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All time \(C^\infty\)-regularity of the interface in degenerate diffusion: A geometric approach. (English) Zbl 1017.35052
Authors’ abstract: We study the connection between the geometry and all time regularity of the interface in degenerated diffusion. Our model considers the porous medium equation \(u_t=\Delta u^m\), \(m>1\), with initial data \(u_0\) nonnegative, integrable, and compactly supported. We show that if the initial pressure \(f_0=u_0^{m-1}\) is smooth up to the interface and in addition it is root-concave and also satisfies the nondegeneracy condition \(|Df_0 |\neq 0\) at \(\partial\overline {\text{supp}} f_0\), then the pressure \(f= u^{m-1}\) remains \(C^\infty\)-smooth up to the interface and root-concave, for all time \(0<t <\infty\). In particular, the free boundary is \(C^\infty\)-smooth for all time.

MSC:
35K65 Degenerate parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs
35K15 Initial value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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