Asymptotic behavior for the filtration equation in domains with noncompact boundary. Issue 3 (4th March 2017)

• Record Type: Journal Article
• Authors: Andreucci, Daniele; Tedeev, Anatoli F.
• Is Part Of: Communications in partial differential equations. Volume 42:Issue 3(2017)
• Page Start: 347
• Page End: 365
• Publication Date: 2017-03-04
• Subjects: Porous media equation -- asymptotic profile -- non-compact domain -- fundamental solution; 35K55, 35K65, 35B40; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2017.1278770
• Languages: English
• ISSNs: 0360-5302

Abstract

ABSTRACT: We consider the initial value boundary problem with zero Neumann data for an equation modeled after the porous media equation, with variable coefficients. The spatial domain is unbounded and shaped like a (general) paraboloid, and the solution u is integrable in space and nonnegative. We show that the asymptotic profile for large times of u is one dimensional and given by an explicit function, which can be regarded as the fundamental solution of a one-dimensional differential equation with weights. In the case when the domain is a cone or the whole space (Cauchy problem), we obtain a genuine multidimensional profile given by the well-known Barenblatt solution.