Ukrainian Mathematical Bulletin

Volume 1, Issue 4, 2004 pp. 523-536.

Semilinear Parabolic Equations with Superlinear Reaction Terms, and Application to Some Convection-diffusion Problems

Authors: Andrea Dall'Aglio, Daniela Giachetti and Sergio Segura de Le'on
Author institution: A. Dall'Aglio and D. Giachetti, Dipartimento di Metodi e Modelli Matematici, Universit`a di Roma ``La Sapienza'', Via Antonio Scarpa 16, I-00161 Roma, Italy S. Segura de Le'on, Departament d'An`alisi Matem`atica, Universitat de Val`encia, Dr. Moliner 50, 46100 Burjassot, Val`encia, Spain

Summary: We are interested in the existence of distributional solutions for two types of nonlinear evolution problems, whose models are and below. In the first one the nonlinear reaction term depends on the solution with a slightly superlinear growth. In the second one we consider a first order term depending also on the gradient of the solution in a quadratic way. The two problems are strictly related from the point of view of the a priori estimates we can obtain on their solutions. We point out that no boundedness is assumed on the data of the problems. This implies that the methods involving sub/super-solutions do not apply, and we have to use some convenient test-function to prove the a priori estimates.

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