaystyle\!\;\!\;\!\bigl{\{}\;\!x\in\mathbb{R}^{n}\!\!\;\!:\!\;\!\;\!|\;\!x\;\!|<\mbox{\footnotesizeR}\;\!\bigr{\}}}, and setting {\displaystyle\!\;\!\;\!\mbox{\small\boldmath\tilde{f}}\!\!\;\!\;\!:=\mbox{\small\boldmathf}+\mbox{\small\boldmathg}} ​:

{\displaystyle\int_{\mbox{}_{\scriptstyle\!\>\!B_{\mbox{}_{\!\>\!R}}}}\!\!\!\!\;\!u(x,\mbox{\footnotesizeT})^{2}\,\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}\!\;\!(x)\;\!\;\!dx\;+\;2\!\int_{\mbox{}_{\scriptstyle\!\>\!t_{\mbox{}_{0}}}}^{\>\!T}\!\!\!\mu(t)\!\int_{\mbox{}_{\scriptstyle\!\>\!B_{\mbox{}_{\!\>\!R}}}}\!\!\!\!\;\!|\,\nabla u\,|^{\>\!p}\,\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}\!\;\!(x)\;\!\;\!dx\,dt\;\!\;\!\;\!=}

          ${\displaystyle=\;\!\int_{\mbox{}_{\scriptstyle\!\>\!B_{\mbox{}_{\!\>\!R}}}}\!\!\!\!\;\!u(x,t_{0})^{2}\,\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}\!\;\!(x)\;\!\;\!dx\;-\;2\!\>\!\int_{\mbox{}_{\scriptstyle\!\>\!t_{\mbox{}_{0}}}}^{\>\!T}\!\!\!\mu(t)\!\int_{\mbox{}_{\scriptstyle\!\>\!B_{\mbox{}_{\!\>\!R}}}}\!\!\!\!\;\!u(x,t)\,|\,\nabla u\,|^{\>\!p-2}\;\!\langle\,\nabla u,\!\;\!\;\!\nabla\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}\!\;\!(x)\;\!\rangle\;\!\;\!dx\,dt}$

{\displaystyle+\;\!\;\!\;\!2\!\>\!\int_{\mbox{}_{\scriptstyle\!\>\!t_{\mbox{}_{0}}}}^{\>\!T}\!\!\int_{\mbox{}_{\scriptstyle\!\>\!B_{\mbox{}_{\!\>\!R}}}}\!\!\!\!\;\!\langle\,\mbox{\boldmath\tilde{f}}(x,t,u),\!\;\!\;\!\nabla u\;\!\rangle\,\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}\!\;\!(x)\;\!\;\!dx\,dt\;+\;2\!\>\!\int_{\mbox{}_{\scriptstyle\!\>\!t_{\mbox{}_{0}}}}^{\>\!T}\!\!\!\int_{\mbox{}_{\scriptstyle\!\>\!B_{\mbox{}_{\!\>\!R}}}}\!\!\!\!\;\!u(x,t)\;\!\langle\,\mbox{\boldmath\tilde{f}}(x,t,u),\!\;\!\;\!\nabla\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}\!\;\!(x)\;\!\rangle\;\!\;\!dx\,dt}

{\displaystyle\leq\;\!\;\!\mbox{\footnotesizeM}_{\!\>\!1}(\mbox{\footnotesizeT})\,\mbox{\footnotesizeM}_{\!\;\!\infty}(\mbox{\footnotesizeT})\!\;\!\;\!+\!\!\;\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!t_{\mbox{}_{0}}}}^{\>\!T}\!\!\!\mu(t)\!\int_{\mbox{}_{\scriptstyle\!\>\!B_{\mbox{}_{\!\>\!R}}}}\!\!\!\!\;\!|\,\nabla u\,|^{\>\!p}\;\!\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}\!\;\!(x)\;\!\;\!dx\,dt\;\!\;\!+\;\!\;\!\mbox{\small{\displaystyle\frac{,2^{>!p}}{p}}}\!\int_{\mbox{}_{\scriptstyle\!\>\!t_{\mbox{}_{0}}}}^{\>\!T}\!\!\!\mu(t)\!\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!B_{\mbox{}_{\!\>\!R}}}}\!\!\!\!\;\!|\,u\,|^{\>\!p}\,\frac{\,|\;\!\nabla\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}\;\!|^{\>\!p}}{\;\mbox{\small{\displaystyle\zeta_{\mbox{}_{!>!R,;!{\scriptstyle\epsilon}}}^{;!p;!->!1}}}}\;\!\;\!dx\,dt}

{\displaystyle+\;\;\!2\!\>\!\int_{\mbox{}_{\scriptstyle\!\>\!t_{\mbox{}_{0}}}}^{\>\!T}\!\!\!\mbox{\footnotesizeF}(t)^{\mbox{}^{\scriptstyle\!\;\!\frac{\scriptstyle p}{\scriptstyle p\;\!-\>\!1}}}\!\;\!\mu(t)^{\mbox{}^{\scriptstyle\!\!\!-\,\frac{\scriptstyle 1}{\scriptstyle p\;\!-\>\!1}}}\!\!\!\!\;\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!B_{\mbox{}_{\!\>\!R}}}}\!\!\!\!\;\!|\,u\,|^{\mbox{}^{\scriptstyle\!\;\!(1\>\!+\;\!\kappa)\,\frac{\scriptstyle p}{\scriptstyle p\;\!-\>\!1}}}\!\;\!\;\!\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}\!\;\!\;\!dx\,dt\;\!\;\!+\;\!\!\;\!\;\!2\!\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!t_{\mbox{}_{0}}}}^{\>\!T}\!\!\!\!\;\!\mbox{\footnotesizeF}(t)\!\int_{\mbox{}_{\scriptstyle\!\>\!B_{\mbox{}_{\!\>\!R}}}}\!\!\!\!\;\!|\,u\,|^{\mbox{}^{\scriptstyle\>\!2\;\!+\;\!\kappa}}\,|\,\nabla\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}\;\!|\;\!\;\!dx\,dt}

{\displaystyle+\;\,4\;\mbox{\footnotesizeG}(\mbox{\footnotesizeT})\!\int_{\mbox{}_{\scriptstyle\!\>\!t_{\mbox{}_{0}}}}^{\>\!T}\!\!\int_{\mbox{}_{\scriptstyle\!\>\!B_{\mbox{}_{\!\>\!R}}}}\!\!\!\!\;\!\;\!|\,u(x,t)\,|\;|\,\nabla\zeta_{\mbox{}_{R,\;\!{\scriptstyle\epsilon}}}\;\!|\;dx\,dt}

by (1.2), (1.3) and Young’s inequality (see e.g. [8], p. 622), where \;\!\mbox{\footnotesizeM}_{\mbox{}_{\!1}}\!\;\!, \!\;\!\mbox{\footnotesizeM}_{\mbox{}_{\!\infty}}\!\;\! are given in (1.3) and {\displaystyle\!\;\!\;\!\mbox{\footnotesizeG}(\mbox{\footnotesizeT})\!\;\!\;\!=\;\!\sup\;\{\;\!\;\!|\;\!\;\!\mbox{\boldmathg}(t,\mbox{\small v})\,|\!\!\;\!\;\!:\,0<t<\mbox{\footnotesizeT},\;|\,\mbox{\small v}\,|\!\;\!\;\!<\!\;\!\;\!\mbox{\footnotesizeM}_{\mbox{}_{\!\infty}}\!\;\!(\mbox{\footnotesizeT})\;\!\}}. Letting \mbox{\footnotesizeR}\;\!\mbox{\scriptsize\nearrow}\,\infty, \epsilon\,\mbox{\scriptsize\searrow}\,0 and t_{0}\!\;\!\;\!\mbox{\scriptsize\searrow}\,0 (in this order), we then obtain, by (1.3$a$) and since {\displaystyle\;\!|\,\nabla\zeta_{\mbox{}_{\!\;\!R,\;\!{\scriptstyle\epsilon}}}\>\!|^{\>\!p}/\;\!\zeta_{\mbox{}_{\!\;\!R,\;\!{\scriptstyle\epsilon}}}^{\;\!p\;\!-\>\!1}\!\;\!\leq\bigl{(}\>\!p\,\epsilon\>\!\bigr{)}^{\scriptstyle p}\;\!e^{\scriptstyle-\,p\;\!\epsilon\;\!\sqrt{\>\!1\,+\,|\;\!x\;\!|^{\>\!2}\;\!}}\!},

