For semilinear systems with distinct Caputo fractional orders, an L^p-Carathéodory right-hand side yields unique continuous solutions exactly when p exceeds every reciprocal of the fractional orders.
On the Initial Boundary Value Problem to the Time-Fractional Wave Equation with Acoustic Boundary Conditions
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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 $\Omega \subset {\mathbb{R}^{n}}$, $n \geq 2$, which includes simply connected regions. The boundary of $\Omega$ is made up of two disjoint pieces $\Gamma_{0}$ and $\Gamma_{1}.$ Homogeneous Dirichlet conditions are enforced on $\Gamma_0$, while acoustic boundary conditions are considered on $\Gamma_1$. 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-Lindel\"of theorem.
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A General Version of Carath\'{e}odory's Existence and Uniqueness Theorem
For semilinear systems with distinct Caputo fractional orders, an L^p-Carathéodory right-hand side yields unique continuous solutions exactly when p exceeds every reciprocal of the fractional orders.