Asymptotic behavior of solutions for partial differential equations with degenerate diffusion and logistic reaction

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Abstract

Global existence and asymptotic behavior of solutions for degenerate parabolic equations including ut = λ div (|∇u|p-2 ∇u) + |u|q-2u(1 - |u|r) are studied, where λ is a positive parameter; p > 2, q ≥ 2 and r > 0 are constants. In particular, the behavior of solutions for the initial data close to a maximal stationary solution is discussed. It is shown that the maximal stationary solution is asymptotically stable if p ≥ q and stable if p < q. For the latter case, some remarks on the attractivity of maximal stationary solution are also given.

Original languageEnglish
Pages (from-to)1715-1724
Number of pages10
JournalNonlinear Analysis, Theory, Methods and Applications
Volume47
Issue number3
DOIs
Publication statusPublished - 2001 Aug
Externally publishedYes

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Partial differential equations
Logistics

Keywords

  • Asymptotic behavior
  • Degenerate parabolic equations
  • Logistic reaction
  • p-Laplacian
  • Stability

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Mathematics(all)

Cite this

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abstract = "Global existence and asymptotic behavior of solutions for degenerate parabolic equations including ut = λ div (|∇u|p-2 ∇u) + |u|q-2u(1 - |u|r) are studied, where λ is a positive parameter; p > 2, q ≥ 2 and r > 0 are constants. In particular, the behavior of solutions for the initial data close to a maximal stationary solution is discussed. It is shown that the maximal stationary solution is asymptotically stable if p ≥ q and stable if p < q. For the latter case, some remarks on the attractivity of maximal stationary solution are also given.",
keywords = "Asymptotic behavior, Degenerate parabolic equations, Logistic reaction, p-Laplacian, Stability",
author = "Shingo Takeuchi",
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N2 - Global existence and asymptotic behavior of solutions for degenerate parabolic equations including ut = λ div (|∇u|p-2 ∇u) + |u|q-2u(1 - |u|r) are studied, where λ is a positive parameter; p > 2, q ≥ 2 and r > 0 are constants. In particular, the behavior of solutions for the initial data close to a maximal stationary solution is discussed. It is shown that the maximal stationary solution is asymptotically stable if p ≥ q and stable if p < q. For the latter case, some remarks on the attractivity of maximal stationary solution are also given.

AB - Global existence and asymptotic behavior of solutions for degenerate parabolic equations including ut = λ div (|∇u|p-2 ∇u) + |u|q-2u(1 - |u|r) are studied, where λ is a positive parameter; p > 2, q ≥ 2 and r > 0 are constants. In particular, the behavior of solutions for the initial data close to a maximal stationary solution is discussed. It is shown that the maximal stationary solution is asymptotically stable if p ≥ q and stable if p < q. For the latter case, some remarks on the attractivity of maximal stationary solution are also given.

KW - Asymptotic behavior

KW - Degenerate parabolic equations

KW - Logistic reaction

KW - p-Laplacian

KW - Stability

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