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# Well-posedness results for fractional semi-linear wave equations

• * Corresponding author: Arran Fernandez

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

• 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.

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

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