\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 141, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/141\hfil Behavior of the maximal solution] {Behavior of the maximal solution of the Cauchy problem for some nonlinear pseudoparabolic equation as $|x|\to\infty$} \author[T. Kavitova\hfil EJDE-2012/141\hfilneg] {Tatiana Kavitova} \address{Tatiana Kavitova \newline Department of Mathematics, Vitebsk State University, Moskovskii pr. 33, 210038 Vitebsk, Belarus} \email{KavitovaTV@tut.by} \thanks{Submitted March 22, 2012. Published August 20, 2012.} \subjclass[2000]{35B40, 35B51, 35K70} \keywords{Pseudoparabolic equation; comparison principle; stabilization} \begin{abstract} We prove a comparison principle for solutions of the Cauchy problem of the nonlinear pseudoparabolic equation $u_t=\Delta u_t+ \Delta\varphi(u) +h(t,u)$ with nonnegative bounded initial data. We show stabilization of a maximal solution to a maximal solution of the Cauchy problem for the corresponding ordinary differential equation $\vartheta'(t)=h(t,\vartheta)$ as $|x|\to\infty$ under certain conditions on an initial datum. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this article we consider the Cauchy problem for the pseudoparabolic equation $$\label{in:eq} u_t=\Delta u_t+ \Delta\varphi(u) +h(t,u),\quad x\in\mathbb{R}^n,\;t>0,$$ subject to the initial condition $$\label{in:id} u(x,0)=u_0(x),\quad x\in\mathbb{R}^n.$$ Put $R_{+}=(0,+\infty)$ and $\Pi_T=\mathbb{R}^n\times[0,T]$, $n\geq1,\;T>0$. Throughout this paper we suppose that the functions $\varphi$ and $h$ satisfy the following conditions: $$\label{in:koof} \parbox{10cm}{ $$\varphi(p)$$ is defined for $$p\geq0$$, $$h(t,p)$$ is defined for $$t\geq0$$ and $$p\geq0$$, \varphi(p)\in C^2(\overline R_{+})\cap C^3(R_{+}), h(t,p)\in C^{0,\alpha}_{\rm loc}(\overline R_{+}\times\overline R_{+}) \cap C^{0,1+\alpha}_{\rm loc}(\overline R_{+}\times R_+), 0<\alpha<1, h(t,0)=0, t\in \overline R_{+}, \varphi(p)+h(t,p) does not decrease in $$p$$ for all $$t\in \overline R_{+}$$.}$$ Assume that one of the following conditions is satisfied: $$\label{in:koof+h(x,p)} h(t,p)\geq0,\quad t\in\overline R_{+},\;p\in\overline R_{+},$$ or $$\label{in:koof-h(x,p)} h(t,p) \text{ does not increase in $$p$$ for all $$t\in \overline R_{+}$$}.$$ Let the initial data have the following properties: \begin{gather} u_0(x)\in C^2(\mathbb{R}^n), \quad 0\leq u_0(x)\leq M \,(M\geq0), \; x\in\mathbb{R}^n,\label{in:u0(x)}\\ \lim_{|x|\to\infty}u_0(x)=M.\label{in:limu0(x)} \end{gather} Equations $u_t=\Delta u_t+ \Delta u^p+u^q$ and $u_t=\Delta u_t+ \Delta (u^l+u^p)-u^p$, where $p,l\geq2,\;q>0$, are typical examples of equation~\eqref{in:eq} satisfying~\eqref{in:koof} under conditions~\eqref{in:koof+h(x,p)} and~\eqref{in:koof-h(x,p)} respectively. If we suppose $u_0(x)\equiv M$ in~\eqref{in:id} then a solution of the Cauchy problem for the corresponding ordinary differential equation $$\label{in:odu} \vartheta'(t)=h(t,\vartheta), \quad \vartheta(0)=M$$ will be a solution of~\eqref{in:eq}, \eqref{in:id}. \begin{remark} \label{rmk1.1} \rm We note that problem~\eqref{in:odu} may have more than