{\displaystyle\|\,u(\cdot,\mbox{\footnotesizeT})\,\|_{\mbox{}_{\scriptstyle L^{2}(\mathbb{R}^{n})}}^{\>\!2}+\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!0}}^{\>\!T}\!\!\!\mu(t)\!\int_{\mbox{}_{\scriptstyle\!\;\!\mathbb{R}^{n}}}\!\!\!|\,\nabla u\,|^{\>\!p}\,dx\,dt\;\leq\;\mbox{\footnotesizeM}_{\mbox{}_{\!\infty}}\!\!\;\!\;\!(\mbox{\footnotesizeT})\,\|\,u_{0}\;\!\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R}^{n})}}\!\;\!\>\!+\!\!\;\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!0}}^{\>\!T}\!\!\!\!\;\!\mbox{w}(t)\,\|\,u(\cdot,t)\,\|_{{\scriptstyle L^{q^{\prime}}\!(\mathbb{R}^{n})}}^{\>\!q^{\prime}}dt}

where {\displaystyle\,\mbox{w}(t)\!\;\!\;\!=\!\;\!\;\!2\;\!\!\;\!\;\!\mbox{\footnotesizeF}(t)^{\mbox{}^{\scriptstyle\!\;\!\frac{\scriptstyle p}{\scriptstyle p\;\!-\>\!1}}}\!\!\;\!\;\!\mu(t)^{\mbox{}^{\scriptstyle\!\!\!-\,\frac{\scriptstyle 1}{\scriptstyle p\;\!-\>\!1}}}\!} and ${\displaystyle\>\!q^{\prime}\!\>\!=\;\!(1+\kappa)\;\!p\>\!/(p-1)}$. This shows (2.3). $\Box$

The next result gives one form of the basic energy inequalities that can be obtained for weak solutions {\displaystyle\;\!u(\cdot,t)\in C^{0}([\;\!0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!L^{1}_{\mbox{\scriptsize loc}}(\mathbb{R}^{n}))\>\!\cap\>\!L^{p}_{\mbox{\scriptsize loc}}((\!\;\!\;\!0,\;\!\mbox{\smallT}_{\!\ast}),\>\!W^{1,\,p}_{\mbox{\scriptsize loc}}(\mathbb{R}^{n}))\;\!\;\!\cap} {\displaystyle L^{{}^{\infty}}_{\mbox{\scriptsize loc}}([\;\!0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!L^{1}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n}))} of problem (1.1), (1.2), which plays a key role in [4].

Proposition 2.2. *​​ ​Under the same assumptions of  Proposition 2.1​ ​ above, we have, for each $\!\;\!\;\!q\geq 2$, that ${\displaystyle\,\|\,u(\cdot,t)\,\|_{\scriptstyle L^{q}(\mathbb{R}^{n})}^{\>\!q}\!\>\!}$ is absolutely continuous in \;\!t\in(\!\;\!\;\!0,\!\;\!\;\!\mbox{\smallT}_{\!\ast}). ​ ​Moreover, there exists \!\;\!\;\!E_{q}\subset(\>\!0,\mbox{\smallT}_{\!\ast}) with zero Lebesgue measure such that *

[TABLE]

*for all {\displaystyle\,t\in(\>\!0,\>\!\mbox{\smallT}_{\!\ast})\!\;\!\setminus\!\;\!E_{2}\!\;\!} $($if $\>\!q=2\>\!)$, and *

[TABLE]

(2.5)

[TABLE]

*for all {\displaystyle\,t\in(\>\!0,\>\!\mbox{\smallT}_{\!\ast})\!\;\!\setminus\!\;\!E_{q}\!\;\!} $($if $\>\!q>2\>\!)$, where \mbox{\smallF}(t)\!\;\! is given in $(1.2)\!\;\!$ above.

**Proof. Given \>\!0<t_{0}<t<\mbox{\footnotesizeT}_{\!\!\;\!\;\!\ast}, \mbox{\footnotesizeR}>0, let \>\!\zeta_{\mbox{}_{R}}\!\;\!(x)=\zeta(x/\mbox{\footnotesizeR}), where $\>\!\zeta\in C^{1}(\mathbb{R}^{n})$ is such that $\zeta(x)=1\>\!$ if $\>\!|\>\!x\>\!|\leq 1/2$, $\zeta(x)=0\>\!$ if $\>\!|\>\!x\>\!|>1$, $\>\!0\leq\zeta\leq 1$ for all $\>\!x\in\mathbb{R}^{n}\!$. ​ ​We begin with $\!\;\!\;\!q>2\>\!$: Taking $S\in C^{2}(\mathbb{R})$ such that $S^{\prime}(\mbox{u})=-\;\!1\>\!$ if  ​$\mbox{u}\leq-\;\!1$, $\>\!S^{\prime}(\mbox{u})=1\>\!$ if $\>\!\mbox{u}\geq 1$, $\>\!S(0)=0\>\!$ and $\>\!S^{\prime}(\mbox{u})\geq 0\>\!$ for all $\>\!\mbox{u}\in\mathbb{R}$, let $L(\mbox{u}):=\int_{0}^{\>\!\mbox{\scriptsize u}}\!\;\!S(\mbox{v})\,d\mbox{v}$, and, for each $\delta>0$, $L_{\delta}(\mbox{u}):=\delta\,L(\mbox{u}/\delta)$. ( ​This gives $L_{\delta}(\mbox{u})\rightarrow|\;\!\mbox{u}\;\!|$ as $\delta\rightarrow 0$, uniformly in $\mbox{u}\in\mathbb{R}$.) Setting ${\displaystyle\;\!\Phi_{\delta}(\mbox{u}):=L_{\delta}(\mbox{u})^{q}\!\;\!}$, let us take in (2.2) ${\displaystyle\phi(x)=\Phi_{\delta}^{\prime}(u_{\mbox{}_{\scriptstyle h}}\!(x,t))\,\zeta_{\mbox{}_{R}}\!\;\!(x)}$. ​Integrating (2.2) in $(\!\;\!\!\;\!\;\!t_{0},\!\;\!\!\;\!\;\!t)$ and letting $h\rightarrow 0$, $\delta\rightarrow 0$ and then \>\!\mbox{\footnotesizeR}\rightarrow\infty, we get, by (1.3) and (2.3) above,

**

[TABLE]

[TABLE]

from which the result is obtained from (1.2), (2.3) and Lebesgue’s differentiation theorem. For the case $\>\!q=2\>\!$ we proceed similarly, using ${\displaystyle\>\!\phi(x)=u_{\mbox{}_{\scriptstyle h}}\!(x,t)\,\zeta_{\mbox{}_{R}}\!\;\!(x)\>\!}$ in (2.2) above. $\Box$

Sometimes (as in Propositions 2.3, 2.4 below) the following extra assumption on  $g$ is also needed: given any \>\!\mbox{\smallT}>0, there exists some constant C(\mbox{\smallT}) such that

[TABLE]

Proposition 2.3. *​​ ​Under the same assumptions of  Proposition 2.1​ ​ above, we have *

[TABLE]

