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

Article

Abstract

We prove several uniform \( L^1 \) -estimates on solutions of a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of degenerate type. They are uniform in the sense that they don’t depend on the coefficients, 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 coefficients and possibly non homogeneous boundary conditions. Moreover, an application to the asymptotic decay of the \( L^1 \) -norm of solutions, as t → +∞, is also given.

Keywords

Uniform gradient estimates quasilinear parabolic onedimensional systems uniform L1-estimates independent on the spatial domain 

Mathematics Subject Classifications

35K45 35K65 

Estimaciones sobre el gradiente y otras propiedades cualitativas de las soluciones de sistemas parabólicos no lineales no aut ónomos

Resumen

En este artículo se obtienen varias estimaciones uniformes en \( L^1 \) para las soluciones y su derivada espacial de ciertos sistemas parabólicos no lineales que pueden estar acoplados en los términos de difusión y que, de hecho, puede ser de tipo degenerado.

Tales estimaciones son uniformes en el sentido de que no dependen de los coeficientes del sistema, ni del tamaño del dominio espacial. Las estimaciones se refieren a la norma \( L^1 \) de la propia solución o/y de su gradiente espacial. Este trabajo extiende, al caso de coeficientes no autónomos y a posibles condiciones de contorno no homogéneas, ciertos resultados previos de los autores. Además, se ofrece una aplicación al estudio del decaimiento de la norma \( L^1 \) de la solución, cuando t → +∞.

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Copyright information

© Springer 2009

Authors and Affiliations

  1. 1.CMAFUniversidade de LisboaPortugal
  2. 2.Departamento de Matemática AplicadaUniversidad Complutense de MadridSpain

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