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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.

MSC:
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
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