On the existence, uniqueness and stability results for time-fractional parabolic integrodifferential equations

Abstract

We study the time-fractional parabolic integrodifferential equations with the Caputo fractional time derivative of order $\alpha \in \left(0,1\right)$ in a bounded convex polygonal domain in ${ℝ}^d$. We prove the existence and uniqueness of the solution using the eigenfunction expansion and establish a priori bounds for the solution under various regularity assumptions on the initial data and the source function. Our study includes the initial data in the spaces ${Ḣ}^2\left(\mathrm{\Omega }\right)$, $H_0^1\left(\mathrm{\Omega }\right)$ and $L^2\left(\mathrm{\Omega }\right)$ while the source function belongs to the class of Hölder continuous and bounded functions. It is shown that the solution of the corresponding homogeneous problem is infinitely differentiable with respect to time $t$ when the initial function is an element of $L^2\left(\mathrm{\Omega }\right)$. Finally, we derive a general stability results for the solution of the homogeneous problem.