The Fujita phenomenon in exterior domains under dynamical boundary conditions

Abstract

The Fujita phenomenon for nonlinear parabolic problems ∂_tu=Δu+u^p in an exterior domain of R^N under dissipative dynamical boundary conditions σ ∂_tu+∂_νu=0 is investigated in the superlinear case. As in the case of Dirichlet boundary conditions (see Trans. Amer. Math. Soc. 316 (1989), 595–622 and Israel J. Math. 98 (1997), 141–156), it turns out that there exists a critical exponent p=1+2/N such that blow-up of positive solutions always occurs for subcritical exponents, whereas in the supercritical case global existence can occur for small non-negative initial data.