The Fujita phenomenon in exterior domains under dynamical boundary conditions

Article type: Research Article

Authors: Rault, Jean-François

Affiliations: LMPA Joseph Liouville, FR 2956 CNRS, Université du Littoral Côte d'Opale, 50 rue F. Buisson, B.P. 699, F-62228 Calais Cedex, France. E-mail: jfrault@lmpa.univ-littoral.fr

Abstract: The Fujita phenomenon for nonlinear parabolic problems ∂tu=Δu+up in an exterior domain of RN 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.

Keywords: nonlinear parabolic problems, dynamical boundary conditions, global solutions

DOI: 10.3233/ASY-2009-0954

Journal: Asymptotic Analysis, vol. 66, no. 1, pp. 1-8, 2010