Analysis and computation for quenching solution to the time-space fractional Kawarada problem

Abstract

This study focuses on the existence, uniqueness, and quenching behavior of solution to the time-space fractional Kawarada problem, where the time derivative is the Caputo-Hadamard derivative and the spatial derivative is the fractional Laplacian. The mild solution represented by Fox H-function, based on the fundamental solution, is considered in space \(C\left( [a, T], L^r(\mathbb {R}^d)\right) \). We use the fractional maximum principles to prove \(u(\textrm{x},t)\ge u_a(\textrm{x})\) for the positive initial value. Then the relationship between quenching phenomena and the size of domain is examined. Finally, the finite difference scheme is established for solving the quenching solution to the considered problem in one and two space dimensions. The numerical simulations show the effectiveness and feasibility of the theoretical analysis.