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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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