On existence and uniqueness of solutions for semilinear fractional wave equations

By Yavar Kian, Masahiro Yamamoto. 2015

116 citations recorded by the source corpus.

Abstract Let Ω be a 𝒞2-bounded domain of ℝd, d = 2,3, and fix Q = (0,T)× Ω with T ∈ (0,+∞]. In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation ∂tαu+Au=fb(u)inQwhere1<α<2,∂tα $\partial_t^\alpha u+\mathcal A u=f_b(u)\,\,\text{in}\, Q\, \text{where} \,\,1\lt\alpha\lt2, \,\partial_t^\alpha$ corresponds to the Caputo fractional derivative of order α, 𝒜 is an elliptic operator and the nonlinearity fb ∈ 𝒞1 (ℝ) satisfies fb(0) = 0 and |fb′(u)|⩽C|u|b−1 $|{f_b'(u)}| \leqslant C\,|{u}|^{b-1}$ for some b > 1. We first provide a definition of local weak solutions of this problem by applying some properties of the associated linear equation ∂tαu+Au=f(t,x) $\partial_t^\alpha u+\mathcal A u=f(t,x)$ in Q. Then, we prove existence of local solutions of the semilinear fractional wave equation for some suitable values of b > 1. Moreover, we obtain an explicit dependence of the time of existence of solutions with respect to the initial data that allows longer time of existence for small initial data.

Subfields: Control and Systems Engineering, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/af689bcc09c94312386f045596fcc746df268c1e

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In Control and Systems Engineering

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Citations in the corpus

Global uniqueness in an inverse problem for time fractional diffusion equations (2016, background): “me fractional derivatives. More speciﬁcally, the well-posedness of problem (1.1) with time-independent coeﬃcients is examined in [4, 25, 52], and recently, weak solutions to (1.1) have been deﬁned in [36]. There is a wide mathematical literature for inverse coeﬃcients problems associated with the sy”