\documentclass[reqno]{amsart} %\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 77, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/77\hfil Blow-up of solutions] {Blow-up of solutions to a nonlinear wave equation} \author[Svetlin G. Georgiev \hfil EJDE-2004/77\hfilneg] {Svetlin Georgiev Georgiev} \address{University of Sofia, Faculty of Mathematics and Informatics, Department of Differential Equations, Bulgaria} \email{sgg2000bg@yahoo.com} \date{} \thanks{Submitted March 16, 2004. Published May 26, 2004.} \subjclass[2000]{35L10, 35L50} \keywords{Wave equation, blow-up, hyperbolic space} \begin{abstract} We study the solutions to the the radial 2-dimensional wave equation \begin{gather*} \chi_{tt}-{1\over r}\chi_r-\chi_{rr}+{{\sinh2\chi}\over {2r^2}}=g, \\ \chi(1, r)=\chi_{\circ}\in {\dot H}^{\gamma}_{\rm rad},\quad \chi_t(1, r)=\chi_1 \in {\dot H}^{\gamma-1}_{\rm rad}, \end{gather*} where $r=|x|$ and $x$ in $\mathbb{R}^2$. We show that this Cauchy problem, with values into a hyperbolic space, is ill posed in subcritical Sobolev spaces. In particular, we construct a function $g(t, r)$ in the space $L^p([0,1]L_{\rm rad}^q)$, with ${1\over p}+{2\over q}=3-\gamma$, $0<\gamma<1$, $p\geq 1$, and $1 1$ \item[(H4)] $0<\alpha\leq 2-q$, $\beta>0$ \item[(H5)] Either $\beta>\alpha$ with $\frac{\beta}{\alpha}<\frac{q}{2p(q-1)}$ or $\beta<\alpha$ with $\frac{\beta}{\alpha} > \frac{q(2p-1)}{2p}$. \end{itemize} Note that when (H3) holds, $q\leq 2$; because if $q>2$ then $3-\gamma={1\over p}+{2\over q}<1+1=2$, from where $\gamma>1$ which contradicts $0<\gamma<1$. Let $z^{1/\alpha}={r\over {t^{\beta/\alpha}}}$. Then $z^{2/\alpha}={r^2\over t^{2\beta/\alpha}}$, \begin{gather*} {{\partial z^{2/\alpha}}\over {\partial t}}=-{{2\beta}\over\alpha} {1\over t}z^{2/\alpha},\quad {{\partial^2 z^{2/\alpha}}\over {\partial t^2}}= {{2\beta(\alpha+2\beta)}\over{\alpha^2}} {1\over {t^2}}z^{2/\alpha}, \\ {{\partial z^{2/\alpha}}\over {\partial r}}= {2\over {t^{{\beta}\over\alpha}}}z^{1\over\alpha},\quad {{\partial^2 z^{2/\alpha}}\over {\partial r^2}}= {2\over {t^{2\beta\over\alpha}}}. \end{gather*} Let $f$ be a function satisfying (H1)--(H2) and let \begin{gather} \chi_{\circ}= \begin{cases} f(r^2)&\mbox{for } r\leq 1,\\ 0& \mbox{for } r\geq 1, \end{cases} \label{e3} \\ \chi_1= \begin{cases} -{{2\beta}\over \alpha}r^2f'(r^2)&\mbox{for } r\leq 1,\\ 0 &\mbox{for } r\geq 1, \end{cases} \label{e4}\\ B_1={{4\beta^2}\over {\alpha^2}}z^{4/\alpha}f''\bigl(z^{2/\alpha}\bigr) +{{2\beta(\alpha+2\beta)}\over {\alpha^2}}z^{2/\alpha} f'\bigl(z^{2/\alpha}\bigr), \label{e5} \\ B_2=z^{2/\alpha}f''(z^{2/\alpha})+f'(z^{2/\alpha}), \label{e6} \\ g=\begin{cases} {{B_1}\over {t^2}}-{4\over {t^{{2\beta}/\alpha}}}B_2+ {{\sinh(2{\bar\chi})}\over {2r^2}}&\mbox{for } r\leq t^{\beta/\alpha},\\ 0 &\mbox{for } r\geq t^{\beta/\alpha}, \end{cases} \label{e7} \end{gather} and let $${\bar\chi}=\begin{cases} f(z^{2/\alpha})&\mbox{for } r\leq t^{\beta/\alpha}\\ 0&\mbox{for } r\geq t^{\beta/\alpha}. \end{cases} \label{e8}$$ Note that ${\bar \chi}$ is a solution of \eqref{e1}--\eqref{e2}. Indeed, for $z\leq 1$ we have \begin{gather*} {\bar \chi}_t=-{{2\beta}\over {t\alpha}}z^{2/\alpha}f'(z^{2/\alpha}), \\ {\bar \chi}_{tt}={{4\beta^2}\over {\alpha^2}}{1\over {t^2}}z^{4\over\alpha} f''(z^{2/\alpha})+{{2\beta(\alpha+2\beta)}\over {\alpha^2}} {{z^{2/\alpha}}\over {t^2}}f'(z^{2/\alpha}), \\ {\bar\chi}_r={2\over {t^{\beta/\alpha}}}z^{1\over\alpha}f'(z^{2/\alpha}), \\ {\bar\chi}_{rr}={4\over {t^{{2\beta}\over {\alpha}}}}z^{2/\alpha} f''(z^{2/\alpha})+{2\over {t^{{2\beta}\over\alpha}}}f'(z^{2/\alpha}). \end{gather*} Then, for $z\leq 1$, we have $${\bar\chi}_{tt}-{1\over r}{\bar\chi}_r-{\bar\chi}_{rr}+ {{\sinh{2{\bar\chi}}}\over {2r^2}}= {{B_1}\over {t^2}}-{4\over {t^{{2\beta}\over\alpha}}}B_2+ {{\sinh(2{\bar\chi})}\over {2r^2}}\equiv g,$$ \begin{lemma} \label{lm1} Under Assumptions (H1)-(H5), the function $g$ defined by \eqref{e7} is in the space $L^p([0, 1]L^q_{\rm rad})$. \end{lemma} \begin{proof} Note that $$\|g\|_{L^q_{\rm rad}}^q=\int_0^{t^{\beta/\alpha}}|g|^qr\,dr =\int_0^{t^{\beta/\alpha}}\Bigr|{{B_1}\over {t^2}} -{4\over {t^{{2\beta}\over\alpha}}}B_2+ {{\sinh(2{\bar\chi})}\over {2r^2}}\Bigl|^qr\,dr.$$ By making the change of variable $r=z^{1\over\alpha}t^{\beta/\alpha}$, $$dr=t^{\beta/\alpha}{1\over\alpha}z^{{1\over\alpha}-1}dz,\quad r\,dr=t^{{2\beta}\over\alpha}{1\over\alpha}z^{{2\over\alpha}-1}dz$$ (with $t$ fixed) and \begin{align*} \|g\|_{L^q_{\rm rad}}^q &={{t^{{2\beta}\over\alpha}}\over\alpha} \int_0^1\Bigl|B_1{1\over {t^2}}-{4\over {t^{{2\beta}\over\alpha}}}B_2+ {{sinh(2{\bar\chi})}\over {2t^{{2\beta}\over\alpha}z^{2/\alpha}}} \Bigl|^qz^{{2\over\alpha}-1}dz\\ &\leq c_1 {{t^{{{2\beta}\over\alpha}-2q}}\over\alpha}\int_0^1|B_1|^q z^{{2\over\alpha}-1}dz+c_2 {{t^{{{2\beta}\over\alpha}(1-q)}}\over\alpha} \int_0^1|B_2|^qz^{{2\over\alpha}-1}dz \\ &\quad + c_3 {{t^{{{2\beta}\over\alpha}(1-q)}}\over\alpha} \int_0^1\Bigl|{{\sinh(2{\bar\chi})}\over {2z^{2/\alpha}}}\Bigl|^q z^{{2\over\alpha}-1}dz. \end{align*} Let $I_1=\int_0^1|B_1|^qz^{{2\over\alpha}-1}dz,\quad I_2=\int_0^1|B_2|^qz^{{2\over\alpha}-1}dz,\quad I_3=\int_0^1\Bigl|{{\sinh(2{\bar\chi})}\over {2z^{2/\alpha}}}\Bigl|^q z^{{2\over\alpha}-1}dz.$ From the definition of $B_1$ and $B_2$ and since (H1) and (H4) hold (we note that $\alpha\in (0, 1)$ because \$1