Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem

Pages: 221 - 234, Volume 14, Issue 1, January 2006

doi:10.3934/dcds.2006.14.221       Abstract        Full Text (270.7K)       Related Articles

Philippe Souplet - Département de Mathématiques, Université de Picardie, INSSET, 02109 St-Quentin, France (email)
Juan-Luis Vázquez - Departamento de Matemáticas, Universidad Autónoma de Madrid, 28046 Madrid, Spain (email)

Abstract: This paper is devoted to analyze a case of singularity formation in infinite time for a semilinear heat equation involving linear diffusion and superlinear convection. A feature to be noted is that blow-up happens not for the main unknown but for its derivative. The singularity builds up at the boundary. The formation of inner and outer regions is examined, as well as the matching between them. As a consequence, we obtain the precise exponential rates of blow-up in infinite time.

Keywords:  Gradient blow-up, nonlinear heat equations, viscous Hamilton-Jacobi equation, asymptotic behavior, matched asymptotics.
Mathematics Subject Classification:  35K60, 35K65, 35B45.

Received: October 2004;      Revised: February 2005;      Published: October 2005.