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Regularity and wellposedness of a problem to one parameter and its behavior at the limit. (English) Zbl 1393.35199

Summary: In this article we prove that the Cauchy problem associated to a model of waves in a viscous fluid, proposed by R. J. Iorio jun. and V. de Magalh√£es Iorio [Fourier analysis and partial differential equations. Cambridge: Cambridge University Press (2001; Zbl 0970.35001)], is globally well posed. We do this in an intuitive way using Fourier theory and in a fine version using semigroups theory, getting \(H^\infty\) regularity.
Also, we analyze the behavior of solutions of Cauchy problems to one parameter and prove that their limit is solution of a Cauchy problem whose associated semigroup is the restriction of a group.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35G10 Initial value problems for linear higher-order PDEs
47D06 One-parameter semigroups and linear evolution equations
35B40 Asymptotic behavior of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35Q35 PDEs in connection with fluid mechanics
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness

Citations:

Zbl 0970.35001
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