*provided that $\;\!($i$)$ $\kappa\>\!\geq\>\!1-2/p$,  ​or that $\;\!($ii$)$ $p\>\!\geq\>\!n\>\!$ and $\;\!(2.6)$ holds. *

**Proof. Let $L_{\delta}\in C^{3}(\mathbb{R})$, $\zeta_{\mbox{}_{R}}\!\in C^{1}(\mathbb{R}^{n})$ be constructed as in the proof of Proposition 2.2, and take (2.2) with ${\displaystyle\phi(x)=L_{\delta}^{\prime}(u_{\mbox{}_{\scriptstyle h}}\!(x,t))\,\zeta_{\mbox{}_{R}}\!\;\!(x)}$. If $\kappa\geq 1-2/p$, we may proceed as follows: integrating (2.2) in $(t_{0},t)$ and letting $\>\!h\rightarrow 0$, $t_{0}\rightarrow 0$ and \mbox{\footnotesizeR}\rightarrow\infty, we obtain

**

[TABLE]

from which (2.7) is obtained by letting $\;\!\delta\rightarrow 0$ (because $\>\!(\kappa+1)\;\!p\;\!/(p-1)\>\!\geq\>\!2\;\!$ in this case). In case (ii), we let instead $\>\!h\rightarrow 0$, $t_{0}\rightarrow 0$ and $\delta\rightarrow 0$, which gives, by (2.6),

[TABLE]

{\displaystyle+\;\,\mbox{\footnotesizeK}\!\;\!(\mbox{\footnotesizeM}\!\;\!,\>\!t)\!\>\!\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\mbox{\footnotesize>!t}}\!\!\;\!\int_{\mbox{}_{\scriptstyle\!B_{\mbox{}_{\!\;\!R}}}}\!\!\!\!\;\!|\,u(x,\tau)\,|^{\mbox{}^{\scriptstyle\!\frac{\scriptstyle p\,-1}{\scriptstyle p}}}|\,\nabla\zeta_{\mbox{}_{R}}\!\;\!(x)\,|\;dx\;\!\;\!d\tau}

for some constant {\displaystyle\>\!\mbox{\footnotesizeK}\!=\mbox{\footnotesizeK}\!\;\!(\mbox{\footnotesizeM}\!\;\!,\>\!t)} depending upon $M$  (the maximum size of ${\displaystyle\;\!\|\,u(\cdot,\tau)\,\|_{\mbox{}_{\scriptstyle L^{\infty}(\mathbb{R}^{n})}}\!\;\!}$, $0\leq\tau\leq t\>\!$) and $\>\!t$. Letting \mbox{\footnotesizeR}\rightarrow\infty, this gives (2.7), since we are now assuming $\;\!p\geq n$. $\Box$

Remark 2.1. ​In addition to conditions (i) and (ii) of Proposition 2.3, if $g$ satisfies (2.6) with exponent 1 (cf. (2.9) below), then all solutions to (1.1), (1.2) constructed by parabolic regularization satisfy (2.7) when $\>\!p\geq 3$: see [9], Ch. 2, and Remark 2.3.

Remark 2.2. When (2.7) is valid, it follows more generally that we have, by the same argument: ${\displaystyle\|\,u(\cdot,t)\,\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R}^{n})}}\leq\;\!\|\,u(\cdot,t_{0})\,\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R}^{n})}}\!\;\!}$ for all \;\!0\leq t_{0}\leq t<\mbox{\smallT}_{\!\ast},  ​so that ${\displaystyle\|\,u(\cdot,t)\,\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R}^{n})}}\!\;\!}$ is then monotonically decreasing in $\>\!t$.

Proposition 2.4. *​Let {\displaystyle\,u(\cdot,t)\in C^{0}(\,\![\,0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!L^{1}_{\mbox{\scriptsize\em loc}}(\mathbb{R}^{n})\,\!)\!\>\!\cap\!\>\!L^{\infty}_{\mbox{\scriptsize\em loc}}(\,\![\,0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!L^{1}(\mathbb{R}^{n})\!\>\!\cap\!\>\!L^{\infty}(\mathbb{R}^{n})\,\!)} {\displaystyle\cap\;\!L^{p}_{\mbox{\scriptsize\em loc}}(\,\!(\!\;\!\;\!0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!W^{1,\,p}_{\mbox{\scriptsize\em loc}}(\mathbb{R}^{n})\,\!)} be any solution to $(1.1)$, $(1.2)$. ​If $\,p\geq n$ and $(2.6)$ holds, then {\displaystyle\;\!u(\cdot,t)\in C^{0}(\,\![\,0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!L^{1}(\mathbb{R}^{n})\,\!)}. ​ ​$(\!\;\!\;\!$In particular, ${\displaystyle\;\!\|\,u(\cdot,t)-u_{0}\;\!\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R}^{n})}}\!\rightarrow\>\!0}$ as \>\!t\;\!\mbox{\footnotesize\searrow}\;\!0.$)$ Moreover, the solution mass is conserved, i.e.,

[TABLE]

**Proof. We begin by showing that {\displaystyle\;\!u(\cdot,t)\in C^{0}(\,\![\,0,\;\!\mbox{\smallT}_{\!\ast}),\!\;\!\;\!L^{1}(\mathbb{R}^{n})\,\!)}. The following argument is adapted from [3], Theorem 2.1. Since $\;\!u(\cdot,t)\;\!$ is already known to be continuous in $L^{1}_{\mbox{\scriptsize loc}}(\mathbb{R}^{n})$, it is sufficient to show that, given \;\!0<\mbox{\footnotesizeT}<\mbox{\footnotesizeT}_{\!\ast} arbitrary, we have ${\displaystyle\|\,u(\cdot,t)\,\|_{\scriptstyle\>\!L^{1}(|\,x\,|\,>\,R)}}$ uniformly small (say, $O(\epsilon)$) for all \;\!0<t\leq\mbox{\footnotesizeT} provided that we choose {\displaystyle\!\;\!\;\!\mbox{\footnotesizeR}=\mbox{\footnotesizeR}(\epsilon,\mbox{\footnotesizeT})\gg 1}. Let then $\epsilon>0$, \;\!0<\mbox{\footnotesizeT}<\mbox{\footnotesizeT}_{\!\ast} be given, and let $\zeta_{\scriptstyle R,\>\!S}\!\>\!\in C^{1}(\mathbb{R}^{n})$ be a cut-off function satisfying: $0\leq\zeta_{\scriptstyle R,\>\!S}\leq 1\>\!$ everywhere, and $\>\!\zeta_{\scriptstyle R,\>\!S}(x)=0\;\!$ if \>\!|\,x\,|<\mbox{\footnotesizeR}/2, $\>\!\zeta_{\scriptstyle R,\>\!S}(x)=1$ if \>\!\mbox{\footnotesizeR}<|\,x\,|<\mbox{\footnotesizeR}+\mbox{\footnotesizeS}, $\>\!\zeta_{\mbox{}_{\scriptstyle R,\>\!S}}(x)=0\;\!$ if \>\!|\,x\,|>\mbox{\footnotesizeR}+2\>\!\mbox{\footnotesizeS}, with \>\!|\;\!\nabla\zeta_{R,\>\!S}(x)\,|\leq\mbox{\footnotesizeC}/\mbox{\footnotesizeR}\;\! if \>\!|\,x\,|<\mbox{\footnotesizeR} and \>\!|\;\!\nabla\zeta_{R,\>\!S}(x)\,|\leq\mbox{\footnotesizeC}/\mbox{\footnotesizeS}\;\! if \>\!\mbox{\footnotesizeR}+\mbox{\footnotesizeS}<|\,x\,|<\mbox{\footnotesizeR}+2\>\!\mbox{\footnotesizeS}, for some constant $C$ independent of \mbox{\footnotesizeR},\mbox{\footnotesizeS}>0. Given 0<t_{0}<t\leq\mbox{\footnotesizeT}, $h>0$, $\delta>0$, let $L_{\delta}\in C^{3}(\mathbb{R})$ be the regularized absolute value function introduced in the proof of Proposition 2.2. Taking $\phi(x)=L_{\delta}^{\prime}(u_{\mbox{}_{\scriptstyle h}}\!(x,t))\;\!\zeta_{\mbox{}_{\scriptstyle R,\>\!S}}\!\;\!(x)$ in (2.2), and integrating the result in $\,\!(\,\!t_{0},\>\!t\,\!)$, we get, letting $\>\!h\rightarrow 0$, $t_{0}\rightarrow 0$ and $\delta\rightarrow 0$,