one solution. Indeed, we put $h(t,\vartheta)=\vartheta^p, \, 00,\;(x,t)\in \Pi_T$. Obviously, the function $w(x,t)=u_2(x,t)-u_1(x,t)$ satisfies the problem \begin{gather}\label{CP:urav1} w_t=\Delta{w_t}+\Delta{(aw)}+bw, \quad (x,t)\in \mathbb{R}^n \times (0,T),\\ \label{CP:nach1} w(x,0)=u_{02}(x)-u_{01}(x),\quad x\in \mathbb{R}^n. \end{gather} Here \begin{equation*} a(x,t)=\int_0^1{\varphi'(z(\theta))\,d\theta},\quad b(x,t) =\int_0^1{h_{z(\theta)}(t,z(\theta))\,d\theta}, \end{equation*} where $z(\theta)=\theta u_2(x,t)+(1-\theta)u_1(x,t)$. By~\eqref{in:koof} the functions $a(x,t)$ and $b(x,t)$ have the following properties: $$\label{CP:ab} \begin{gathered} a(x,t)\in C^{2,0}(\Pi_T ),\quad b(x,t)\in C^{\alpha,0}_{\rm loc}(\Pi_T ),\\ a(x,t)+b(x,t)\geq0,\quad |a(x,t)|+|b(x,t)|\leq m,\quad (x,t)\in\Pi_T, \end{gathered}$$ where $m$ is some positive constant. \begin{lemma}\label{CP:uniq:th} Let $a(x,t)$ and $b(x,t)$ be functions such that conditions~\eqref{CP:ab} are satisfied. Then a solution of~\eqref{CP:urav1}, \eqref{CP:nach1} is unique. \end{lemma} The proof of the above lemma is analogous to the proof the same statement for problem~\eqref{in:eq}, \eqref{in:id} with $h(t,p)=0$ in~\cite{Gladkov2}. Let $$Q$$ be a bounded domain in $$\mathbb{R}^n$$ for $n\geq1$ with a smooth boundary $\partial Q$. We denote $Q_{T}=Q\times(0,T)$ and $S_{T}=\partial Q\times(0,T)$. Let us consider the equation $$\label{CP:Fur_eq} u_t=\Phi(x,t,u)+F(u(\cdot,t)),\quad (x,t)\in Q_T,$$ subject to the initial data $$\label{CP:Fur_in_datum} u(x,0)=u_0(x),\quad x\in Q,$$ where the function $\Phi(x,t,\xi)$ is defined on the set $\overline Q\times[0,T]\times R$ and $F(u(\cdot,t))$ is a nonlinear integral operator. \begin{definition} \label{CP:sub-super} \rm We shall say that a function $\sigma^+(x,t)\in C^{0,1}(Q_T)$, $-\infty0$. Let the functions of the sequence $w_l(x,t)\;(l=1,2,\dots)$ satisfy equation~\eqref{CP:int-diff-equat} in $Q_{l,T}=Q_l\times(0,T)$ and initial data~\eqref{CP:nach12} in $Q_l$. According to Lemma~\ref{CP:exist:lemma} there exists a solution $w_l(x,t)$ of~\eqref{CP:int-diff-equat}, \eqref{CP:nach12} in $Q_{l,T}$ such that $$\label{CP:ocenkaWl} 0\leq w_l(x,t)\leq M_1e^{2mt},\quad (x,t)\in Q_{l,T}.$$ Differentiating \eqref{CP:int-diff-equat} with respect to $x_i$ $(i=1,\dots,n)$ we obtain \begin{align*} w_{lt{x_i}}(x,t) &=-a_{x_i}(x,t)w_l(x,t)-a(x,t)w_{lx_i}(x,t)\\ &\quad +\int_{Q_l}G_{nx_i}(x,\xi,l)[a(\xi,t)+b(\xi,t)]w_l(\xi,t)\,d\xi,\quad (x,t)\in Q_{l,T}, \end{align*} from which we find that $$\label{CP:dif} w_{lx_i}(x,t)=e^{-\int_0^t{a(x,\tau)}\,d\tau} \Big[u_{02}(x)-u_{01}(x)+\int_0^t{p_l(x,\tau)e^{\int_0^\tau{a(x,\tau_1)\; d\tau_1}}\,d\tau}\Big],$$ where $$p_l(x,t)=-a_{x_i}(x,t)w_l(x,t)+\int_{Q_l}{G_{nx_i}(x,\xi,l)[a(\xi,t) +b(\xi,t)]w_l(\xi,t)\,d\xi}.