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September  2003, 9(5): 1277-1292. doi: 10.3934/dcds.2003.9.1277

A priori estimates of global solutions of superlinear parabolic problems without variational structure

Pavol Quittner 1, and  Philippe Souplet 2,

1. 

Institute of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava, Slovak Republic
2. 

Département de Mathématiques, Université de Picardie, INSSET, 02109 St-Quentin, France

Received  December 2001 Revised  February 2003 Published  June 2003

We consider various classes of superlinear parabolic problems, including reaction-diffusion systems and scalar reaction-diffusion equations with convective or dissipative gradient terms. For these problems we prove uniform a priori estimates for all nonnegative global solutions. The existence of an energy functional for these problems is not known, so that traditional methods for a priori estimates do not apply. We use a different approach based on scaling and Fujita-type theorems. In the case of reaction-diffusion systems, we also obtain some universal bounds, i.e. a priori estimates independent of the initial data.
Keywords: universal bound., semilinear parabolic problem, a priori estimate.
Mathematics Subject Classification: Primary: 35B45, 35K45, 35K5.
Citation: Pavol Quittner, Philippe Souplet. A priori estimates of global solutions of superlinear parabolic problems without variational structure. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1277-1292. doi: 10.3934/dcds.2003.9.1277
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