\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 63, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2003 Southwest Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2003/63\hfil Weak solutions] {Weak solutions for a viscous $p$-Laplacian equation} \author[Changchun Liu\hfil EJDE--2003/63\hfilneg] {Changchun Liu} \address{Changchun Liu \hfill\break Department of Mathematics, Nanjing Normal University \\ Nanjing 210097, China \hfill\break Department of Mathematics, Jilin University, Changchun 130012, China} \email{mathlcc@21cn.com} \date{} \thanks{Submitted August 5, 2002. Published June 10, 2003.} \subjclass[2000]{35G25, 35Q99, 35K55, 35K70} \keywords{Pseudo-parabolic equations, existence, uniqueness} \begin{abstract} In this paper, we consider the pseudo-parabolic equation $u_t-k\Delta u_t=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$. By using the time-discrete method, we establish the existence of weak solutions, and also discuss the uniqueness. \end{abstract} \maketitle \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} This paper concerns the study of the viscous $p$-Laplacian equation $$\frac{\partial u}{\partial t} -k\frac{\partial\Delta u}{\partial t} =\mathop{\rm div}(|\nabla u|^{p-2}\nabla u),\quad x\in\Omega,\; p>2, \eqno(1.1)$$ with boundary condition $$u\big|_{\partial\Omega}=0, \eqno(1.2)$$ and initial condition $$u(x, 0)=u_0(x),\quad x\in\Omega. \eqno(1.3)$$ Here $\Omega$ is a bounded domain in $\mathbb{R}^N$ and $k>0$ is the viscosity coefficient. The term $k\frac{\partial\Delta u}{\partial t}$ in (1.1) is interpreted as due to viscous relaxation effects, or viscosity; hence, the equation (1.1) is called viscous $p$-Laplacian equations''. The well-known $p$-Laplacian equation is obtained by setting $k=0$. Equation (1.1) arises as a regularization of the pseudo-parabolic equation $$\frac{\partial u}{\partial t}-k\frac{\partial\Delta u}{\partial t}=\Delta u, \eqno(1.4)$$ which arises in various physical phenomena. (1.4) can be assumed as a model for diffusion of fluids in fractured porous media \cite{b1,d1,c2}, or as a model for heat conduction involving a thermodynamic temperature $\theta=u-k\Delta u$ and a conductive temperature $u$ \cite{t1,c1}. Equation (1.4) has been extensively studied, and there are many outstanding results concerning existence, uniqueness, regularity, and special properties of solutions, see for example \cite{c2,d1,d2,n1,r1,s1,t2}. To derive (1.4), B. D. Coleman, R. J. Duffin and V. J. Mizel considered a special kinematical situation, of nonsteady simple shearing flow \cite{c2}. In fact, when the influence of many factors, such as the molecular and ion effects, are considered, one has the nonlinear relation $\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ in stead of $\Delta u$ in right-hand side of (1.4). Hence, we obtain (1.1). Equation (1.1) is something like the $p$-Laplacian equation, but many methods which are useful for the $p$-Laplacian equation are no longer valid for this equation. Because of the degeneracy, problem (1.1)-(1.3) does not admit classical solutions in general. So, we study weak solutions in the sense of following \noindent{\bf Definition} A function $u$ is said to be a weak solution of (1.1)-(1.3), if the following conditions are satisfied: \begin{enumerate} \item $u\in L^\infty(0, T;W^{1, p}_0(\Omega))\cap C(0,T;H^1(\Omega))$, $\frac{\partial u}{\partial t}\in L^\infty(0, T;W^{-1, p'}(\Omega))$, where $p'$ is conjugate exponent of $p$. \item For $\varphi\in C_0^\infty(Q_T)$ and $Q_T=\Omega\times(0,T)$, $$\iint_{Q_T}u\frac{\partial\varphi}{\partial t} dx\,dt+k\iint_{Q_T}\nabla u \frac{\partial \nabla \varphi}{\partial t} dx\,dt -\iint_{Q_T}|\nabla u|^{p-2}\nabla u\nabla \varphi dx\,dt=0\,.