Global solutions of a semilinear parabolic equation. (English) Zbl 0953.35065

Fila, Marek; Poláčik, Peter

Adv. Differ. Equ. 4, No. 2, 163-196 (1999).

The authors study radial solutions of the initial Dirichlet problem for the equation \(u_t=\Delta u+\lambda e^u\) on the \(N\)-dimensional unit ball for \(3\leq N\leq 9\). It is shown that every global classical solution is uniformly bounded while unbounded global \(L^1\)-solutions are constructed for some \(\lambda\). The proofs are based on zero number arguments. Similar results for the equation \(u_t=\Delta u^m+u^p\) with supercritical \(p\) were obtained by V. A. Galaktionov and J. L. Vázquez [Commun. Pure Appl. Math. 50, No. 1, 1-67 (1997; Zbl 0874.35057)].

Reviewer: P.Quittner (Bratislava)

MSC:

35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

global existence; blow-up; radial solutions; initial Dirichlet problem; zero number arguments

Citations:

Zbl 0874.35057