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The Cauchy problem for a Sobolev-type equation with a power nonlinearity. (English) Zbl 1079.35024
This paper deals with the study of the large-time asymptotic behaviour of solutions of the Cauchy problem for a nonlinear nonlocal equation of Sobolev type with dissipation. In the case when the initial data are small the approach is based on a detailed study of the Green’s function of the linear problem and the use of the contraction-mapping method. The authors also consider the case when the initial data are large. In the supercritical case the asymptotic is quasilinear. The asymptotic behaviour of solutions in the critical case differs from the behaviour of solutions of the corresponding linear equation by a logarithmic correction. In the subcritical case the authors prove that the principal term of the large-time asymptotics of the solution can be represented by a self-similar solution if the initial data have nonzero total mass.
MSC:
35B40Asymptotic behavior of solutions of PDE
35K55Nonlinear parabolic equations
35A08Fundamental solutions of PDE