On local existence and blowup of solutions for a time‐space fractional diffusion equation with exponential nonlinearity

Abstract

In this paper, we are concerned with local existence and blowup of a unique solution to a time‐space fractional evolution equation with a time nonlocal nonlinearity of exponential growth. At first, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. Then, the blowup result of the solution in finite time is established by the test function method with a judicious choice of the test function.

Sub-diffusion equations with Mittag-Leffler nonlinearity

“…In addition, the system case was investigated by Ahmad et al in [2]. Note also that the time-space fractional case of equation ( 2) has been studied in [6,8,12].…”

Existence of solutions to a nonlinear fractional diffusion equation with exponential growth

“…Mahouachi and Saanouni [21] derived the well-posed and ill-posed results for a wave equation with exponential growth. In terms of the fractional derivatives, Bekkai et al [3] discussed the local existence and blow-up of solution for a space-time fractional diffusion equation with nonlocal nonlinearity of the form f (u) ∼ J 1−α t (e u ), where J 1−α t represents the Riemann-Liouville fractional integral operator. Alsaedi et al [2] proved the existence and uniqueness of the local mild solution for a system of space-time fractional evolution equations with nonlocal nonlinearities of exponential growth.…”