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Decay Estimates for Solutions of Porous Medium Equations with Advection

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Abstract

In this paper, we show that bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the form

$$ u_{t} + \operatorname{div}f(x,t,u) = \operatorname{div}\bigl( |u|^{\alpha } \nabla u\bigr), \quad x \in \mathbb{R}^{n} , \ t > 0, $$

where \(\alpha > 0 \) is constant, decrease to zero, under fairly broad conditions on the advection flux \(f\). Besides that, we derive a time decay rate for these solutions.

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Authors and Affiliations

  1. Instituto Federal de Educação, Ciência e Tecnologia do Rio Grande do Sul, Canoas, RS, 92412, Brazil

    Nicolau M. L. Diehl

  2. Universidade Federal de Santa Maria - Cachoeira do Sul, RS, 96501, Cachoeira do Sul, Brazil

    Lucineia Fabris

  3. Universidade Federal do Rio Grande do Sul, RS, 91509, Porto Alegre, Brazil

    Juliana S. Ziebell

Authors
  1. Nicolau M. L. Diehl
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  2. Lucineia Fabris
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  3. Juliana S. Ziebell
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Correspondence to Nicolau M. L. Diehl.

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Diehl, N.M.L., Fabris, L. & Ziebell, J.S. Decay Estimates for Solutions of Porous Medium Equations with Advection. Acta Appl Math 165, 149–162 (2020). https://doi.org/10.1007/s10440-019-00246-4

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  • DOI: https://doi.org/10.1007/s10440-019-00246-4

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