Well-posedness results for fractional semi-linear wave equations

Jean-Daniel Djida; Arran Fernandez; Iván Area

doi:10.3934/dcdsb.2019255

Article Contents

2020, Volume 25, Issue 2: 569-597. Doi: 10.3934/dcdsb.2019255

This issue Previous Article Existence of homoclinic solutions for a nonlinear fourth order

$ p $

-Laplacian difference equation Next Article Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities

Well-posedness results for fractional semi-linear wave equations

1.
Departamento de Análise Matemática, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
2.
Department of Mathematics, Eastern Mediterranean University, Famagusta, Northern Cyprus, via Mersin-10, Turkey
3.
Departamento de Matemática Aplicada II, E.E. Aeronáutica e do Espazo, Universidade de Vigo, 32004-Ourense, Spain

^* Corresponding author: Arran Fernandez
^* Corresponding author: Arran Fernandez

Dedicated to Prof. Juan J. Nieto on the occasion of his 60th birthday

Received: March 31, 2019

Revised: April 30, 2019

Early access: November 2019

Published: February 2020

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Abstract

This work is concerned with well-posedness results for nonlocal semi-linear wave equations involving the fractional Laplacian and fractional derivative operator taken in the sense of Caputo. Representations for solutions, existence of classical solutions, and some $ L^{p} $-estimates are derived, by considering a quasi-stationary elliptic problem that comes from the realisation of the fractional Laplacian as the Dirichlet-to-Neumann map for a non-uniformly elliptic problem posed on a semi-infinite cylinder. We derive some properties such as existence of global weak solutions of the extended semi-linear integro-differential equations.

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Mathematics Subject Classification: Primary: 26A33, 74G20, 74G25; Secondary: 35R11, 35G31, 35B65.

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