On existence and uniqueness of solutions for semilinear fractional wave equations

Abstract

Let $\Omega$ be a ${\mathcal{C}}^2$-bounded domain of ${\mathbb{R}}^d$, $d=2,3$, and fix $Q=(0,T)\times \Omega$ with $T\in (0,+\infty ]$. In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation ${\partial }_t^{\alpha }u+\mathcal{A}u=f_b(u)$ in $Q$ where $1<\alpha <2$, ${\partial }_t^{\alpha }$ corresponds to the Caputo fractional derivative of order $\alpha$, $\mathcal{A}$ is an elliptic operator and the nonlinearity $f_b\in {\mathcal{C}}^1(\mathbb{R})$ satisfies $f_b(0)=0$ and $|f_b^{\prime }(u)|\le C|u{|}^{b-1}$ 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 ${\partial }_t^{\alpha }u+\mathcal{A}u=f(t,x)$ 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.

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