\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 53, pp. 1--5.\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--2003/53\hfil Blow up of solutions] {Blow up of solutions to semilinear wave equations} \author[Mohammed Guedda\hfil EJDE--2003/53\hfilneg] {Mohammed Guedda} \address{Mohammed Guedda \hfill\break Lamfa, CNRS UMR 6140, Universit\'e de Picardie Jules Verne, Facult\'e de Math\'ematiques et d'Informatique, 33, rue Saint-Leu 80039 Amiens, France} \email{guedda@u-picardie.fr} \date{} \thanks{Submitted November 15, 2002. Published May 3, 2003.} \subjclass[2000]{35L70, 35B40, 35L15} \keywords{Blow up, conformal compactification} \begin{abstract} This work shows the absence of global solutions to the equation $$u_{tt} = \Delta u + p^{-k}|u|^m,$$ in the Minkowski space $\mathbb{M}_0=\mathbb{R}\times\mathbb{R}^N$, where $m > 1$, $(N-1)m < N+1$, and $p$ is a conformal factor approaching 0 at infinity. Using a modification of the method of conformal compactification, we prove that any solution develops a singularity at a finite time. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \section{Introduction } This note presents nonexistence results of the problem $$\label{eq:1.1} u_{tt} = \Delta u + p^{-k}|u|^m,$$ posed in the Minkowski space $\mathbb{M}_0=\mathbb{R}\times\mathbb{R}^N,$ $N \geq 1$, with the initial condition $$\label{eq:1.2} u(0,x) = u_0(x),\quad u_t(0,x) = u_1(x), \quad x \in \mathbb{R}^N\,.$$ Here $p$ is a conformal factor approaching $0$ at infinity, the parameter $m > 1$ satisfies $(N-1)m < N+1$. The constant $k=sm-(N+3)/2$, where $s = (N-1)/2$. The initial data $u_0,u_1$ belong to $X := \{f: f \in C^\infty_0(\mathbb{R}^N); 0\not\equiv f \geq 0 \}$. Note that the factor $p^{-k}$ approaches 0 as $|x|$ tends to infinity for $(N-1)m < N+1$. This work is motivated by a recent paper by Belchev, Kepka and Zhou \cite{ABC} in which Problem (\ref{eq:1.1}),(\ref{eq:1.2}) with $1 < m < 1 +(2/N)$ is considered. The authors proved the following theorem using a modification of the technique of conformal compactification due to Penrose \cite{P} and developed by Christodolou \cite{C} and Baez {\it et al.} \cite{B}. \begin{theorem} \label{thm1.1} Let $1 < m < 1 + (2/N)$ and $u$ be a solution to \eqref{eq:1.1},\eqref{eq:1.2} with $u_0,u_1 \in X$. Then $u$ blows up in finite time. \end{theorem} Attention will be given to show that {\rm (\ref{eq:1.1}),(\ref{eq:1.2})} does not possess global solutions for $m > 1$ and $(N-1)m < N+1$, complementing in this way the results in \cite{ABC}. Theorem \ref{thm1.1} is also announced in \cite{As} and the proof is similar to the one given in \cite{ABC}. Our main result is the following: \begin{theorem} \label{thm1.2} Let $m > 1, (N-1)m < N+1$ and $u$ be a solution to \eqref{eq:1.1},\eqref{eq:1.2} with $u_0,u_1 \in X$. Then $u$ blows up in finite time. \end{theorem} The proof of this theorem is given in Section 2 which contains also a result of the nonexistence of global solutions in the case $u_1 \leq 0$. \section{Proof of the main result} \subsection*{Notation and preliminary results} To clarify the proof, we consider as in \cite{ABC} the conformal map $c$ from the Minskowski space $\mathbb{M}_0$ to the Einstein universe $\mathbb{E}:=\mathbb{R}\times S^N$. Here $S^N$ is the unit sphere in $\mathbb{R}^{N+1}$ and $c(t,x) := c(t,x_1,x_2,\dots ,x_N) = (T,Y_1,Y_2,\dots ,Y_{N+1}),$ where \begin{gather*} \sin T = pt,\; \cos T = p\big(1-\frac{t^2-x^2}{4}\big), \quad T \in (-\pi,\pi),\\ Y_j =px_j,\; j=1,\dots ,N,\quad Y_{N+1}= p\big(1+\frac{t^2-x^2}{4}\big),\\ p=\Big(t^2+ \big(1-\frac{t^2-x^2}{4}\big)^2\Big)^{-1/2}. \end{gather*} The space $\mathbb{M}_0$ is equipped with the Minkowski metric: $g = dt^2-dx^2,$ and the space $\mathbb{E}$ with the metric $\tilde{g} = dT^2-dS^2,$ where $dS^2$ is the canonical metric on $S^N$. Therefore, $c$ is a conformal map between the Lorentz manifolds $(\mathbb{M}_0,g)$ and $(\mathbb{E},\tilde{g})$, with the conformal factor $p$; that is, $c^\star \tilde{g} = p^2g$. Next, we consider as in \cite{ABC}, the function $v$ defined in $\mathbb{E}$ by $u = R^{-2/(m-1)}p^{s}v, \quad R > 0,\; s =\frac{N-1}{2},$ where $u$ is a solution to (\ref{eq:1.1}), (\ref{eq:1.2}). Then $v$ satisfies $$\label{eq:2.1} \begin{gathered} (\mathcal{L}_c+s^2)v = |v|^m,\quad \mbox{on } \mathbb{E},\\ v(0,.) = R^{2/(m-1)}p_0^{-s}u_0\circ c^{-1},\\ v_T(0,.) = R^{(m+1)/(m-1)}p_0^{-(s+1)}u_1\circ c^{-1}, \end{gathered}$$ where $p_0 = \cos^2\frac{\rho}{2}$, $\rho \in [0,\pi)$ is the distance on $S^N$ from the north pole $T=Y_j=0,\; j=1,\dots ,N,\quad Y_{N+1}=1$ and $\mathcal{L}_c$ denotes the d'Alembertien in $\mathbb{E}$ relative to the metric $\tilde{g}$. Then the function $H(T) =\int_{S^N}v(T,.)dS$ satisfies (see \cite{ABC}) $$\label{eq:H} H'' \geq (C_0|H|^{m-1}-s^2)|H|,$$ for some positive constant $C_0$ independent of the parameter $R$. At the origin we have \label{eq:H0} \begin{aligned} H(0) &= R^{2/(m-1)-N}\int_{\mathbb{R}^N} \big(1+\frac{r^2}{4R^2}\big)^{-(N+1)/2}u_0dx, \\ & \geq R^{2/(m-1)-N}\int_{\mathbb{R}^N}\big(1+\frac{r^2}{4}\big)^{-(N+1)/2}u_0dx, \end{aligned} and \label{eq: H1} \begin{aligned} H'(0) &= R^{(m+1)/(m-1)-N}\int_{\mathbb{R}^N} \big(1+\frac{r^2}{4R^2}\big)^{-(N-1)/2}u_1dx,\\ & \geq R^{(m+1)/(m-1)-N}\int_{\mathbb{R}^N} \big(1+\frac{r^2}{4}\big)^{-(N-1)/2}u_1dx ,\quad r = |x|,\, R \geq 1. \end{aligned} \begin{proposition} \label{prop2.1} Let $H$ be a solution to \eqref{eq:H} where $H'(0) \geq 0$ and $H(0) > (\frac{s^2}{C_0})^{1/(m-1)}$. Then $H$ cannot be a global solution. \end{proposition} \begin{proof} By contradiction and assume that $H$ is global. By (\ref{eq:H}) we have $H''(0) > 0$. It follows that $H' > 0$ and then $H > (\frac{s^2}{C_0})^{1/(m-1)}$ on $(0,\varepsilon), \varepsilon$ small. Arguing in the same way, we deduce that $H' > 0$ and $H > (\frac{s^2}{C_0})^{1/(m-1)}$ on $(\varepsilon, \varepsilon+\varepsilon^\star)$. This shows, in particular that $H'(T) > 0,\quad H(T) > \big(\frac{s^2}{C_0}\big)^{1/(m-1)} \quad\mbox{and}\quad H''(T) > 0,$ for all $T > 0$. Next we claim that $H(T)$ goes to infinity with $T.$ First note that $H(T)$ has a limit as $T$ tends to infinity. Assume that this limit is finite. Since $H''$ is positive, $H'(T)$ goes to 0 as $T$ tends to infinity. Integrating inequality (\ref{eq:H}) over $(0,T)$ and passing to the limit yield $-H'(0) \geq \int_0^\infty (C_0H^{m-1}-s^2)HdT.