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%_ ************************************************************************** %_ * The TeX source for AMS journal articles is the publishers TeX code * %_ * which may contain special commands defined for the AMS production * %_ * environment. Therefore, it may not be possible to process these files * %_ * through TeX without errors. To display a typeset version of a journal * %_ * article easily, we suggest that you retrieve the article in DVI, * %_ * PostScript, or PDF format. * %_ ************************************************************************** % Author Package file for use with AMS-LaTeX 1.2 \controldates{12-OCT-2001,12-OCT-2001,12-OCT-2001,12-OCT-2001} \documentclass{era-l} \issueinfo{7}{12}{}{2001} \dateposted{October 15, 2001} \pagespan{87}{93} \PII{S 1079-6762(01)00098-1} \copyrightinfo{2001}{American Mathematical Society} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \newcommand{\p}{\partial} \newcommand{\R}{\mathbf{R}} \newcommand{\N}{\mathbf{N}} \newcommand{\loc}{\text{\rm loc}} \newcommand{\divv}{\text{\rm div\,}} \begin{document} \title[Nonexistence results]{Some nonexistence results for higher-order evolution inequalities in cone-like domains} \author{Gennady G. Laptev} \address{Department of Function Theory, Steklov Mathematical Institute, Gubkina Street 8, Moscow, Russia} \email{laptev@home.tula.net} \thanks{The author was supported in part by RFBR Grant \#01-01-00884.} \commby{Guido Weiss} \date{April 7, 2001} \subjclass[2000]{Primary 35G25; Secondary 35R45, 35K55, 35L70} \begin{abstract} Nonexistence of global (positive) solutions of semilinear higher-order evolution inequalities \begin{equation*} \frac{\partial^k u}{\partial t^k}-\Delta u^m\ge |u|^q,\quad \frac{\partial^k u}{\partial t^k}-\Delta u\ge |x|^\sigma u^q,\quad \frac{\partial^ku}{\partial t^k}-\text{\rm div}\, (|x|^\alpha Du)\ge u^q \end{equation*} with $k=1,2,\dots$, in cone-like domains is studied. The critical exponents $q^*$ are found and the nonexistence results are proved for $10$ denotes the set $\{x\in K : |x|>R\}$ with full surface $\p K_R$. Recall that the Laplace operator $\Delta$ in polar coordinates $(r,\omega)$ has the form \begin{equation*} \Delta=\frac1{r^{N-1}}\frac\p{\p r}\left(r^{N-1}\frac\p{\p r}\right)+ \frac1{r^2}\Delta_\omega=\frac{\p^2}{\p r^2}+\frac{N-1}r \frac\p{\p r}+\frac1{r^2}\Delta_\omega, \end{equation*} where $\Delta_\omega$ denotes the Laplace--Beltrami operator on the unit sphere $S^{N-1}\subset\R^N$. We shall use the first Helmholtz eigenvalue $\lambda_\omega\equiv \lambda_1(K_\omega)>0$ and corresponding eigenfunction~$\Phi(\omega)$ for the Dirichlet problem of $\Delta_\omega$ in $K_\omega$, \begin{equation}\label{e:lambda} \begin{cases} \Delta_\omega\Phi+\lambda\Phi=0&\text{ in }K_\omega,\cr \Phi|_{\p K_\omega}=0.