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Zbl 0576.35068
Friedman, Avner; McLeod, Bryce
Blow-up of positive solutions of semilinear heat equations.
(English)
[J] Indiana Univ. Math. J. 34, 425-447 (1985). ISSN 0022-2518

The following problem is discussed: \par $u\sb t=\Delta u+f(u)$ in $\Omega$ $\times (0,T)$, $f\in C\sp 1$, $f(s)>0$ if $s>0$, $\Omega \subset R\sp n$; $u(x,0)=\phi (x)$ if $x\in \Omega$, $\phi \in C\sp 1({\bar \Omega})$, $\phi\ge 0$, $\phi =0$ on $\partial \Omega$, $u(x,t)=0$, $x\in \partial \Omega$, $0<t<T.$ \par As $U(t)=\max\sb{x\in \Omega} u(x,t)$ grows with t, it is assumed that $T<\infty$ is the supremum of all $\sigma$ such that the solution to the problem above exists for $t<\sigma$, and that $U(T-)=+\infty:x\in \Omega$ is a blow-up point if there is $\{(x\sb mt\sb m)\}$, $t\sb m\uparrow T$, $x\sb m\to X$, and $u(x\sb m,t\sb m)\to \infty$, $m\to \infty.$ \par A partial outline of the results can be divided into: \par Case (i): $\Omega$ is a ball, u(.,t) are radial functions, $\phi\sb r\le 0$. The authors prove that the only blow-up point is $x=0$. For the case $f(u)=(u+\lambda)\sp p$, $p>1$, $\lambda\ge 0$, they obtain: $\vert u(r,t)\vert \le C/r\sp{2/(\lambda -1)}$, any $\gamma <p$, $\lim\sb{t\to T} \sup \Vert u(.,t)\Vert\sb{L\sp q(\Omega)}<\infty$, $q<n(p-1)/2$, $\lim\sb{t\to T} \inf \Vert u(.,t)\Vert\sb{L\sp q(\Omega)}=\infty$, $q>n(p-1)/2$; if moreover $\Delta \phi +f(\phi)\ge 0$ and $n=1,2$; or $n\ge 3$ and $p\le (n+2)/n-2$, $(T-t)\sp{1/(p-1)}u(r,t)\to (1/p- 1)\sp{1/p-1}$ as $t\uparrow T$ provided $r\le C(T-t)\sp{1/2}$, some $C>0.$ \par Case (ii): Non symmetric, $\Omega$ is a convex domain: the blow-up points lie in a compact subset of $\Omega$. For the particular $f(u)=(u+\lambda)\sp p$, u(x,t)$\le\sp C/(T-t)\sp{1/(p-1)}$, all $x\in \Omega$, and $\lim\sb{t\to T} \inf \Vert u(.,t)\Vert\sb{L\sp q(\Omega)}=\infty$ if $q>n(p-1)/2.$ \par In the one-dimensional case $n=1$, if $\phi$ ' changes sign just once, then the solution blow-up at a single point. The authors extend some of the results to the case of a boundary condition $\partial u/\partial \nu +\beta u=0$, $x\in \partial \Omega$, $0<t<T$, $\beta\ge 0$ and $\nu$ the outward normal vector.
[J.E.Bouillet]
MSC 2000:
*35K60 (Nonlinear) BVP for (non)linear parabolic equations
35B40 Asymptotic behavior of solutions of PDE
35K20 Second order parabolic equations, boundary value problems
35B05 General behavior of solutions of PDE

Keywords: positive solutions; semilinear heat equations; blow-up

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