On the Initial Boundary Value Problem to the Time-Fractional Wave Equation with Acoustic Boundary Conditions

This paper is concerned with the study of the well-posedeness for the initial boundary value problem to the time-fractional wave equation with acoustic boundary conditions. The problem is considered in a bounded and connected domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, which includes simply connected regions. The boundary of $\mathrm{\Omega }$ is made up of two disjoint pieces ${\mathrm{\Gamma }}_0$ and ${\mathrm{\Gamma }}_1\mathrm{.}$ Homogeneous Dirichlet conditions are enforced on ${\mathrm{\Gamma }}_0$, while acoustic boundary conditions are considered on ${\mathrm{\Gamma }}_1$. To establish our main result, we employ the Faedo-Galerkin method and successfully solve a general system of time-fractional ordinary differential equations which extends the scope of the classical Picard-Lindelöf theorem.