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Global-in-time Gevrey regularity solution for a class of bistable gradient flows. (English) Zbl 1350.35050

Summary: In this paper, we prove the existence and uniqueness of a Gevrey regularity solution for a class of nonlinear bistable gradient flows, where with the energy may be decomposed into purely convex and concave parts. Example equations include certain epitaxial thin film growth models and phase field crystal models. The energy dissipation law implies a bound in the leading Sobolev norm. The polynomial structure of the nonlinear terms in the chemical potential enables us to derive a local-in-time solution with Gevrey regularity, with the existence time interval length dependent on a certain \(H^m\) norm of the initial data. A detailed Sobolev estimate for the gradient equations results in a uniform-in-time-bound of that \(H^m\) norm, which in turn establishes the existence of a global-in-time solution with Gevrey regularity.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
35K30 Initial value problems for higher-order parabolic equations
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