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% 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