**

[TABLE]

by (1.2), (1.3) and (2.3), where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Recalling that $\>\!p\geq n$ (by hypothesis), we observe that

[TABLE]

and similarly for {\displaystyle J_{\mbox{}_{\scriptstyle\!\;\!2}}\!\;\!(\mbox{\footnotesizeR},\,\!\mbox{\footnotesizeS})}, {\displaystyle H_{\mbox{}_{\scriptstyle\!\>\!1}}\!\>\!(\mbox{\footnotesizeR})} and {\displaystyle H_{\mbox{}_{\scriptstyle\!\;\!2}}\!\;\!(\mbox{\footnotesizeR},\,\!\mbox{\footnotesizeS})}. This gives, letting \>\!\mbox{\footnotesizeS}\rightarrow\infty,

                {\displaystyle\int_{\mbox{}_{\scriptstyle\!\>\!|\,x\,|\,>\,R}}|\,u(x,t)\,|\;\!\;\!dx\;\leq\;\!\int_{\mbox{}_{\scriptstyle\!\;\!|\,x\,|\,>\,R/2}}|\,u_{0}(x)\,|\;\!\;\!dx\;+\;\mbox{\footnotesize{\displaystyle\frac{;!2;!\mbox{\footnotesize$C$}}{\mbox{\footnotesize$R$}}}}\!\;\!\int_{\mbox{}_{\scriptstyle 0}}^{\>\!T}\!\!\!\mbox{\footnotesizeF}(\tau)\!\!\;\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!|\,x\,|\,>\,R/2}}|\,u(x,t)\,|^{\>\!\kappa\;\!+\;\!1}\,dx\,d\tau}

                                             {\displaystyle+\;\;\epsilon^{\scriptstyle\!\>\!-\,\frac{\scriptstyle 1}{\scriptstyle\;\!p\;\!-\;\!1\>\!}}\!\!\>\!\int_{\mbox{}_{\scriptstyle 0}}^{\>\!T}\!\!\!\mu(\tau)\!\!\;\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!|\,x\,|\,>\,R/2}}|\,\nabla u\,|^{\>\!p}\;dx\,d\tau\,+\,\mbox{\footnotesizeK}_{\mbox{}_{\scriptstyle\!\>\!n}}\!\;\!\;\!\epsilon\;\!\;\!\Bigl{\{}\;\!1+\!\int_{\mbox{}_{\scriptstyle 0}}^{\>\!T}\!\!\!\mu(\tau)\;\!\;\!d\tau\Bigr{\}}}

                                                                 ${\displaystyle+\;\;\epsilon^{\scriptstyle\!\>\!-\,\frac{\scriptstyle 1}{\scriptstyle\;\!p\;\!-\;\!1\>\!}}\!\!\>\!\int_{\mbox{}_{\scriptstyle 0}}^{\>\!T}\!\!\;\!\int_{\mbox{}_{\scriptstyle\!\>\!|\,x\,|\,>\,R/2}}|\,u(x,\tau)\,|\;dx\,d\tau}$

for every \;\!0<t\leq\mbox{\footnotesizeT}, where {\displaystyle\!\;\!\;\!\mbox{\footnotesizeK}_{\mbox{}_{\scriptstyle\!\;\!n}}\!\;\!} is some constant depending on $n$, $C$ only (and not on $R$), and where we have used (2.6) and the assumption $\!\;\!\;\!p\geq n$. Therefore, by (1.3) and (2.3), we can choose \>\!\mbox{\footnotesizeR}>0 sufficiently large (depending on $\epsilon$, $T$) such that

[TABLE]

Since $\;\!\epsilon>0\>\!$ is arbitrary, and the constant \mbox{\footnotesizeK}_{\mbox{}_{\scriptstyle\!\>\!n}}\!\>\! in the estimate above is independent of $\>\!\epsilon$, this gives {\displaystyle\;\!u(\cdot,t)\in C^{0}(\,\![\,0,\;\!\mbox{\footnotesizeT}_{\!\ast}),\;\!L^{1}(\mathbb{R}^{n})\,\!)}, as claimed in the first part of Proposition 2.4.

Finally, to show the second part (i.e., mass conservation), we proceed in a similar way, but taking this time $\phi(x)=\zeta_{\mbox{}_{R}}\!\;\!(x)$ in (2.2), where $\zeta_{\mbox{}_{R}}\!\;\!(\cdot)$ is the cut-off function considered in the proof of Proposition 2.2. This completes the proof of Proposition 2.4. $\Box$

Remark 2.3. In a similar way, in the remaining case $\;\!p<n\>\!$ mass conservation can be obtained from (2.2) with $\phi(x)=\zeta_{\mbox{}_{R}}\!\;\!(x)$ provided that we have, instead of (2.6), the stronger condition

[TABLE]

and that we have {\displaystyle\;\!|\,\nabla u\>\!(\cdot,t)\,|\,\mbox{\small\in}\,L^{q}_{\mbox{\scriptsize loc}}([\;\!0,\>\!\mbox{\smallT}_{\!\ast}),\>\!L^{q}(\mathbb{R}^{n})\,\!)} for some \;\!q\,\mbox{\small\in}\,[\;\!p-1,p) satisfying $\;\!q\;\!\leq\>\!(\,\!p-1)\;\!n\,\!/\,\!(\,\!n-1)$. For still other conditions, see [9], Ch. 2.

3. $L^{1}\!\>\!$ contraction and comparison properties

The results obtained in this section, where we introduce a few extra assumptions (see (3.1) - (3.4) below), serve to establish the uniqueness of solutions to (1.1), (1.2), among other important properties [10, 14]. Upon  $f$  and $g$, it will be required one of the following sets of conditions: for every given \>\!\mbox{\smallM}>0, \>\!0<\mbox{\smallT}<\mbox{\smallT}_{\!\ast}, one must have (1.6) and (1.7) satisfied, that is,

    {\displaystyle|\,\mbox{\boldmathf}(x,t,\mbox{u})\;\!-\>\!\mbox{\boldmathf}(x,t,\mbox{v})\,|\;\leq\>\mbox{\smallK}_{\!\,\!f}\,\!(\mbox{\smallM}\!\;\!,\>\!\mbox{\smallT})\,|\,\mbox{u}-\mbox{v}\,|^{\scriptstyle\>\!1\,-\,\frac{\scriptstyle 1}{\scriptstyle p}}\quad\;\,\forall\;\,x\in\mathbb{R}^{n}\!\>\!,\>0\>\!\leq\>\!t\leq\>\!\mbox{\smallT}\!\;\!}, (3.1)

                                                                                      {\displaystyle|\,\mbox{u}\,|\>\!\leq\>\!\mbox{\smallM}\!\;\!,\;|\,\mbox{v}\,|\>\!\leq\>\!\mbox{\smallM}\!\;\!},

