ON BLOW-UP FOR A TIME–SPACE FRACTIONAL PARTIAL DIFFERENTIAL EQUATION WITH EXPONENTIAL KERNEL IN TEMPORAL DERIVATIVE

Abstract

In this paper, we study the blow-up of the solution to a semilinear time–space fractional diffusion equation, where the time derivative is the Caputo–type derivative with the exponential kernel (“the exponential Caputo derivative” for brevity) of order \(\alpha \in (0,1)\) and the spatial derivative is the fractional Laplacian of order \(s\in (0, 1)\). We first define a mild solution of the considered equation in terms of convolution form, where the fundamental solutions are denoted by Fox H-functions. The local existence and uniqueness of a mild solution are further obtained by using a fixed point argument. Then, a weak solution is defined by the test function and it is proved to be a mild solution. Finally, the finite time blow-up is shown. Besides, the global existence of the solution is shown too where the critical index is determined.