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Regularity and time behavior of the solutions of linear and quasilinear parabolic equations. (English) Zbl 1390.35115

Summary: In this paper, we study the regularity, the uniqueness and the asymptotic behavior of the solutions to a class of nonlinear operators in dependence of the summability properties of the datum \(f\) and of the initial datum \(u_0\). The case of only summable data \(f\) and \(u_0\) is allowed. We prove that these equations satisfy surprising regularization phenomena. Moreover, we prove estimates (depending continuously from the data) that for zero datum \(f\) become well known decay (or ultracontractive) estimates.

MSC:

35K10 Second-order parabolic equations
35K59 Quasilinear parabolic equations
35D30 Weak solutions to PDEs
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Full Text: Euclid