Asymptotic behavior for the filtration equation in domains with noncompact boundary |
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Authors: | Daniele Andreucci Anatoli F. Tedeev |
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Affiliation: | 1. Department of Basic and Advanced Sciences for Engineering, Sapienza University of Rome, Rome, Italydaniele.andreucci@sbai.uniroma1.it;3. South Mathematical Institute of VSC RAS, Vladikavkaz, Russia |
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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. |
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Keywords: | Porous media equation asymptotic profile non-compact domain fundamental solution |
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