& \end{cases} \end{equation} It is well known that $\Phi(\omega)>0$ for $\omega\in K_\omega$. We assume $\Phi(\omega)\le1$. Let $\Omega$ be an unbounded domain in $\R^{N+1}$ with piecewise smooth boundary. We shall use the anisotropic Sobolev spaces $W^{2,k}_q(\Omega)$, and the local space $L_{q,\loc}(\Omega)$, whose elements belong to $L_q(\Omega')$ for any compact subset $\Omega'$: $\overline{\Omega'}\subset\Omega$. Denote the space of continuous functions by $C(\overline\Omega)$. The expression $\int_{\p K}\frac{\p u}{\p n}\,dx$ denotes the integral of the directional derivative of $u$ with respect to the outward normal $n$ to the cone lateral surface $\p K$. \section{Main results}\label{sec:2} Let $k\in\N$. Our model problem has the form \begin{equation}\label{e:1} \begin{cases} \frac{\p^ku}{\p t^k}-\Delta u\ge |u|^q& \text{in } K\times(0,\infty), \\ u|_{\p K\times[0,\infty)}\ge0,\\ \frac{\p^{k-1}u}{\p t^{k-1}}|_{t=0}\ge0,& u\not\equiv0. \end{cases} \end{equation} \begin{definition}\label{dfn:1} Let $u(x,t)\in C(\overline K\times[0,\infty))$ and suppose the locally summable traces $\frac{\p^i u}{\p t^i}$, $i=1,\dots,k-1$, as $t=0$, are well defined. The function $u(x,t)$ is called a weak solution of~\eqref{e:1} if for any nonnegative test function $\varphi(x,t)\in W^{2,k}_\infty(K\times(0,\infty))$ with compact support, such that $\varphi|_{\p K\times(0,\infty)}=0$, the following inequality holds: \begin{multline*} \int_0^\infty\int_{\p K}u\frac{\p\varphi}{\p n}\,dxdt +\int_0^\infty\int_{K} u\left((-1)^k\frac{\p^k\varphi}{\p t^k}-\Delta\varphi\right)\,dxdt \\ \ge\int_0^\infty\int_{K} |u|^q\varphi\,dxdt +\sum_{i=1}^{k-1}(-1)^i\int_K\frac{\p^{k-1-i}u}{\p t^{k-1-i}}(x,0) \frac{\p^i\varphi}{\p t^i}(x,0)\,dx \\ +\int_{K} \frac{\p^{k-1} u}{\p t^{k-1}}(x,0)\varphi(x,0)\,dx. \end{multline*} \end{definition} Let us introduce the parameters \begin{equation}\label{e:2} s^*=\frac{N-2}2+\sqrt{\left(\frac{N-2}2\right)^2+\lambda_\omega},\qquad s_*=-\frac{N-2}2+\sqrt{\left(\frac{N-2}2\right)^2+\lambda_\omega}; \end{equation} here $\lambda_\omega$ is the first eigenvalue of the problem~\eqref{e:lambda} introduced above. It is evident that $s^*-s_*=N-2$. These parameters go back to Kondrat'ev~\cite{Kondratiev:1967}. \begin{theorem}\label{thm:1} Let \begin{equation*} 11$, $q_2>1$ and \begin{equation*} \max\{\gamma_1,\gamma_2\}\ge\frac{s^*+2/k}{2}, \quad\text{ where }\quad \gamma_1=\frac{q_1+1}{q_1q_2-1},\quad \gamma_2=\frac{q_2+1}{q_1q_2-1}. \end{equation*} Then~\eqref{e:En.1} has no nontrivial global solution. \end{theorem} In this theorem the concept of solution is understood in the weak sense of Definition~\ref{dfn:1} (with some evident corrections; see for example~\cite{Laptev:2000,Laptev:2001}). For the parabolic system ($k=1$) \begin{equation*} \begin{cases} \frac{\p u}{\p t}-\Delta u\ge |v|^{q_1}& \text{in } K\times(0,\infty), \\ \frac{\p v}{\p t}-\Delta v\ge |u|^{q_2}& \text{in } K\times(0,\infty), \\ u|_{\p K\times[0,\infty)}\ge0,\quad v|_{\p K\times[0,\infty)}\ge0,\\ u|_{t=0}\ge0,\quad v|_{t=0}\ge0,\qquad &u\not\equiv0,\quad v\not\equiv0. \end{cases} \end{equation*} we obtain from Theorem~\ref{t:new4.1} the well-known condition~\cite{Levine:1990,DengLevine:2000} \begin{equation*} \max\left\{\frac{q_1+1}{q_1q_2-1},\frac{q_2+1}{q_1q_2-1}\right\} \ge\frac{s^*+2}2. \end{equation*} Let