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Zbl 0648.35042
Friedman, Avner; Giga, Yoshikazu
A single point blow-up for solutions of semilinear parabolic systems. (English)
[J] J. Fac. Sci., Univ. Tokyo, Sect. I A 34, 65-79 (1987). ISSN 0040-8980

Consider the system $$ (1.1)\quad u\sb t-\alpha u\sb{xx}=f(v)\ (-a<x<a,\ t>0), $$ $$ (1.2)\quad v\sb t-\beta v\sb{xx}=g(u)\ (-a<x<a,\ t>0) $$ with $$ (1.3)\quad u(\pm a,t)=0\ (t>0),\ u(x,0)=\phi (x)\ (-a<x<a), $$ $$ (1.4)\quad v(\pm a,t)=0\quad (t>0),\ v(x,0)=\psi (x)\ (-a<x<a), $$ where $\alpha >0$, $\beta >0$, and assume:\par (1.5) $\phi(x)=\phi(-x)$, $\phi(x)\ge 0$, $\phi\in C\sp 1[- a,a];\phi'(x)\le 0$ if $0<x<a$, $\phi (a)=0$; $\psi (x)=\psi (-x)$, $\psi(x)\ge 0$, $\psi\in C\sp 1[-a,a];\psi'(x)\le 0$ if $0<x<a$, $\psi(a)=0,$ \par (1.6) $f,g\in C\sp 1(R\sp 1),$ $f(s)>0$, $g(s)>0$ if $s>0$; $f'(s)>0$, $g'(s)>0$ if $s>0.$ \par Set $$ H\sb{\alpha}w=w\sb t-\alpha w\sb{xx},\quad Q\sb{\sigma}=\{(x,t);\quad -a<x<a,\quad 0<t<\sigma \}. $$ Then there exists a unique classical solution of (1.1)-(1.4) in some $Q\sb{t\sb 0}$, and $u\ge 0$, $v\ge 0$ by the maximum principle. Let $T=\sup t\sb 0$, for all $t\sb 0$ as above. We claim $$ (1.7)\quad \sup\sb{Q\sb{\sigma}} u\to \infty \ if\ \sigma \to T. $$ Further we assume that, for some $M>1$, $$ (2.1)\quad pf(v)\le vf'(v)\ if\ v>M,\ p>1;\ qg(u)\le ug'(u)\ if\ u>M,\ q>1 $$ and that the solution (u,v) satisfies the estimates: $$ (2.2)\quad u\le C(v\sp{\gamma}+1);\quad v\le C(u\sp{1/\gamma}+1),\ C>0,\ \gamma >0,\ p>\gamma,\ q>1/\gamma. $$ Then we see: Suppose that u and v solves (1.1), (1.2) with (1.3)-(1.6). If the conditions (2.1), (2.2) are satisfied, then there is a single blow-up point.
[Y.Ebihara]

MSC 2000:: *35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions of PDE
35A05 General existence and uniqueness theorems (PDE)
35K15 Second order parabolic equations, initial value problems
35K45 Systems of parabolic equations, initial value problems

Keywords: semilinear; classical solution; maximum principle; single blow-up point

Cited in: Zbl 1171.35059

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