American Institute of Mathematical Sciences

February  2020, 25(2): 569-597. doi: 10.3934/dcdsb.2019255

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

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

Received  April 2019 Revised  May 2019 Published  November 2019

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.

Citation: Jean-Daniel Djida, Arran Fernandez, Iván Area. Well-posedness results for fractional semi-linear wave equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 569-597. doi: 10.3934/dcdsb.2019255
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