Asymptotic Complexity in Filtration Equations |
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Authors: | J A Carrillo J L Vázquez |
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Institution: | 1.ICREA (Institució Catalana de Recerca i Estudis Avan?ats) and Departament de Matemàtiques,Universitat Autònoma de Barcelona,Bellaterra,Spain;2.Departamento de Matemáticas,Universidad Autònoma de Madrid,Madrid,Spain |
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Abstract: | We show that the solutions of nonlinear diffusion equations of the form u
t
= ΔΦ(u) appearing in filtration theory may present complicated asymptotics as t → ∞ whenever we alternate infinitely many times
in a suitable manner the behavior of the nonlinearity Φ. Oscillatory behaviour is demonstrated for finite-mass solutions defined
in the whole space when they are renormalized at each time t > 0 with respect to their own second moment, as proposed in Tos05, CDT05]; they are measured in the L
1 norm and also in the Euclidean Wasserstein distance W
2. This complicated asymptotic pattern formation can be constructed in such a way that even a chaotic behavior may arise depending
on the form of Φ.
In the opposite direction, we prove that the assumption that the asymptotic normalized profile does not depend on time implies
that Φ must be a power-law function on the appropriate range of values. In other words, the simplest asymptotic behavior implies
a homogeneous nonlinearity. |
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Keywords: | |
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