          {\displaystyle|\,\mbox{\boldmathg}(t,\mbox{u})\;\!-\>\!\mbox{\boldmathg}(t,\mbox{v})\,|\;\leq\,\mbox{\smallK}_{\!\,\!g}\,\!(\mbox{\smallM}\!\;\!,\>\!\mbox{\smallT})\;\!\;\!|\,\mbox{u}-\mbox{v}\,|^{\scriptstyle\>\!1\,-\,\frac{\scriptstyle 1}{\scriptstyle p}}\hskip 46.94687pt\forall\;\,0\>\!\leq\>\!t\leq\>\!\mbox{\smallT}\!\;\!}, (3.2)

                                                                                      {\displaystyle|\,\mbox{u}\,|\>\!\leq\>\!\mbox{\smallM}\!\;\!,\;|\,\mbox{v}\,|\>\!\leq\>\!\mbox{\smallM}\!\;\!},

or the stronger assumptions

           {\displaystyle|\,\mbox{\boldmathf}_{\!\,\!\mbox{\scriptsize u}}(x,t,\mbox{u})\,|\;\leq\>\mbox{\smallF}_{\!\!\;\!\mbox{}_{\mbox{\scriptsize u}}}(\mbox{\smallM}\!\;\!,\>\!\mbox{\smallT})\,|\,\mbox{u}\,|^{\>\!\kappa}\qquad\;\,\forall\;\,x\in\mathbb{R}^{n}\!\>\!,\;0\>\!\leq\>\!t\>\!\leq\>\!\mbox{\smallT}\!\;\!,\>|\,\mbox{u}\,|\>\!\leq\>\!\mbox{\smallM}\!\;\!}, (3.3)

             {\displaystyle|\,\mbox{\boldmathg}_{\!\;\!\mbox{\scriptsize u}}(t,\mbox{u})\,|\;\leq\>\mbox{\smallG}_{\!\;\!\mbox{}_{\mbox{\scriptsize u}}}\!\;\!(\mbox{\smallM}\!\>\!,\>\!\mbox{\smallT})\,|\,\mbox{u}\,|^{\>\!\gamma}\hskip 57.61665pt\forall\;\,0\>\!\leq\>\!t\>\!\leq\>\!\mbox{\smallT}\!\;\!,\>|\,\mbox{u}\,|\>\!\leq\>\!\mbox{\smallM}\!\;\!}, (3.4)

with constants \;\!\mbox{\smallK}_{\!\!\;\!f}\,\!(\mbox{\smallM}\!,\!\;\!\;\!\mbox{\smallT}),\;\!\mbox{\smallK}_{\!\!\;\!g}\,\!(\mbox{\smallM}\!,\!\;\!\;\!\mbox{\smallT}),\;\!\mbox{\smallF}_{\!\!\;\!\mbox{}_{\mbox{\scriptsize u}}}\!\;\!(\mbox{\smallM}\!,\!\;\!\;\!\mbox{\smallT}),\;\!\mbox{\smallG}_{\!\!\;\!\;\!\mbox{}_{\mbox{\scriptsize u}}}\!\;\!(\mbox{\smallM}\!,\!\;\!\;\!\mbox{\smallT}) depending on \>\!\mbox{\smallM}\!\,\!\,\!,\;\!\mbox{\smallT}\!\,\!, where \mbox{\boldmathf}_{\!\,\!\mbox{\scriptsize u}}\!\,\!=\>\!\partial\mbox{\boldmathf}\!\;\!/\!\;\!\>\!\partial\>\!\mbox{u}, \mbox{\boldmathg}_{\!\;\!\mbox{\scriptsize u}}\!\,\!=\>\!\partial\!\;\!\>\!\mbox{\boldmathg}\!\;\!/\!\;\!\partial\>\!\mbox{u}. ​ ​We note that (3.3) - (3.4) are satisfied in the prototype model given by \mbox{\boldmathf}(x,t,\mbox{u})\!\;\!=\!\;\!\;\!\mbox{\boldmathb}(x,t)\;\!|\;\!\mbox{u}\;\!|^{\>\!\kappa}\!\;\!\;\!\mbox{u}, \;\!\mbox{\boldmathg}(t,\mbox{u})\!\;\!=\!\;\!\;\!\mbox{\boldmathc}(t)\;\!|\;\!\mbox{u}\;\!|^{\>\!\gamma}\!\;\!\;\!\mbox{u}.

Again, as in the previous section, solutions to (1.1), (1.2) are always meant in the space {\displaystyle\,\!\,\!C^{0}([\;\!0,\>\!\mbox{\smallT}_{\!\ast}),L^{1}_{\mbox{\scriptsize loc}}(\mathbb{R}^{n}))} {\displaystyle\cap\,L^{p}_{\mbox{\scriptsize loc}}([\;\!0,\>\!\mbox{\smallT}_{\!\ast}),W^{1\!\;\!,\>p}_{\mbox{\scriptsize loc}}(\mathbb{R}^{n}))\!\;\!\cap L^{\infty}_{\mbox{\scriptsize loc}}([\;\!0,\>\!\mbox{\smallT}_{\!\ast}),\>\!L^{1}(\mathbb{R}^{n})\!\;\!\cap\!L^{\infty}(\mathbb{R}^{n}))}, with its maximal existence interval given by [\;\!0,\>\!\mbox{\smallT}_{\!\ast}\!\;\!).

Proposition 3.1. *Let $\;\!u(\cdot,t),\;\!v(\cdot,t)$, 0<t\leq\mbox{\smallT}\!\;\!, be given solutions of $\,\!(1.1a)$, $(1.2)$ corresponding to initial states ${\displaystyle\;\!u_{0},\>\!v_{0}\in L^{1}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})}$, respectively. Then

[TABLE]

*provided that $:$ $($i$)$ $p\geq n$, and $f$, $g$ satisfy $\!\;\!\;\!(3.1)\!\;\!$ and $\!\;\!\;\!(3.2)$ above, or $($when $\;\!2<p<n\!\;\!\;\!)\!\!:$ $($ii$)$ $\;\!\kappa\geq 1-2/p$, $\;\!\gamma\geq 1-2/p$,  ​and  $f$ , $g$  satisfy $\!\;\!\;\!(3.3)\!\;\!$ and $\!\;\!\;\!(3.4)$, respectively.

Proof. Given $\>\!h>0$, $\delta>0$, \>\!\mbox{\footnotesizeR}>0, let $\>\!\zeta_{\mbox{}_{R}}\!\in C^{1}(\mathbb{R}^{n})\>\!$ be the cut-off function considered in the proof of Proposition 2.2. ​Let $\,\!u_{h}(\cdot,t)$, $v_{h}(\cdot,t)\,\!$ be the time Steklov regularizations of $u(\cdot,t)$, $v(\cdot,t)$, respectively. Let $L_{\delta}\!\;\!\in C^{3}(\mathbb{R}^{n})$ be defined as in the proof of Proposition 2.2, and let $\>\!\theta(\cdot,t)\!\;\!:=u(\cdot,t)-v(\cdot,t)$, $\theta_{\!\;\!h}\!\;\!(\cdot,t)\!\;\!:=u_{h}\!\;\!(\cdot,t)-v_{h}\!\;\!(\cdot,t)$. ​Taking ${\displaystyle\;\!\phi(x)=L_{\delta}^{\prime}(\theta_{\!\;\!h}\!\;\!(x,t))\;\!\zeta_{\mbox{}_{R}}\!\;\!(x)}$ in the equations (2.2) for $\>\!u_{h}\!\;\!(\cdot,t)$, $v_{h}\!\;\!(\cdot,t)$, subtracting one from the other and integrating the result in the interval $(\,\!t_{0},\>\!t\,\!)$, where $\>\!0<t_{0}<t$, we get, letting \>\!h\;\!\mbox{\scriptsize\searrow}\;\!0\>\! and \>\!t_{0}\>\!\mbox{\scriptsize\searrow}\;\!0,

