## Complete blow-up after $$T_{\max}$$ for the solution of a semilinear heat equation.(English)Zbl 0653.35037

Let $$\Omega$$ be a bounded open subset of $${\mathbb{R}}^ N$$with a smooth boundary $$\partial \Omega$$. Consider the problem $(P)\quad u_ t- \Delta u=f(u)\quad in\quad \Omega \times (0,T),$
$u=0\quad on\quad \partial \Omega \times (0,T),\quad u(x,0)=u_ 0(x)\quad for\quad all\quad x\in \Omega,$ where f: $${\mathbb{R}}$$ $$+\to {\mathbb{R}}$$ $$+$$ is locally Lipschitz, nondecreasing and $$f(0)=0$$. If $$u_ 0$$ is a continuous function on $${\bar \Omega}$$, there exists a unique classical solution u of (P) defined on $$[0,T_{\max})$$ and such that $$u\in {\mathcal C}^{2,1}({\bar \Omega}\times (0,T_{\max}))\cap {\mathcal C}(\Omega \times [0,T_{\max}))$$ with $$\lim_{t\to T_{\max}} \| u\|_{\infty}=\infty$$ if $$T_{\max}<\infty$$. A well-known result asserts that if u is large enough and $$f(u)=u$$ p, $$p>1$$, for example, then $$T_{\max}<\infty$$ (this is the case when $$()| \nabla u_ 0| \quad 2-1/(p+1)\int_{\Omega}| u_ 0|^{p+1}<0).$$
In what follows, we suppose that $$T_{\max}<+\infty$$. Assume $$f_ n: {\mathbb{R}}$$ $$+\to {\mathbb{R}}$$ $$+$$ is a sequence of functions such that (a) for each n, $$u\to f_ n(u)$$ is globally Lipschitz, non decreasing, $$f_ n(0)=0$$, (b) for each u, $$n\to f_ n(u)$$ is increasing and converges to f(u). Let $$u_ n$$ be the unique global classical solution of $(P_ n)\quad u_{nt}-\Delta u_ n=f_ n(u_ n)\quad in\quad \Omega \times (0,+\infty),$
$u_ n=0\quad on\quad \partial \Omega \times (0,+\infty),\quad u_ n(x,0)=u_ 0(x)\quad for\quad all\quad x\in \Omega.$ We say that f satisfies (h) if: (h) f is convex and $$\exists \gamma >1$$, $$a\geq 0$$ such that $$u\to f(u)/u^{\gamma}$$ is nondecreasing on $$(a,+\infty)$$. Our main result is Theorem 1. Let $$u_ 0\in L^{\infty}(\Omega)$$, $$u_ 0\geq 0$$. Suppose that one of the following hypotheses holds:
(H1) $$\Omega$$ convex and if $$N\geq 2$$, there exists $$p\in (1,N/(N-2))$$ and $$c>0$$ such that $$0\leq f'(u)\leq C(u^{p-1}+1)$$ for all $$u\geq 0$$, $$u_ 0\in W_ 0^{1,1}(\Omega)$$, $$\Delta u_ 0+f(u_ 0)\geq 0$$ in $${\mathcal D}'(\Omega)$$. (No hypothesis on f for $$N=1.)$$
(H2) f satisfies (h) and $$u_ 0\in W_ 0^{1,1}(\Omega)$$, $$\Delta u_ 0+f(u_ 0)\geq 0$$ in $${\mathcal D}'(\Omega).$$
(H3) f is convex and there exists $$p\in (1,(N+2)/(N-2))$$ such that $$0\leq \lim_{u\to \infty} (f(u)/u$$ $$p)<\infty.$$
Then (i) $$\lim_{n\to \infty}u_ n(x,t)=u(x,t)$$ for all $$(x,t)\in \Omega \times [0,T_{\max})$$, (ii) $$\lim_{n\to \infty}u_ n(x,t)=\infty$$ for all $$(x,t)\in \Omega \times (T_{\max},\infty)$$.
Reviewer: Y.Ebihara

### MSC:

 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B65 Smoothness and regularity of solutions to PDEs
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