$$ It follows from~\eqref{ÑP:Green}, \eqref{CP:int-diff-equat}, \eqref{CP:ocenkaWl} and \eqref{CP:dif} that absolute values of functions $w_l,\;w_{lt},\;w_{l{x_i}}\;(i=1,2,\dots,n)$ are uniformly bounded with respect to $l$ on each set $\overline Q_{k,T}$, where $k$ is an arbitrary fixed natural number, $k 0$ and~\eqref{in:koof+h(x,p)} hold then problem~\eqref{in:eq}, \eqref{in:id} has an unique solution. \end{remark} Indeed, in the same way as it was done in~\cite{Kavitova} we can show the existence of the solution $u(x,t)$ of problem~\eqref{in:eq}, \eqref{in:id} such that $u(x,t)\geq m>0$. \section{Auxiliary statements}\label{Beh} Let condition~\eqref{in:koof+h(x,p)} hold. We consider the Cauchy problem for equation~\eqref{in:eq} subject to the initial condition $$\label{Beh:id} u(x,0)=u_0(x)+\varepsilon,\quad x\in\mathbb{R}^n.$$ If we suppose $u_0(x)\equiv M$ in~\eqref{Beh:id} then a solution of the Cauchy problem for the corresponding ordinary differential equation $$\label{Beh:zCodu} \vartheta'(t)=h(t,\vartheta),\quad \vartheta(0)=M+\varepsilon$$ will be a solution of~\eqref{in:eq}, \eqref{Beh:id}. Suppose that the solution $\vartheta_\varepsilon(t)$ of~\eqref{Beh:zCodu} exists on $[0,T_{0,\varepsilon})$, $T_{0,\varepsilon}\leq+\infty$. It is easy to show (see~\cite{Kavitova}) that a solution $u_\varepsilon(x,t)$ of the integral equation \label{Beh:Ue} \begin{aligned} u_\varepsilon(x,t) &=u_0(x)+\varepsilon-\int_0^t{\varphi(u_\varepsilon(x,\tau))}\,d\tau\\ &\quad +\int_0^t\int_{\mathbb{R}^n}{\mathcal{E}_n(x-\xi) [\varphi(u_\varepsilon(\xi,\tau))+h(\tau,u_\varepsilon(\xi,\tau))]}\,d\xi\,d\tau \end{aligned} for any $T_\varepsilon0$. By the induction assumption for any $\delta_0>0$ there exists a constant $A_0=A_0(\delta_0,\varepsilon,T_{*,\varepsilon},k_0)$ such that if $|x|>A_0$ and $0\leq t\leq T_{*,\varepsilon}$ then \begin{equation*} |u_{k_0,\varepsilon}(x,t)-\vartheta_{\varepsilon}(t)|<\delta_0. \end{equation*} From \eqref{Beh:ve1} and \eqref{Beh:seqU} we have \begin{align*} &|u_{k_0+1,\varepsilon}(x,t)-\vartheta_{\varepsilon}(t)|\\ &=|u_0(x)+\varepsilon-\int_0^t{\varphi(u_{k_0,\varepsilon}(x,\tau))}\,d\tau +\int_0^t\int_{\mathbb{R}^n} \mathcal{E}_n(x-\xi) \Big[\varphi(u_{k_0,\varepsilon}(\xi,\tau))\\ &\quad +h(\tau,u_{k_0,\varepsilon}(\xi,\tau)) \Big]\,d\xi\,d\tau -M-\varepsilon-\int_0^t{h(\tau,\vartheta_{\varepsilon}(\tau))}\,d\tau| \\ &\leq|u_0(x)-M|+\int_0^t{|\varphi'(\theta_1(x,\tau))|\cdot|u_{k_0,\varepsilon} (x,\tau)-\vartheta_\varepsilon(\tau)|}\,d\tau\\ &\quad +\int_0^t\int_{|\xi|\leq A_0}{\mathcal{E}_n(x-\xi) (|\varphi'(\theta_2(\xi,\tau))|+|h_{\theta_3}(\tau,\theta_3(\xi,\tau))|) |u_{k_0,\varepsilon}(\xi,\tau)-\vartheta_\varepsilon(\tau)|}\,d\xi\,d\tau\\ &\quad +\int_0^t\int_{|\xi|>A_0}{\mathcal{E}_n(x-\xi) (|\varphi'(\theta_2(\xi,\tau))|+|h_{\theta_3}(\tau,\theta_3(\xi,\tau))|) |u_{k_0,\varepsilon}(\xi,\tau)-\vartheta_\varepsilon(\tau)|}\,d\xi\,d\tau, \end{align*} where $\frac{\varepsilon}{2}\leq\theta_i\leq M+\frac{3\varepsilon}{2} +\vartheta_\varepsilon(T_{*,\varepsilon})$, $i=1,2,3$. By \eqref{in:limu0(x)} for any $\delta_1>0$ there exists a constant $A_1=A_1(\delta_1)$ such that $|u_0(x)-M|<\delta_1$ if $|x|>A_1$. Using the property of the fundamental solution $\mathcal{E}_n$ and \eqref{Beh:rav_ogr} we obtain that for any $\delta_2>0$ there exists a constant $A_2=A_2(\delta_2,\varepsilon)$ such that if $|x|>A_2$ then \begin{align*} &\int_0^t\int_{|\xi|\leq