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


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
Full Text: DOI


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