{\displaystyle\int_{\mbox{}_{\scriptstyle|\,x\,|\,<\,R}}L_{\delta}(\theta(x,t))\,\zeta_{\mbox{}_{R}}\!\;\!(x)\,dx\;+\>\!\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\>\!\mu(\tau)\!\int_{\mbox{}_{\scriptstyle|\,x\,|\,<\,R}}L_{\delta}^{\prime\prime}(\theta)\;\!\>\!\langle\;\!\;\!\mbox{\boldmatha}(u,v),\>\!\nabla\theta\;\!\;\!\rangle\>\zeta_{\mbox{}_{R}}\!\;\!(x)\;\!\;\!dx\,d\tau}

          {\displaystyle\leq\,\int_{\mbox{}_{\scriptstyle|\,x\,|\,<\,R}}L_{\delta}(\theta_{0}(x))\,\zeta_{\mbox{}_{R}}\!\;\!(x)\,dx\;+\>\!\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\>\!\mu(\tau)\!\int_{\mbox{}_{\scriptstyle\!\>\!R/2\,<\,|\,x\,|\,<\,R}}|\,L_{\delta}^{\prime}(\theta)\,|\cdot|\,\mbox{\boldmatha}(u,v)\,|\cdot|\,\nabla\zeta_{\mbox{}_{R}}\!\;\!(x)\,|\;dx\,d\tau\;\;\!+}

{\displaystyle\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\>\!\mu(\tau)\!\int_{\mbox{}_{\scriptstyle|\,x\,|\,<\,R}}L_{\delta}^{\prime\prime}(\theta)\,\,|\,[\,\mbox{\boldmath\tilde{f}}\,]\,|\cdot|\,\nabla\theta\,|\;\zeta_{\mbox{}_{R}}\!\;\!(x)\;dx\,d\tau\;\!\;\!+\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\>\!\mu(\tau)\!\int_{\mbox{}_{\scriptstyle\!\,\!R/2\,<\,|\,x\,|\,<\,R}}|\,L_{\delta}^{\prime}(\theta)\,|\cdot|\,[\,\mbox{\boldmath\tilde{f}}\,]\,|\cdot|\,\nabla\zeta_{\mbox{}_{R}}\!\;\!(x)\,|\;dx\,d\tau}

in view of (2.3), where $\;\!\theta_{0}=u_{0}\!\;\!-v_{0}$, \>\![\,\mbox{\boldmath\tilde{f}}\,]\equiv[\,\mbox{\boldmath\tilde{f}}\,](x,\tau)=\mbox{\boldmath\tilde{f}}(x,\tau,u(x,\tau))-\mbox{\boldmath\tilde{f}}(x,\tau,v(x,\tau)), {\displaystyle\mbox{\boldmath\tilde{f}}=\mbox{\boldmathf}+\mbox{\boldmathg}}, and {\displaystyle\;\!\mbox{\boldmatha}(u,v)\>\!=\;\!|\,\nabla u(x,\tau)\,|^{\>\!p-2}\;\!\nabla u(x,\tau)\,\!-\>\!|\,\nabla v(x,\tau)\,|^{\>\!p-2}\;\!\nabla v(x,\tau)}. ​ ​Noticing that

             {\displaystyle\langle\;\!\;\!\mbox{\boldmatha}(u,v),\>\!\nabla\theta\;\!\;\!\rangle\>=\;\>\!\mbox{\footnotesize{\displaystyle\frac{1}{2}}}\;\!\bigl{(}\>|\,\nabla u\,|^{\>\!p-2}+\;\!|\,\nabla v\,|^{\>\!p-2}\;\!\bigr{)}\,\>\!|\,\nabla\theta\,|^{\>\!2}\;\;\!+}

                                                                    {\displaystyle+\;\,\mbox{\footnotesize{\displaystyle\frac{1}{2}}}\;\!\bigl{(}\>|\,\nabla u\,|^{\>\!p-2}-\;\!|\,\nabla v\,|^{\>\!p-2}\;\!\bigr{)}\;\!\bigl{(}\>|\,\nabla u\,|^{\>\!2}-\;\!|\,\nabla v\,|^{\>\!2}\;\!\bigr{)}}

                                     ${\displaystyle\geq\;\frac{1}{\;2^{\mbox{}^{\scriptstyle\>\!p-1}}}\>|\,\nabla\theta\,|^{\>\!p}}$

and that {\displaystyle\;\!|\,\mbox{\boldmatha}(u,v)\,|\;\!\leq\,|\,\nabla u\,|^{\>\!p-1}+\;\!|\,\nabla v\,|^{\>\!p-1}\!}, we then have

               {\displaystyle\int_{\mbox{}_{\scriptstyle|\,x\,|\,<\,R}}L_{\delta}(\theta(x,t))\,\zeta_{\mbox{}_{R}}\!\;\!(x)\,dx\;+\;\Bigl{(}\,1-\frac{2}{p}\,\Bigr{)}\,\frac{1}{\;2^{\mbox{}^{\scriptstyle\>\!p-1}}}\!\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\>\!\mu(\tau)\!\int_{\mbox{}_{\scriptstyle|\,x\,|\,<\,R}}L_{\delta}^{\prime\prime}(\theta)\;\!\>\!|\,\nabla\theta\,|^{\>\!p}\,\zeta_{\mbox{}_{R}}\!\;\!(x)\;\!\;\!dx\,d\tau}

         {\displaystyle\leq\;\!\;\!\;\!\|\,u_{0}-\;\!v_{0}\;\!\|_{\mbox{}_{\scriptstyle L^{1}(\mathbb{R}^{n})}}\>\!+\!\;\!\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\>\!\mu(\tau)\!\int_{\mbox{}_{\scriptstyle\!\>\!R/2\,<\,|\,x\,|\,<\,R}}|\,L_{\delta}^{\prime}(\theta)\,|\>\bigl{(}\,|\,\nabla u\,|^{\>\!p-1}\!\;\!+\;\!|\,\nabla v\,|^{\>\!p-1}\>\!\bigr{)}\,|\,\nabla\zeta_{\mbox{}_{R}}\!\;\!(x)\,|\;\,\!dx\,d\tau\;\;\!+}

{\displaystyle+\;\;\!\;\!2\!\,\!\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\>\!\mu(\tau)^{\mbox{}^{\scriptstyle\!\!-\,\frac{\scriptstyle 1}{\scriptstyle\;\!p\>\!-1\>\!}}}\!\!\!\>\!\int_{\mbox{}_{\scriptstyle|\,x\,|\,<\,R}}L_{\delta}^{\prime\prime}(\theta)\;\!\;\!|\,[\,\mbox{\boldmathf}\,]\,|^{\mbox{}^{\scriptstyle\frac{\scriptstyle p}{\scriptstyle\;\!p\>\!-1\>\!}}}\!\;\!\zeta_{\mbox{}_{R}}\!\;\!(x)\;dx\,d\tau\,+\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\;\!\int_{\mbox{}_{\scriptstyle\!\,\!R/2\,<\,|\,x\,|\,<\,R}}|\,L_{\delta}^{\prime}(\theta)\,|\cdot|\,[\,\mbox{\boldmathf}\,]\,|\cdot|\,\nabla\zeta_{\mbox{}_{R}}\!\;\!(x)\,|\;dx\,d\tau}

