Asymptotic Properties of Solutions to the Cauchy Problem for Degenerate Parabolic Equations with Inhomogeneous Density on Manifolds

Abstract

We consider the Cauchy problem for doubly nonlinear degenerate parabolic equations with inhomogeneous density on noncompact Riemannian manifolds. We give a qualitative classification of the behavior of the solutions of the problem depending on the behavior of the density function at infinity and the geometry of the manifold, which is described in terms of its isoperimetric function. We establish for the solutions properties as: stabilization of the solution to zero for large times, finite speed of propagation, universal bounds of the solution, blow up of the interface. Each one of these behaviors of course takes place in a suitable range of parameters, whose definition involves a universal geometrical characteristic function, depending both on the geometry of the manifold and on the asymptotics of the density at infinity.

Qualitative Properties of Solutions of Degenerate Parabolic Equations via Energy Approaches

“…Let us begin with some estimates for large times. Theorem 7.2 (Finite speed of propagation ( [18])). Let u !…”

“…0 be a solution of ð7:1Þ-ð7:2Þ, with p þ m À 3 > 0 and supp u 0 & B 0 , 0 < 0 < þ1. Then we have for large enough t ZðtÞ :¼ inff > 0 j uðx; tÞ ¼ 0; jxj > g t qÀð pþmÀ2Þ L : ð7:4Þ Theorem 7.3 (Mass decay ( [18])). Under the same assumptions of Theorem 7.2 we have if q < q à : kuðtÞk 1;R N t ÀA ; ð7:5Þ if q ¼ q à : kuðtÞk 1;R N jlog tj 2) In the case q > q à on combining ð7:5Þ with the known estimate…”

“…Theorem 7.6 (Asymptotically positive mass ( [18])). Let u be a solution to ð7:1Þ-ð7:2Þ, with q > q à .…”

“…The approach used in the proof of Theorem 3.3 is taken from the work [13] (see also [16]) and is similar to the one introduced in [11] with the only difference that instead of Gagliardo-Nirenberg inequalities, the Faber-Krahn's inequality is used. The latter is a flexible tool for research on Riemannian manifolds (see [43]).…”

“…References for this section: [7], [8], [9], [10], [11], [13], [15], [16], [17], [19], [28], [33], [34], [35], [41], [43], [44], [45], [49], [51], [52], [53], [54], [58], [61], [66], [68], [70].…”