A_0}{\mathcal{E}_n(x-\xi)(|\varphi' (\theta_2(\xi,\tau))|+|h_{\theta_3}(\tau,\theta_3(\xi,\tau))|) |u_{k_0,\varepsilon}(\xi,\tau)-\vartheta_\varepsilon(\tau)|}\,d\xi\,d\tau\\ &<\delta_2. \end{align*} Hence, we obtain \begin{equation*} |u_{k_0+1,\varepsilon}(x,t)-\vartheta_{\varepsilon}(t)| <\delta_1+\delta_2+T_{*,\varepsilon}(2\lambda+\nu)\delta_0, \end{equation*} where $$\lambda=\max_{\frac{\varepsilon}{2}\leq\theta\leq M+\frac{3\varepsilon}{2} +\vartheta_{\varepsilon}(T_{\varepsilon})}|\varphi'(\theta)|, \quad \nu=\max_{0\leq t\leq T_{\varepsilon},\,\frac{\varepsilon}{2}\leq\theta \leq M+\frac{3\varepsilon}{2}+\vartheta_{\varepsilon}(T_{\varepsilon})} |h_\theta(t,\theta)|.$$ Let $\delta_0=\frac{\delta}{3T_{*,\varepsilon}(2\lambda+\nu)}$, $\delta_1=\frac{\delta}{3},\;\delta_2=\frac{\delta}{3}$ and $A=\max(A_0,A_1,A_2)$ then $$|u_{k_0+1,\varepsilon}(x,t)-\vartheta_{\varepsilon}(t)|<\delta$$ if $0\leq t\leq T_{*,\varepsilon}$ and $|x|>A$. It follows that for any $\delta>0$ by suitable choosing $k$ and $A$ we obtain \begin{align*} |u_\varepsilon(x,t)-\vartheta_\varepsilon(t)| &=|u_\varepsilon(x,t)-u_{k,\varepsilon}(x,t)+u_{k,\varepsilon}(x,t) -\vartheta_\varepsilon(t)|\\ &\leq|u_\varepsilon(x,t)-u_{k,\varepsilon}(x,t)| +|u_{k,\varepsilon}(x,t)-\vartheta_\varepsilon(t)|< \delta \end{align*} for $0\leq t\leq T_{*,\varepsilon}$ and $|x|>A$. \end{proof} \begin{lemma}\label{Beh:ue-ve:lemm} Let~\eqref{in:koof}, \eqref{in:koof+h(x,p)}, \eqref{in:u0(x)} and \eqref{in:limu0(x)} hold. Then for any $T_\varepsilon0$ and $00$ such that for any $\varepsilon<\varepsilon_1$ the inequality $T0$ such that for any $\varepsilon<\varepsilon_2$, \begin{gather}\label{vspom1} |u_{\varepsilon}(x,t)-u(x,t)|<\frac{\delta}{3},\quad (x,t)\in\Pi_T, \\ \label{vspom2} |\vartheta_{\varepsilon}(t)-\vartheta(t)|<\frac{\delta}{3},\quad t\in[0,T]. \end{gather} Put $\varepsilon_0=\min(\varepsilon_1,\varepsilon_2)$. From Lemma~\ref{Beh:ue-ve:lemm} there exists the constant $A_0=A_0(\delta,\varepsilon_0,T)$ such that for any $|x|>A_0$ we obtain $$\label{vspom3} |u_{\varepsilon_0}(x,t)-\vartheta_{\varepsilon_0}(t)| <\frac{\delta}{3},\quad (x,t)\in\Pi_T.$$ By \eqref{vspom1}--\eqref{vspom3} we conclude that by suitable choosing $\varepsilon=\varepsilon_0$ and $A=A_0$, \begin{align*} |u(x,t)-\vartheta(t)| &=|u(x,t)-u_\varepsilon(x,t)+u_\varepsilon(x,t)+\vartheta_\varepsilon(t) -\vartheta_\varepsilon(t)-\vartheta(t)|\\ &\leq|u_\varepsilon(x,t)-u(x,t)|+|u_\varepsilon(x,t)-\vartheta_\varepsilon(t)| +|\vartheta_\varepsilon(t)-\vartheta(t)|<\delta \end{align*} for $0\leq t\leq T$ and $|x|>A$. Let~\eqref{in:koof-h(x,p)} hold. Consider the Cauchy problems $$\label{in:eqvsp} \begin{gathered} \omega_t=\Delta \omega_t+ \Delta\varphi(\omega) +h(t,\omega) -h(t,\varepsilon),\quad x\in\mathbb{R}^n,\;t>0,\\ \omega(x,0)=u_0(x)+\varepsilon,\quad x\in\mathbb{R}^n, \end{gathered}$$ and $$\label{Beh:zCoduvsp} g'(t)=h(t,g)-h(t,\varepsilon),\quad g(0)=M+\varepsilon.$$ We suppose that the maximal nonnegative solution $g_\varepsilon(t)$ of~\eqref{Beh:zCoduvsp} exists on $[0,T_{0,\varepsilon})$, $T_{0,\varepsilon}\leq+\infty$. It is easy to show (see~\cite{Kavitova}) that for any \$T_\varepsilon