Blowup for semilinear fractional diffusion system with Caputo–Hadamard derivative

Abstract

The main aim of this paper is to study the blowing-up behavior of the solution for semilinear fractional diffusion system with the Caputo-Hadamard derivative and the fractional Laplacian. We construct a mild solution of the semilinear system by using the fundamental solutions and then prove the local existence and uniqueness of the mild solution by virtue of the fixed point argument. Next, the definition of a weak solution is introduced by the test function, and the mild solution can be verified to be a weak solution. Finally, we show the finite time blowup and global solution to the considered system in terms of the contraction mapping principle.

Reconstruct an unknown source term on the right hand side of the time-fractional diffusion equation with Caputo-Hadamard derivative

“…Ting Wei used the Modified Quasi Boundary value method [11,12] to deal with the inverse source problem and obtain two kinds of convergence rates by using an a priori and an a posteriori regularization parameter choice rule, respectively. Fang Yang [13] with the Simplified Tikhonov, see in [14], and the truncation method, see [15].…”