$$ \item $u(x,0)=u_0(x)$. \end{enumerate} In this paper, we discuss first the existence of weak solutions. Most proofs of existence for (1.4) are based on the Yoshida approximations \cite{d2}, but these methods do not apply to (1.1). Our method for proving the existence of weak solutions is based on a time discrete method that constructs approximate solutions. Later on, we discuss the uniqueness of a solution. For simplicity we set $k=1$ in this paper. \section{Existence of weak solutions} \begin{theorem} \label{thm2.1} If $u_0\in W_0^{1, p}(\Omega)$ with $p>2$, then problem (1.1)-(1.3) has at least one solution. \end{theorem} We use the a discrete method for constructing an approximate solution. First, divide the interval $(0,T)$ in $N$ equal segments and set $h=\frac T N$. Then consider the problem \begin{gather*} \frac1h(u_{k+1}-u_k)-\frac1h(\Delta u_{k+1}-\Delta u_k)=\mathop{\rm div}(|\nabla u_{k+1}|^{p-2}\nabla u_{k+1}), \tag{2.1}\\ u_{k+1}|_{\partial\Omega}=0, \quad k=0, 1, \dots, N-1, \tag{2.2} \end{gather*} where $u_0$ is the initial value. \begin{lemma} \label{lm2.1} For a fixed $k$, if $u_k\in H^1_0(\Omega)$, problem (2.1)-(2.2) admits a weak solution $u_{k+1}\in W_0^{1, p}(\Omega)$, such that for any $\varphi\in C_0^\infty(\Omega)$, have $$\frac1h\int_\Omega(u_{k+1}-u_k)\varphi dx +\frac1h\int_\Omega(\nabla u_{k+1}-\nabla u_k)\nabla\varphi dx +\int_\Omega|\nabla u_{k+1}|^{p-2}\nabla u_{k+1}\nabla \varphi dx=0. \eqno(2.3)$$ \end{lemma} \begin{proof} On the space $W^{1,p}_0(\Omega)$, we consider the functionals \begin{gather*} \Phi_1[u]=\frac1p\int_\Omega|\nabla u|^p dx, \\ \Phi_2[u]=\frac12\int_\Omega|u|^2 dx, \\ \Phi_3[u]=\frac12\int_\Omega|\nabla u|^2 dx, \\ \Psi[u]=\Phi_1[u]+\frac1h\Phi_2[u]+\frac1h\Phi_3[u]-\int_\Omega fu dx, \end{gather*} where $f\in H^{-1}(\Omega)$ is a known function. Using Young's inequality, there exist constants $C_1, C_2>0$, such that \begin{align*} \Psi[u]&=\frac1p\int_\Omega|\nabla u|^p dx+\frac1{2h}\int_\Omega|u|^2 dx +\frac1{2h}\int_\Omega|\nabla u|^2 dx-\int_\Omega f u \,dx\\ &\geq C_1\int_\Omega|\nabla u|^p dx-C_2\|f\|_{-1}; \end{align*} hence $\Psi[u]\to\infty$, as $\|u\|_{1, p}\to+\infty$. Here $\|u\|_{1,p}$ denotes the norm of $u$ in $W^{1,p}_0(\Omega)$. Since the norm is lower semi-continuous and $\int_\Omega fudx$ is a continuous functional, $\Psi[u]$ is weakly lower semi-continuous on $W_0^{1, p}(\Omega)$ and satisfying the coercive condition. From \cite{c3} we conclude that there exists $u_*\in W^{1,p}_0(\Omega)$, such that $$\Psi[u_*]=\inf\Psi[u],$$ and $u_*$ is the weak solutions of the Euler equation corresponding to $\Psi[u]$, $$\frac1h u-\frac1h\Delta u-\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)=f.$$ Taking $f=(u_k-\Delta u_k)/h$, we obtain a weak solutions $u_{k+1}$ of (2.1)--(2.2). The proof is complete. \end{proof} Now, we need to establish a priori estimates, for the weak solutions $u_{k+1}$ of (2.1)--(2.2). First, we define the weak solutions of (1.1)--(1.3) as follows: \begin{gather*} u^h(x, t)=u_k(x), \quad kh1$;$\int_Rj(s)ds=1$. For$h>0$, define$j_h(s)=\frac1hj(\frac s h)$and $$\eta_h(t)=\int_{t-t_2+2h}^{t-t_1-2h}j_h(s)ds.