On gradient estimates and other qualitative properties of solutions of nonlinear non autonomous parabolic systems

We prove several uniforme L1 estimates on solutions of a general class of one-dimensional parabolic systems, mainly coupled in the di¤usion term, which, in fact, can be of degenerate type. They are uniform in the sense that they don’t depend on the coe¢ cients, nor on the size of the spatial domain. The estimates concern the own solution or/and its spatial gradient. This paper extends some previous results by the authors to the case of nonautonomous coe¢ cients and possibly non homogeneous boundary conditions. Moreover, an application to the asymptotic decay of the L1 norm of solutions, as t! +1; is also given. Presentado por Amable Liñán. Recibido: 13 de enero de 2009. Aceptado: ¿¿¿¿¿¿¿¿¿¿¿¿¿¿. Palabras clave / Keywords: uniform gradient estimates, quasilinear parabolic onedimensional systems, uniform L-estimates independent on the spatial domain Mathematics Subject Classi…cations: 35K45, 35K65 Estimaciones sobre el gradiente y otras propiedades cualitativas de las soluciones de sistemas parabólicos no lineales no autónomos