Interaction between nonlinear diffusion and geometry of domain. (English) Zbl 1243.35096
Summary: Let \(\Omega \) be a domain in \(\mathbb R^N\), where \(N \geqslant 2\) and \(\partial \Omega \) is not necessarily bounded. We consider nonlinear diffusion equations of the form \(\partial _tu=\Delta \varphi (u)\). Let \(u=u(x,t)\) be the solution of either the initial-boundary value problem over \(\Omega \), where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial data is the characteristic function of the set \(\mathbb R^N \setminus \Omega \).
We consider an open ball \(B\) in \(\Omega \) whose closure intersects \(\partial \Omega \) only at one point, and we derive asymptotic estimates for the content of substance in \(B\) for short times in terms of geometry of \(\Omega \). Also, we obtain a characterization of the hyperplane involving a stationary level surface of \(u\) by using the sliding method due to H. Berestycki, L. A. Caffarelli and L. Nirenberg [Commun. Pure Appl. Math. 50, No. 11, 1089–1111 (1997; Zbl 0906.35035)]. These results tell us about interactions between nonlinear diffusion and geometry of domain.
We consider an open ball \(B\) in \(\Omega \) whose closure intersects \(\partial \Omega \) only at one point, and we derive asymptotic estimates for the content of substance in \(B\) for short times in terms of geometry of \(\Omega \). Also, we obtain a characterization of the hyperplane involving a stationary level surface of \(u\) by using the sliding method due to H. Berestycki, L. A. Caffarelli and L. Nirenberg [Commun. Pure Appl. Math. 50, No. 11, 1089–1111 (1997; Zbl 0906.35035)]. These results tell us about interactions between nonlinear diffusion and geometry of domain.
MSC:
| 35K59 | Quasilinear parabolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K65 | Degenerate parabolic equations |
Keywords:
initial-boundary value problem; Cauchy problem; initial behavior; stationary level surface; sliding methodCitations:
Zbl 0906.35035References:
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