{\displaystyle+\;\;\!\!\;\!\;\!2\!\,\!\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\>\!\mu(\tau)^{\mbox{}^{\scriptstyle\!\!-\,\frac{\scriptstyle 1}{\scriptstyle\;\!p\>\!-1\>\!}}}\!\!\!\>\!\int_{\mbox{}_{\scriptstyle|\,x\,|\,<\,R}}L_{\delta}^{\prime\prime}(\theta)\;\!\;\!|\,[\,\mbox{\boldmathg}\,]\,|^{\mbox{}^{\scriptstyle\frac{\scriptstyle p}{\scriptstyle\;\!p\>\!-1\>\!}}}\!\;\!\zeta_{\mbox{}_{R}}\!\;\!(x)\;dx\,d\tau\,+\!\;\!\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\;\!\int_{\mbox{}_{\scriptstyle\!\,\!R/2\,<\,|\,x\,|\,<\,R}}|\,L_{\delta}^{\prime}(\theta)\,|\cdot|\,[\,\mbox{\boldmathg}\,]\,|\cdot|\,\nabla\zeta_{\mbox{}_{R}}\!\;\!(x)\,|\;dx\,d\tau},

(3.6)

where, as before, \>\![\,\mbox{\boldmathf}\,]\equiv[\,\mbox{\boldmathf}\,](x,\tau)=\mbox{\boldmathf}(x,\tau,u(x,\tau))-\mbox{\boldmathf}(x,\tau,v(x,\tau)), \>\![\,\mbox{\boldmathg}\,]\equiv[\,\mbox{\boldmathg}\,](x,\tau)=\mbox{\boldmathg}(\tau,u(x,\tau))-\mbox{\boldmathg}(\tau,v(x,\tau)). If $\;\!p\geq n$, we may proceed as in the proof of Proposition 2.4 (using that $|\,L_{\delta}^{\prime}(\vartheta)\,|\leq 1$ for any $\>\!\vartheta\in\mathbb{R}$), letting $\;\!\delta\rightarrow 0\;\!$ and then \>\!\mbox{\footnotesizeR}\rightarrow\infty\>\! to obtain, given $\;\!\epsilon>0\>\!$ arbitrary :

[TABLE]

for each \;\!0<t\leq\mbox{\footnotesizeT}\!\;\!, because of (1.3), (2.3) and (3.1), (3.2) above, where \!\;\!\;\!\mbox{\smallK}_{\mbox{}_{\scriptstyle\!\>\!n}}\!>0\>\! is some appropriate constant depending on the dimension $\,\!\,\!n\,\!\,\!$ but not on $\,\!\,\!\epsilon$. Since this holds for any $\epsilon>0$, (3.5) is obtained in the case $\>\!p\geq n$, as claimed.

When $\;\!2<p<n$, we assume (3.3), (3.4) with $\kappa\geq 0$, $\gamma\geq 0$ satisfying $\kappa\geq 1-2/p$ and $\gamma\geq 1-2/p$, proceeding instead as follows. Because ${\displaystyle\;\!|\,L_{\delta}^{\prime}(\theta)\,|\!\;\!\;\!\leq K\;\!|\;\!\theta\;\!|\;\!/\;\!\delta\;\!}$ for all $\theta\in\mathbb{R}$, $\delta>0$ (and some constant $\!\;\!K$ independent of $\!\;\!\;\!\theta,\delta$), we obtain, letting \mbox{\footnotesizeR}\rightarrow\infty\>\! in (3.6):

[TABLE]

{\displaystyle+\;\,2\!\>\!\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\>\!\mu(\tau)^{\mbox{}^{\scriptstyle\!\!-\,\frac{\scriptstyle 1}{\scriptstyle\;\!p\>\!-1\>\!}}}\!\!\!\>\!\int_{\mbox{}_{\scriptstyle\mathbb{R}^{n}}}\!\!\!\!\;\!L_{\delta}^{\prime\prime}(\theta)\;|\,[\,\mbox{\boldmathg}\,](x,\tau)\,|^{\mbox{}^{\scriptstyle\frac{\scriptstyle p}{\scriptstyle\;\!p\>\!-1\>\!}}}\;\!dx\,d\tau}

(3.7)

by (1.3) and (2.3). Now, because of (3.3) and (3.4), we have

[TABLE]

[TABLE]

for all $\;\!x\in\mathbb{R}^{n}\!\;\!$, 0<\tau\leq\mbox{\footnotesizeT}\!\>\!,  ​where {\displaystyle\;\!\mbox{\footnotesizeM}=\;\!\sup\,\bigl{\{}\;\!\|\,u(\cdot,\tau)\,\|_{L^{\infty}(\mathbb{R}^{n})}\!\;\!,\,\|\,v(\cdot,\tau)\,\|_{L^{\infty}(\mathbb{R}^{n})}\!\!\;\!:\,0<\tau\leq\mbox{\footnotesizeT}\!\;\!\;\!\bigr{\}}}, so that

[TABLE]

and

[TABLE]

for all $\>\!(x,\tau)\>\!$ concerned,  ​where {\displaystyle\;\!\mbox{\footnotesizeK}\!\;\!(\mbox{\footnotesizeM}\!\>\!,\>\!\mbox{\footnotesizeT}\!\>\!,\;\!p,\>\!n)} is some constant that does not depend on $\;\!\delta$. Hence, letting $\;\!\delta\rightarrow 0\;\!$ in (3.7), we obtain

[TABLE]

by Lebesgue’s dominated convergence, since ${\displaystyle\;\!(\kappa\>\!p+1)/(p-1)\geq 1}$, ${\displaystyle(\gamma\>\!p+1)/(p-1)\geq 1}$. ​ ​This shows (3.5) in case (ii), so that the proof of Proposition 3.1 is now complete. $\Box$

Actually, under the same assumptions of Proposition 3.1, a lot more is true, as shown by the next two results (cf. Propositions 3.2 and 3.3 below):

Proposition 3.2. *Let $\;\!u(\cdot,t),\;\!v(\cdot,t)$, 0<t\leq\mbox{\smallT}\!\;\!, be given solutions of $\,\!(1.1a)$, $(1.2)$ corresponding to initial states ${\displaystyle\;\!u_{0},\>\!v_{0}\in L^{1}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})}$, respectively. Then

[TABLE]

and

[TABLE]

*provided that $:$ $($i$)$ $p\geq n$, and $f$, $g$ satisfy $\!\;\!\;\!(3.1)\!\;\!$ and $\!\;\!\;\!(3.2)$ above, or $($when $\;\!2<p<n\!\;\!\;\!)\!\!:$ $($ii$)$ $\;\!\kappa\geq 1-2/p$, $\;\!\gamma\geq 1-2/p$,  ​and  $f$ , $g$  satisfy $\!\;\!\;\!(3.3)\!\;\!$ and $\!\;\!\;\!(3.4)$, respectively.

( ​Here, as usual, $\>\!\theta_{\mbox{}_{\!+}}\!$ and $\>\!\theta_{\mbox{}_{\!-}}\!$ stand for the positive and negative real parts, respectively, of a given number $\theta\in\mathbb{R}$, that is ​: $\theta_{\mbox{}_{\!+}}\!=\>\!(\;\!|\;\!\theta\;\!|+\theta\;\!)/2$, and $\;\!\theta_{\mbox{}_{\!-}}\!=\>\!(\;\!|\;\!\theta\;\!|-\theta\;\!)/2$.)

