Remarks on a nonlinear parabolic equation

Matania Ben-Artzi; Jonathan Goodman; Arnon Levy

doi:10.1090/s0002-9947-99-02336-3

Remarks on a nonlinear parabolic equation

Matania Ben-Artzi^*
, Jonathan Goodman
, Arnon Levy

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

21 Scopus citations

Abstract

The equation u_t = Δu + μ|∇u|, μ ∈ ℝ, is studied in ℝⁿ and in the periodic case. It is shown that the equation is well-posed in L¹ and possesses regularizing properties. For nonnegative initial data and μ < 0 the solution decays in L¹(ℝⁿ) as t → ∞. In the periodic case it tends uniformly to a limit. A consistent difference scheme is presented and proved to be stable and convergent.

Original language	English
Pages (from-to)	731-751
Number of pages	21
Journal	Transactions of the American Mathematical Society
Volume	352
Issue number	2
DOIs	https://doi.org/10.1090/s0002-9947-99-02336-3
State	Published - 2000

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AB - The equation ut = Δu + μ|∇u|, μ ∈ ℝ, is studied in ℝn and in the periodic case. It is shown that the equation is well-posed in L1 and possesses regularizing properties. For nonnegative initial data and μ < 0 the solution decays in L1(ℝn) as t → ∞. In the periodic case it tends uniformly to a limit. A consistent difference scheme is presented and proved to be stable and convergent.

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Remarks on a nonlinear parabolic equation

Abstract

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