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Pointwise gradient estimates of solutions to onedimensional nonlinear parabolic equations

In: Nonlinear Evolution Equations and Related Topics

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  • Philippe Bénilan
  • Jesús Ildefonso Díaz

    (Universidad Complutense de Madrid, Facultad de Matemáticas)

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.

Suggested Citation

  • Philippe Bénilan & Jesús Ildefonso Díaz, 2004. "Pointwise gradient estimates of solutions to onedimensional nonlinear parabolic equations," Springer Books, in: Wolfgang Arendt & Haïm Brézis & Michel Pierre (ed.), Nonlinear Evolution Equations and Related Topics, pages 577-602, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-7924-8_30
    DOI: 10.1007/978-3-0348-7924-8_30
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