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Stabilization of solutions of certain one-dimensional degenerate diffusion equations. (English) Zbl 0619.35064
Die Autoren untersuchen Anfangsrandwertprobleme der Form \[ (*)\quad u_ t=(u^ m)_{xx}+f(u),\quad (x,t)\in (-L,L)\times {\mathbb{R}}^+;\quad u(\pm L,t)=0;\quad u(x,0)=u_ 0(x),\quad x\in [-L,L], \] wobei die Nichtlinearität f(u) gewisse Bedingungen erfüllt (als typisches Beispiel wird \(f(r)=ar^{\beta}-br\) mit \(a,b>0\) und \(m>\gamma >1\) erwähnt). Untersucht werden die stationären Lösungen der obigen Gleichung sowie - mittels der Galerkinmethode - Existenz- und Konvergenzfragen (für \(t\to \infty)\) beim zeitabhängigen Problem.
Reviewer: W.Wendt

MSC:
35K65 Degenerate parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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References:
[1] ARONSON D. G., CRANDALL M. G., PELETIER L. A.: Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Analysis 6, 1982, 1001-1022. · Zbl 0518.35050
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[3] NAKAO M.: Existence, nonexistence and some asymptotic behaviour of global solutions of a nonlinear degenerate parabolic equation. Math. Rep. College General Edc, Kyushu Univ. 14, 1983, 1-21. · Zbl 0563.35038
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