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

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)].

### MSC:

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

Zbl 0874.35057