**Proof. The following argument is adapted from the proof of Proposition 3.1 and [7, 12]: taking $H\in C^{2}(\mathbb{R})$ such that $H^{\prime}(\xi)\geq 0$ for all $\xi\in\mathbb{R}$, $H(\xi)=0$ $\,\forall\;\!\;\!\xi\leq 0$, $H(\xi)=1$ $\forall\;\xi\geq 1$, and given $\delta>0$ (arbitrary), let $H_{\mbox{}_{\scriptstyle\!\>\!\delta}}\!\in C^{2}(\mathbb{R})$ be defined by $H_{\mbox{}_{\scriptstyle\!\>\!\delta}}(\xi)\!\>\!:=H(\xi/\delta)$. Also, given $\>\!h>0$, \>\!\mbox{\footnotesizeR}>0, let $\>\!\zeta_{\mbox{}_{R}}\!\in C^{1}(\mathbb{R}^{n})\>\!$ be the cut-off function used in the proof of Proposition 2.2. Letting $u_{\mbox{}_{\scriptstyle h}}\!(\cdot,t)$, $v_{\mbox{}_{\scriptstyle h}}\!(\cdot,t)$ denote the Steklov regularizations of $u(\cdot,t)$, $v(\cdot,t)$, respectively, and setting $\>\!\theta(\cdot,t)\!\>\!:=u(\cdot,t)-v(\cdot,t)$, $\theta_{\mbox{}_{\scriptstyle\!\;\!h}}\!(\cdot,t)\!\>\!:=u_{\mbox{}_{\scriptstyle\!\;\!h}}\!(\cdot,t)-v_{\mbox{}_{\scriptstyle\!\;\!h}}\!(\cdot,t)$, we may proceed as follows. Taking ${\displaystyle\;\!\phi(x)=H_{\mbox{}_{\scriptstyle\!\>\!\delta}}\!\;\!(\theta_{\mbox{}_{\scriptstyle\!\;\!h}}\!\>\!(x,t))\;\!\zeta_{\mbox{}_{R}}\!\;\!(x)}$ in the equations (2.2) for $\>\!u_{\mbox{}_{\scriptstyle\!\;\!h}}\!(\cdot,t)$, $v_{\mbox{}_{\scriptstyle\!\;\!h}}\!(\cdot,t)$, subtracting one from the other and integrating the result in the interval $(\,\!t_{0},\>\!t\,\!)$, where $\>\!0<t_{0}<t$, we get, letting \>\!h\;\!\mbox{\scriptsize\searrow}\;\!0\>\! and \>\!t_{0}\>\!\mbox{\scriptsize\searrow}\;\!0,

               {\displaystyle\int_{\mbox{}_{\scriptstyle|\,x\,|\,<\,R}}G_{\mbox{}_{\scriptstyle\!\>\!\delta}}\!\;\!(\>\!\theta(x,t))\;\!\;\!\zeta_{\mbox{}_{R}}\!\;\!(x)\,dx\;+\>\!\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\>\!\mu(\tau)\!\int_{\mbox{}_{\scriptstyle|\,x\,|\,<\,R}}H_{\!\>\!\delta}^{\,\prime}(\theta)\;\!\>\!\langle\;\!\;\!\mbox{\boldmatha}(u,v),\>\!\nabla\theta\;\!\;\!\rangle\>\zeta_{\mbox{}_{R}}\!\;\!(x)\;\!\;\!dx\,d\tau}

          {\displaystyle\leq\,\int_{\mbox{}_{\scriptstyle|\,x\,|\,<\,R}}G_{\mbox{}_{\scriptstyle\!\>\!\delta}}\!\;\!(\theta_{0}(x))\;\!\;\!\zeta_{\mbox{}_{R}}\!\;\!(x)\,dx\;+\>\!\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\>\!\mu(\tau)\!\int_{\mbox{}_{\scriptstyle\!\>\!R/2\,<\,|\,x\,|\,<\,R}}|\,H_{\mbox{}_{\scriptstyle\!\>\!\delta}}(\theta)\,|\cdot|\,\mbox{\boldmatha}(u,v)\,|\cdot|\,\nabla\zeta_{\mbox{}_{R}}\!\;\!(x)\,|\;dx\,d\tau\;\;\!+}

{\displaystyle\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\>\!\mu(\tau)\!\int_{\mbox{}_{\scriptstyle|\,x\,|\,<\,R}}H_{\!\>\!\delta}^{\,\prime}(\theta)\,\,|\,[\,\mbox{\boldmath\tilde{f}}\,]\,|\cdot|\,\nabla\theta\,|\;\zeta_{\mbox{}_{R}}\!\;\!(x)\;dx\,d\tau\;\!\;\!+\int_{\mbox{}_{\scriptstyle\!\;\!0}}^{\;\!t}\!\!\>\!\mu(\tau)\!\int_{\mbox{}_{\scriptstyle\!\,\!R/2\,<\,|\,x\,|\,<\,R}}|\,H_{\mbox{}_{\scriptstyle\!\>\!\delta}}(\theta)\,|\cdot|\,[\,\mbox{\boldmath\tilde{f}}\,]\,|\cdot|\,\nabla\zeta_{\mbox{}_{R}}\!\;\!(x)\,|\;dx\,d\tau\!\;\!,}

where $\;\!G_{\mbox{}_{\scriptstyle\!\>\!\delta}}\!\;\!(\eta)=\int_{0}^{\>\!\eta}H_{\mbox{}_{\scriptstyle\!\>\!\delta}}\!\;\!(\xi)\,d\xi$, \;\![\,\mbox{\boldmath\tilde{f}}\,]\equiv[\,\mbox{\boldmath\tilde{f}}\,](x,\tau)=\mbox{\boldmath\tilde{f}}(x,\tau,u(x,\tau))-\mbox{\boldmath\tilde{f}}(x,\tau,v(x,\tau)), {\displaystyle\mbox{\boldmath\tilde{f}}\!:=\!\;\!\mbox{\boldmathf}\!+\mbox{\boldmathg}}, and {\displaystyle\;\!\mbox{\boldmatha}(u,v)\>\!=|\,\nabla u(x,\tau)\,|^{\>\!p-2}\;\!\nabla u(x,\tau)\,\!-|\,\nabla v(x,\tau)\,|^{\>\!p-2}\;\!\nabla v(x,\tau)}, as before. ​​ ​From this point, we repeat the steps in the proof of Proposition 3.1, using now that $\;\!G_{\mbox{}_{\scriptstyle\!\;\!\delta}}\!(\theta)\rightarrow\theta_{\mbox{}_{\!\>\!+}}\!\;\!$ as $\delta\rightarrow 0\!\;\!\;\!$: in case (i), we let $\delta\rightarrow 0$ and \mbox{\footnotesizeR}\rightarrow\infty to obtain (3.8), and in case (ii) we reverse the order, letting this time \mbox{\footnotesizeR}\rightarrow\infty and then $\delta\rightarrow 0$ to arrive at (3.8), as claimed.**

The proof of (3.9) follows exactly the same lines, except that this we take $H\in C^{2}(\mathbb{R})$ satisfying ​: $H^{\prime}(\xi)\leq 0$ for all $\xi\in\mathbb{R}$, $H(\xi)=1$ $\,\forall\;\!\;\!\xi\leq-1$, and $H(\xi)=0$ $\forall\;\xi\geq 0$. $\Box$

A direct consequence of (3.8) (or of (3.9)) is the following comparison principle.

Proposition 3.3. *Let $\;\!u(\cdot,t),\;\!v(\cdot,t)$, 0<t\leq\mbox{\smallT}\!\;\!, be given solutions of $\,\!(1.1a)$, $(1.2)$ corresponding to initial states ${\displaystyle\;\!u_{0},\>\!v_{0}\in L^{1}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})}$, respectively. Then

[TABLE]

*provided that $:$ $($i$)$ $p\geq n$, and $f$, $g$ satisfy $\!\;\!\;\!(3.1)\!\;\!$ and $\!\;\!\;\!(3.2)$ above, or $($when $\;\!2<p<n\!\;\!\;\!)\!\!:$ $($ii$)$ $\;\!\kappa\geq 1-2/p$, $\;\!\gamma\geq 1-2/p$,  ​and  $f$ , $g$  satisfy $\!\;\!\;\!(3.3)\!\;\!$ and $\!\;\!\;\!(3.4)$, respectively.

Acknowledgements. This work was partly supported by CNPq (Ministry of Science and Technology, Brazil), Grant # 154037/2011-7 and by CAPES (Ministry of Education, Brazil), Grant # 1212003/2013. The authors also express their gratitude to Paulo R. Zingano (UFRGS, Brazil) for some helpful suggestions and discussions.