WELL-POSEDNESS RESULTS FOR FRACTIONAL SEMI-LINEAR WAVE EQUATIONS - Djida, Jean-Daniel; Fernandez, Arran; Area, Ivan (Amer Inst Mathematical Sciences-Aims, 2020)
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.
Keywords: Fractional partial differential equations; fractional semi-linear wave equations; local and global existence of solutions; regularity estimates