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A single point blow-up for solutions of semilinear parabolic systems. (English) Zbl 0648.35042
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: (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 $0<x<a$, $\psi(a)=0,$ (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 $s>0.$ 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.
Reviewer: Y.Ebihara

35K55Nonlinear parabolic equations
35B40Asymptotic behavior of solutions of PDE
35A05General existence and uniqueness theorems (PDE) (MSC2000)
35K15Second order parabolic equations, initial value problems
35K45Systems of second-order parabolic equations, initial value problems