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January  2006, 14(1): 221-234. doi: 10.3934/dcds.2006.14.221

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

Philippe Souplet 1, and  Juan-Luis Vázquez 2,

1. 

Département de Mathématiques, Université de Picardie, INSSET, 02109 St-Quentin
2. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28046 Madrid

Received  October 2004 Revised  February 2005 Published  October 2005

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: asymptotic behavior, viscous Hamilton-Jacobi equation, nonlinear heat equations, matched asymptotics., Gradient blow-up.
Mathematics Subject Classification: 35K60, 35K65, 35B4.
Citation: Philippe Souplet, Juan-Luis Vázquez. Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 221-234. doi: 10.3934/dcds.2006.14.221
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