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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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