us consider the inequality of porous medium type, \begin{equation}\label{e:6.1} \begin{cases} \frac{\p^ku}{\p t^k}-\Delta u^m\ge |u|^q& \text{in } K\times(0,\infty), \qquad m\ge1,\quad q>m, \\ u|_{\p K\times[0,\infty)}\ge0,\\ \frac{\p^{k-1}u}{\p t^{k-1}}|_{t=0}\ge0,& u\not\equiv0. \end{cases} \end{equation} \begin{definition}\label{dfn:6.1} Let $u(x,t)\in C(\overline K\times[0,\infty))$ and suppose the locally summable traces $\frac{\p^i u}{\p t^i}$, $i=1,\dots,k-1$, as $t=0$, are well defined. The function $u(x,t)$ is called a weak solution of~\eqref{e:6.1} if for any nonnegative test function $\varphi(x,t)\in W^{2,k}_\infty(K\times(0,\infty))$ with compact support, such that $\varphi|_{\p K\times(0,\infty)}=0$, the following inequality holds: \begin{multline*} \int_0^\infty\int_{\p K}u^m\frac{\p\varphi}{\p n}\,dxdt +(-1)^k\int_0^\infty\int_{K} u\frac{\p^k\varphi}{\p t^k}\,dxdt -\int_0^\infty\int_{K} u^m\Delta\varphi\,dxdt \\ \ge\int_0^\infty\int_{K} |u|^q\varphi\,dxdt +\sum_{i=1}^{k-1}(-1)^i\int_K\frac{\p^{k-1-i}u}{\p t^{k-1-i}}(x,0) \frac{\p^i\varphi}{\p t^i}(x,0)\,dx \\ +\int_{K} \frac{\p^{k-1} u}{\p t^{k-1}}(x,0)\varphi(x,0)\,dx. \end{multline*} \end{definition} \begin{theorem}\label{thm:6.1} Let \begin{equation*} 1\le mm, \\ u|_{\p K\times[0,\infty)}\ge0,\\ u|_{t=0}\ge0,& u\not\equiv0. \end{cases} \end{equation*} has no nontrivial global solution in the sense of Definition~\ref{dfn:6.1}. \end{theorem} Now we turn our attention to the singular problem in cone-like domain $K_R$, $R>0$, \begin{equation}\label{e:9.1} \begin{cases} \frac{\p^ku}{\p t^k}-\divv(|x|^\alpha Du)\ge u^q& \text{ in } K_R\times(0,\infty), \\ u\ge0,\\ \frac{\p^{k-1} u}{\p t^{k-1}}|_{t=0}\ge0,&u\not\equiv0, \end{cases} \end{equation} where $-\infty<\alpha<2$. Remark that we deal with nonnegative solutions. \begin{definition}\label{dfn:9.1} Let $u(x,t)\in C(\overline K_R\times[0,\infty))$ and suppose the locally summable traces $\frac{\p^i u}{\p t^i}$, $i=1,\dots,k-1$, as $t=0$, are well defined. The function $u(x,t)$ is called a weak solution of~\eqref{e:9.1} if for any nonnegative test function $\varphi(x,t)\in W^{2,k}_\infty(K_R\times(0,\infty))$ with compact support, such that $\varphi|_{\p K_R\times(0,\infty)}=0$, the following inequality holds: \begin{multline*} \int_0^\infty\int_{\p K_R}u|x|^\alpha\frac{\p\varphi}{\p n}\,dxdt +\int_0^\infty\int_{K_R} u\left((-1)^k\frac{\p^k\varphi}{\p t^k} -\divv(|x|^\alpha D\varphi)\right)\,dxdt \\ \ge\int_0^\infty\int_{K_R} u^q\varphi\,dxdt +\sum_{i=1}^{k-1}(-1)^i\int_{K_R}\frac{\p^{k-1-i}u}{\p t^{k-1-i}}(x,0) \frac{\p^i\varphi}{\p t^i}(x,0)\,dx \\ +\int_{K_R} \frac{\p^{k-1} u}{\p t^{k-1}}(x,0)\varphi(x,0)\,dx. \end{multline*} \end{definition} Let us introduce the parameters \begin{align*}\label{e:5.2} &s^*_\alpha=\frac{\alpha+N-2} 2+\sqrt{\left(\frac{\alpha+N-2}2\right)^2+\lambda_\omega},\\ &s_{*\alpha}=-\frac{\alpha+N-2}2+ \sqrt{\left(\frac{\alpha+N-2}2\right)^2+\lambda_\omega}. \end{align*} \begin{theorem}\label{thm:9.1} Let $-\infty<\alpha<2$. If \begin{equation*} 1