The Fujita phenomenon in exterior domains under the dynamical boundary conditions

TL;DR

This paper investigates the Fujita phenomenon for nonlinear parabolic equations in exterior domains with dynamical boundary conditions, identifying a critical exponent that determines solution blow-up or global existence.

Contribution

It extends the understanding of the Fujita phenomenon to exterior domains with dynamical boundary conditions, establishing the critical exponent for blow-up and global existence.

Findings

Existence of a critical exponent p=1+2/N for blow-up versus global solutions.

Blow-up occurs for subcritical exponents regardless of initial data.

Global solutions can exist for small initial data in the supercritical case.

Abstract

The Fujita phenomenon for nonlinear parabolic problems $\partial tu = \Delta u + up$ in an exterior domain of RN under the dynamical boundary conditions is investigated in the superlinear case. As in the case of Dirichlet boundary conditions, 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.