Local behaviour near blow-up points for semilinear parabolic equations. (English) Zbl 0798.35023

The paper deals with the solutions of the problem \[ U_ t- U_{xx}= U^ p,\quad p>1\tag{1} \] defined in the cylinder \[ Q_ R= \bigl\{(x,t): - R< x< R, 0< t< 1\bigr\},\quad R>0\tag{2} \] which blow-up at \(t= 1\). The author investigates the asymptotic behaviour of such solutions near a blow-up point as \(t\uparrow 1\). It is proved that if \(U(t,x)\) is a solution of the problem (1), (2) which blows up at \(t=1\) and any blow-up point \(\bar x\) of \(U\) satisfies \(\bar x\in [-\delta,\delta]\), where \(\delta< R\), then the asymptotics are the same as those corresponding to the solutions defined on the whole line which assume bounded initial values. Also, the author investigates the case when the blow-up set reaches the boundary of the interval \((-R,R)\) and he proves that the solutions have different asymptotic behaviour in this case.
Reviewer: E.Minchev (Sofia)


35B40 Asymptotic behavior of solutions to PDEs
35K55 Nonlinear parabolic equations
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