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Remarks on a nonlinear parabolic equation. (English) Zbl 0932.35115
Authors’ abstract: The equation \(u_t=\Delta u+\mu|\nabla u|\), \(\mu\in\mathbb{R}\), is studied in \(\mathbb{R}^n\) and in the periodic case. It is shown that the equation is well-posed in \(L^1\) and possesses regularizing properties. For nonnegative initial data and \(\mu<0\) the solution decays in \(L^1(\mathbb{R}^n)\) as \(t\to\infty\). In the periodic case it tends uniformly to a limit. A consistent difference scheme is presented and proved to be stable and convergent.

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
Full Text: DOI
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