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.
Highlights
We consider the Cauchy problem ρ(x)ut − Δp,m(u) = 0, x ∈ M, t > 0, u(x, 0) = u0(x), x ∈ M. (1.1) (1.2)We consider only non-negative solutions to (1.1) and (1.2)
Still in the slow diffusion case, but when the density function decays fast enough, we investigate the behavior of the solutions for large t; we obtain under different assumptions a universal bound for solutions and a result of interface blow up
Before describing the results of this paper, we recall that parabolic problems in a Euclidean metric with inhomogeneous density were studied in [27,28]; [24,36]; [22,31]
Summary
The various cases recalled above are discriminated in terms of the behavior of a universal function involving the density and the volume growth of the manifold (see Remark 1.4) The interest of this problem appeared first in the case M = R3 with the Euclidean metric, where [25,38] obtained the first surprising results, in symmetric cases, on the qualitative properties of solutions to the porous media equation with inhomogeneous density. Before describing the results of this paper, we recall that parabolic problems in a Euclidean metric with inhomogeneous density were studied in [27,28] (blow up phenomena); [24,36] (asymptotic expansion of the solution of the porous media equation); [22,31] (critical case). See [1] for the Euclidean case; we borrow the energetic setting of [4,6,7,8]; see [41]
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