Pointwise gradient estimates of solutions to onedimensional nonlinear parabolic equations

Philippe Bénilan and Jesús Ildefonso Díaz
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Jesús Ildefonso Díaz: Universidad Complutense de Madrid, Facultad de Matemáticas

A chapter in Nonlinear Evolution Equations and Related Topics, 2004, pp 577-602 from Springer

Abstract: Abstract We present here an improved version of the method introduced by the first author to derive pointwise gradient estimates for the solutions of one-dimensional parabolic problems. After considering a general qualinear equation in divergence form we apply the method to the case of a nonlinear diffusion-convection equation. The conclusions are stated first for classical solutions and then for generalized and mild solutions. In the case of unbounded initial datum we obtain several regularizing effects fort> 0. Some unilateral pointwise gradient estimates are also obtained. The case of the Dirichlet problem is also considered. Finally, we collect, in the last section, several comments showing the connections among these estimates and the study of the free boundaries associated to the solutions of the diffusion-convection equation.

Keywords: 35K65; 3SK10; 35R3S; 76D27; Pointwise gradient estimates; one-dimensional parabolic equations; non linear diffusion-convection equation; regularizing effects; unilateral estimates; interfaces (search for similar items in EconPapers)
Date: 2004
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-7924-8_30

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DOI: 10.1007/978-3-0348-7924-8_30

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