Compactness and asymptotic behavior in nonautonomous linear parabolic equations with unbounded coefficients in \(\mathbb R^{d}\). (English) Zbl 1250.35035
Escher, Joachim (ed.) et al., Parabolic problems. The Herbert Amann Festschrift. Based on the conference on nonlinear parabolic problems held in celebration of Herbert Amann’s 70th birthday at the Banach Center in Bȩdlewo, Poland, May 10–16, 2009. Basel: Birkhäuser (ISBN 978-3-0348-0074-7/hbk; 978-3-0348-0075-4/ebook). Progress in Nonlinear Differential Equations and Their Applications 80, 447-461 (2011).
Summary: We consider a class of second-order linear nonautonomous parabolic equations in \(\mathbb R^{d}\) with time periodic unbounded coefficients. We give sufficient conditions for the evolution operator \(G(t, s)\) be compact in \(C_{b}(\mathbb R^{d})\) for \(t > s\), and describe the asymptotic behavior of \(G(t,s)f\) as \(t - s \rightarrow \infty \) in terms of a family of measures \(\mu_{s}, s \in\mathbb R\), solution of the associated Fokker-Planck equation.
For the entire collection see [Zbl 1220.35003].
For the entire collection see [Zbl 1220.35003].
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35D40 | Viscosity solutions to PDEs |
| 28C10 | Set functions and measures on topological groups or semigroups, Haar measures, invariant measures |
| 47D07 | Markov semigroups and applications to diffusion processes |
| 35K15 | Initial value problems for second-order parabolic equations |
| 35K90 | Abstract parabolic equations |
Keywords:
evolution operator; compactness; asymptotic behavior; time periodic unbounded coefficients; Fokker-Planck equationReferences:
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