$ The left side of the last inequality is non-positive while the right hand side is positive. This is impossible. Now using (\ref{eq:H}) and the fact that $H(\infty) =\infty$, $H'' \geq C_1 H^m,\quad \forall\ T > T_0,$ holds for some $T_0$ large and for some positive constant $C_1$. Therefore, $H$ develops a singularity since $m > 1$. \end{proof} \begin{remark} \label{rmk2.1} {\rm Note that, as inequality (\ref{eq:H}) is autonomous, if there exists $T_0$ such that $H(T_0) > (\frac{s^2}{C_0})^{1/(m-1)}$ and $H'(T_0) \geq 0$ the conclusion of the preceding proposition remains valid. } \end{remark} \begin{remark} \label{rmk2.2} {\rm The condition $H(0) > (\frac{s^2}{C_0})^{1/(m-1)}$ can be replaced by $H(0) \geq (\frac{s^2}{C_0})^{1/(m-1)}$ if $H'(0) > 0$.} \end{remark} \begin{remark} \label{rm2.3}{\rm In the case $1 < m < 1 + \frac{2}{N}$ we have $\ lim_{R\to\infty} R^{2/(m-1)-N}\int_{\mathbb{R}^N}\big(1+ \frac{r^2}{4}\big)^{-(N+1)/2}u_0\,dx =\infty.$ Hence we can choose $R > R_0$ such that $H(0) >(\frac{s^2}{C_0})^{1/(m-1)}$; therefore using Proposition \ref{prop2.1} we deduce Theorem \ref{thm1.1} for $1 < m < 1 +\frac{2}{N}$.} \end{remark} \begin{proof}[Proof of Theorem \ref{thm1.2}] Let $u$ be a local solution to (1.1), (1.2) where $(N-1)m < N+1,$ $m > 1$. Using the fact that $$\label{eq:Q} \lim_{R\to\infty} R^{(m+1)/(m-1)-N}\int_{\mathbb{R}^N}\big(1+\frac{r^2}{4}\big)^{-(N-1)/2}u_1dx = \infty,$$ we deduce from (2.4), that $H'(0) > Q,$ for $R > R_0$ large, where $$\label{e2.6} Q^2 := \frac{m-1}{m+1}C_0^{-2/(m-1)}s^{2(m+1)/(m-1)}.$$ Hence Theorem \ref{thm1.2} is a direct consequence of the following result which is valid for any $m > 1$. \end{proof} \begin{proposition} \label{prop2.2} Let $m > 1$ and $H$ be a solution to {\rm (2.2)} where $H(0) \geq 0$ and $H'(0) > Q$. Then there exists $T_1 > 0$ such that $H(T_1) \geq (\frac{s^2}{C_0})^{1/(m-1)}$, $H'(T_1) > 0$ and hence $H$ is not a global solution. \end{proposition} \begin{proof} Let $H$ be a solution to (2.2) such that $H(0) \geq 0$ and $H'(0) > Q$. Let us suppose that $H(0) < (\frac{s^2}{C_0})^{1/(m-1)}$, otherwise the proof follows from Proposition \ref{prop2.1}. Therefore, there exists $T_0 \leq \infty$ such that $0 < H(T) < (\frac{s^2}{C_0})^{1/(m-1)}$ and $H'(T) >0$ for all $T$ in $(0,T_0)$. Assume first that $T_0$ is finite and $H'(T_0) = 0$. Since the function $F(T) = \frac{1}{2}(H'(T))^2 - \frac{C_0}{m+1}H^{m+1}(T) +\frac{s^2}{2}H^2(T)$ is strictly increasing on $(0,T_0)$, thanks to (\ref{eq:H}), we get $F(T) \leq F(T_0) \leq \frac{1}{2}Q^{2}$, for all $0 \leq T < T_0$, in particular $F(0) \leq \frac{1}{2}Q^2$ which yields to $H'(0) \leq Q$. A contradiction. Next we suppose that $T_0 = \infty$. Since $H$ is monotone and bounded, there exists $0 < L \leq (\frac{s^2}{C_0})^{1/(m-1)}$ such that $\lim_{T\to\infty}H(T) = L$ and then there exists $T_n$ converging to infinity with $n$ such that $H'(T_n) \to 0$, as $n \to \infty$. Using again the function $F$ we deduce that $F(0) \leq \lim_{n\to \infty}F(T_n)$. Hence $H'(0) \leq Q$, a contradiction. Then there exists $T_1 > 0$ such that $H(T_1) > (\frac{s^2}{C_0})^{1/(m-1)}$, $H'(T_1) > 0$ and hence $H$ is not global thanks to Proposition \ref{prop2.1} and Remark \ref{rmk2.1}. \end{proof} \begin{corollary} Let $m > 1$ and let $u_0, u_1$ be in $X$ such that, for some positive $R,$ one of the following two conditions is satisfied \begin{enumerate} \item $R^{2/(m-1)-N}\int_{\mathbb{R}^N}\big(1+\frac{r^2}{4R^2}\big)^{-(N+1)/2}u_0dx > \big(\frac{s^2}{C_0}\big)^{1/(m-1)},$ \item $R^{(m+1)/(m-1)-N}\int_{\mathbb{R}^N}\big(1+\frac{r^2}{4R^2}\big)^{-(N-1)/2}u_ 1d x > Q$. \end{enumerate} Then Problem \eqref{eq:1.1},\eqref{eq:1.2} has no global solution. \end{corollary} \subsection*{Case $u_1 \leq 0$} In what follows we shall see that solutions to (1.1) may blow up in the case where $u_1 \in C^\infty_0(\mathbb{R}^N)$ is non-positive. \begin{theorem} \label{thm2.1} Let $m > 1$ and $u_0, -u_1$ in $X$ be such that $$\label{eq:initialdata} (H'(0))^2 - \frac{2C_0}{m+1}H^{m+1}(0) + s^2H^2(0) \leq{Q},\quad H(0) > \big(\frac{s^2}{C_0}\big)^{1/(m-1)},$$ where $Q$ is given by \eqref{e2.6}, $H(0) =R^{\frac{m+1}{m-1}}\int_{\mathbb{R}^N}\big(R^2+\frac{r^2}{4} \big)^{-\frac{N+1}{2}}u_0dx$ and $H'(0)=R^{2/(m-1)}\int_{\mathbb{R}^N}\big(R^2+\frac{r^2}{4} \big)^{-\frac{N-1}{2}}u_1dx,$ for some fixed $R > 0$. Then Problem \eqref{eq:1.1},\eqref{eq:1.2} has no global solution. \end{theorem} \begin{proof} Assume that $u_0$ and $u_1$ satisfy (\ref{eq:initialdata}) and are such that (\ref{eq:1.1}) has a global solution. Using Proposition \ref{prop2.1} we easily deduce that the function $H$ is strictly decreasing and $H > (\frac{s^2}{C_0})^{1/(m-1)}$ on $(0,T_0),$ for some $0 < T_0 \leq \infty.$ Now, a simple analysis shows that $H(T_0) = (\frac{s^2}{C_0})^{1/(m-1)}.$ Next, since $H' < 0$ the function $F(T) = \frac{1}{2}(H'(T))^2 - \frac{C_0}{m+1}H^{m+1}(T) +\frac{s^2}{2}H^2(T)$ is decreasing on $(0,T_0),$ thanks to (\ref{eq:H}). Therefore $F(0) > F(T_0) \geq\frac{1}{2}{Q},$ which contradicts (\ref{eq:initialdata}). \end{proof} \begin{thebibliography}{00} \bibitem{As} {M. Aassila,} \textit{Non existence de solutions globales de certaines \'equations d'ondes non lin\'eaires,} C. R. Acad. Sci. Paris, Ser. I 334 (2002) 961--966. \bibitem{AF} {C. Antonini, F. Merle,} \textit{Optimal bounds on positive blow up solutions for a semilinear wave equations,} Intern. Math. Res. Notices 21 (2001) 1143--1167. \bibitem{ABC} {E. Belchev, M. Kepka, Z. Zhou,} \textit{Finite-Time Blow-up of Solutions to Semilinear Wave Equations,} J. Funct. Anal. 190 (2002) 233--254. \bibitem{C} {D. Christodolou,} \textit{Global solutions of nonlinear hyperbolic equations for small initial data,} Comm. Pure Appl. Math. 39 (1986) 267--282. \bibitem{B} {J. C. Baez, I. Segal, Z. Zhou,} \textit{The global Goursat problem and Scattering for nonlinear wave equations,} J. Funct. Anal. 93 (1990) 239--269. \bibitem{P} {R. Penrose,} \textit{Conformal treatment of infinity, in relativity, groups and topology,} B. De Witt C. De Witt (Eds.), Gordon and Breach, 1963. \bibitem{Sh} {J. Shatah,} \textit{Weak solutions and the development of singularities in the SU(2) $\sigma$-model,} Comm. Pure Appl. Math. 41 (1988) 459--469. \bibitem{St} {W. Strauss,} \textit{Wave Equations,} American Mathematical Society, Providence, 1989. \end{thebibliography} \end{document}