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A single point blow-up for solutions of semilinear parabolic systems. (English) Zbl 0648.35042

Consider the system

(1·1)u t -αu xx =f(v)(-a<x<a,t>0),
(1·2)v t -βv xx =g(u)(-a<x<a,t>0)

with

(1·3)u(±a,t)=0(t>0),u(x,0)=φ(x)(-a<x<a),
(1·4)v(±a,t)=0(t>0),v(x,0)=ψ(x)(-a<x<a),

where α>0, β>0, and assume:

(1.5) φ(x)=φ(-x), φ(x)0, φC 1 [-a,a];φ ' (x)0 if 0<x<a, φ(a)=0; ψ(x)=ψ(-x), ψ(x)0, ψC 1 [-a,a];ψ ' (x)0 if 0<x<a, ψ(a)=0,

(1.6) f,gC 1 (R 1 ), f(s)>0, g(s)>0 if s>0; f ' (s)>0, g ' (s)>0 if s>0·

Set

H α w=w t -αw xx ,Q σ ={(x,t);-a<x<a,0<t<σ}·

Then there exists a unique classical solution of (1.1)-(1.4) in some Q t 0 , and u0, v0 by the maximum principle. Let T=supt 0 , for all t 0 as above. We claim

(1·7)sup Q σ uifσT·

Further we assume that, for some M>1,

(2·1)pf(v)vf ' (v)ifv>M,p>1;qg(u)ug ' (u)ifu>M,q>1

and that the solution (u,v) satisfies the estimates:

(2·2)uC(v γ +1);vC(u 1/γ +1),C>0,γ>0,p>γ,q>1/γ·

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

MSC:
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