## 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

#### 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

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

#### 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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