$$ Clearly$\eta_h(t)\in C_0^\infty(t_1,t_2)$,$\lim_{h\to0^+}\eta_h(t)=1$, for all$t\in(t_1,t_2)$. In the definition of weak solutions, choose$\varphi=\varphi_k(x,t)\eta_h(t). We have \begin{align*} \int_{t_1}^{t_2}\int_\Omega u\varphi_k j_h(t-t_1-2h)dx\,dt-\int_{t_1}^{t_2}\int_\Omega u\varphi_k j_h(t-t_2+2h)dx\,dt& \\ +\int_{t_1}^{t_2}\int_\Omega \nabla u\nabla\varphi_k j_h(t-t_1-2h)dx\,dt-\int_{t_1}^{t_2}\int_\Omega\nabla u\nabla\varphi_k j_h(t-t_2+2h)\, dx\,dt& \\ +\int_{t_1}^{t_2}\int_\Omega u\varphi_{kt}\eta_h dx\,dt+ \int_{t_1}^{t_2}\int_\Omega \nabla u\nabla\varphi_{kt}\eta_h \,dx\,dt&\\ -\int_{t_1}^{t_2}\int_\Omega |\nabla u|^{p-2} \nabla u\nabla\varphi_k\eta_h \,dx\,dt&=0. \end{align*} Observe that \begin{align*} &\big|\int_{t_1}^{t_2}\int_\Omega u\varphi_k j_h(t-t_1-2h)dx\,dt-\int_\Omega(u\varphi_k)|_{t=t_1}dx\big| \\ &=\big|\int_{t_1+h}^{t_1+3h}\int_\Omega u\varphi_k j_h(t-t_1-2h)dx\,dt -\int_{t_1+h}^{t_1+3h}\int_\Omega (u\varphi_k)|_{t=t_2} j_h(t-t_1-2h)dx\,dt\big| \\ &\leq \sup_{t_1+hT-h. \end{cases} \begin{theorem} \label{thm3.1} Problem (1.1)-(1.3) admits only one weak solution. \end{theorem} \begin{proof} Suppose u_1,u_2 are two solutions of (1.1)-(1.3), then \begin{align*} \int_\Omega(u_1(x,\tau)-u_2(x,\tau))_{h\tau}\varphi(x)dx +\int_\Omega((\nabla u_1-\nabla u_2)_h(x,\tau))_\tau\varphi(x)dx& \\ +\int_\Omega(|\nabla u_1|^{p-2}\nabla u_1 -|\nabla u_2|^{p-2}\nabla u_2)_h(x,\tau)\nabla\varphi dx&=0. \end{align*} For a fixed \tau, we take \varphi(x)=[u_1-u_2]_h\in W^{1,p}_0(\Omega), and hence \begin{align*} &\int_\Omega(u_1(x,\tau)-u_2(x,\tau))_{h\tau}(u_1-u_2)_h dx\\ &+\int_\Omega\nabla(u_1(x,\tau)-u_2(x,\tau))_{h\tau}\nabla(u_1-u_2)_h dx \\ &=-\int_\Omega[(|\nabla u_1|^{p-2}\nabla u_1-|\nabla u_2|^{p-2}\nabla u_2)_h](x,\tau) \nabla(u_1-u_2)_hdx, \end{align*} i.e., \begin{align*} &\int_\Omega(u_1(x,\tau)-u_2(x,\tau))_{h\tau}(u_1-u_2)_h dx\\ &+\int_\Omega\nabla(u_1(x,\tau)-u_2(x,\tau))_{h\tau}\nabla(u_1-u_2)_h dx\\ &=-\int_\Omega[(|\nabla u_1|^{p-2}\nabla u_1 -|\nabla u_2|^{p-2}\nabla u_2)_h](x,\tau)\nabla(u_1-u_2)_hdx. \end{align*} Integrating the above equality with respect to \tau over (0,t), \int_\Omega|(u_1-u_2)_h|^2(x,t)dx+\int_\Omega|\nabla(u_1-u_2)_h|^2(x,t)dx\leq0,$we have$\int_\Omega|(u_1-u_2)_h|^2dx=0$; therefore,$u_1=u_2$. \end{proof} \subsection*{Acknowledgment} The author would like to thank referee for his/her valuable suggestions and for providing the references E. Di Benedtto \& M. Pierre \cite{d1} and E. Di Benedetto \& R. E. Showalter \cite{d2}. \begin{thebibliography}{00} \bibitem{b1} G. I. Barwnblatt, Iv. P. Zheltov, and I. N. Kochina, \textit{Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks}, J. Appl. Math. Mech., 24(1960), 1286-1303. \bibitem{c3} Kungching Chang, \textit{Oritical point theory and its applications}, Shanghai Sci. Tech. Press, Shanghai, 1986. \bibitem{c1} P. J. Chen and M. E. Gurtin, \textit{On a theory of heat conduction involving two temperatures}, Z. Angew. Math. Phys., 19(1968), 614-627. \bibitem{c2} B. D. Coleman, R. J. Duffin, and V. J. Mizel, \textit{Instability, uniqueness and non-existence theorems for the equations,$u_t=u_{xx}-u_{xtx}\$ on a strip}, Arch. Rat. Mech. Anal., 19(1965), 100-116. \bibitem{d1} E. DiBenedtto and M. Pierre, \textit{On the maximum principle for pseudoparabolic equations}, Indiana Univ. Math. J., 30(6)(1981), 821-854. \bibitem{d2} E. DiBenedetto and R. E. Showalter, \textit{Implicit decenerate evolution equations and applications}, SIAM J. Math. Anal., 12(5)(1981), 731-751. \bibitem{n1} A. Novick-Cohen and R. L. Pego, \textit{Stable patterns in a viscous diffusion equation}, Trans. Amer. Math. Soc., 324(1991), 331--351. \bibitem{r1} V. R. G. Rao and T. W. Ting, \textit{Solutions of pseudo-heat equation in whole space}, Arch. Rat. Mech. Anal., 49(1972), 57-78. \bibitem{s1} R. E. Showalter and T. W. Ting, \textit{Pseudo-parabolic partial differential equations}, SIAM J. Math. Anal., 1(1970), 1-26. \bibitem{t1} T. W. Ting, \textit{A cooling process according to two-temperature theory of heat conduction}, J. Math. Anal. Appl., 45(1974), 23-31. \bibitem{t2} T. W. Ting, \textit{Parabolic and pseudoparabolic partial differential equations}, J. Math. Soc. Japan, 21(1969), 440-453. \end{thebibliography} \end{document}