Rocky Mountain Journal of Mathematics

Well-posedness of semilinear strongly damped wave equations with fractional diffusion operators and $C^0$ potentials on arbitrary bounded domains

Joseph L. Shomberg

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Abstract

We examine the well-posedness of a strongly damped wave equation equipped with fractional diffusion operators. Ranges on the orders of the diffusion operators are determined in connection with global well-posedness of mild solutions or the global existence of weak solutions. Local existence proofs employ either semigroup methods or a Faedo-Galerkin scheme, depending on the type of solution sought. Mild solutions arising from semigroup methods are either analytic or of Gevrey class; the former produce a gradient system. We also determine the critical exponent for the nonlinear term depending on the orders of the fractional diffusion operators. Thanks to the nonlocal presentation of the fractional diffusion operators, we are able to work on arbitrary bounded domains. The nonlinear potential is only assumed to be continuous while satisfying a suitable growth condition.

Article information

Source
Rocky Mountain J. Math., Volume 49, Number 4 (2019), 1307-1334.

Dates
First available in Project Euclid: 29 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1567044041

Digital Object Identifier
doi:10.1216/RMJ-2019-49-4-1307

Mathematical Reviews number (MathSciNet)
MR3998923

Zentralblatt MATH identifier
07104719

Subjects
Primary: 35L71: Semilinear second-order hyperbolic equations 35L20: Initial-boundary value problems for second-order hyperbolic equations
Secondary: 35Q74: PDEs in connection with mechanics of deformable solids 74H40.

Keywords
Nonlocal diffusion semilinear strongly damped wave equation global well-posedness uniqueness weak solutions mild solutions regularity $C^0$ potential.

Citation

Shomberg, Joseph L. Well-posedness of semilinear strongly damped wave equations with fractional diffusion operators and $C^0$ potentials on arbitrary bounded domains. Rocky Mountain J. Math. 49 (2019), no. 4, 1307--1334. doi:10.1216/RMJ-2019-49-4-1307. https://projecteuclid.org/euclid.rmjm/1567044041


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