Local and blowing‐up solutions for a space‐time fractional evolution system with nonlinearities of exponential growth

Abstract

Local and blowing‐up solutions for the Cauchy problem for a system of space and time fractional evolution equations with time‐nonlocal nonlinearities of exponential growth are considered. The existence and uniqueness of the local mild solution is assured by the Banach fixed point principle. Then, we establish a blow‐up result by Pokhozhaev capacity method. Finally, under some suitable conditions, an estimate of the life span of blowing‐up solutions is established.

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

“…associated with an initial condition u(0, x) = u 0 (x), x ∈ R d , d 1, where f is the exponential growth function, like asymptotic growth f (u) ∼ e 4π|u| 2 and with a vanishing behavior at zero and ∂ α t standing for the Caputo fractional partial derivative of order α ∈ (0, 1) defined by…”

“…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. They also established a blow-up result by applying Pokhozhaev capacity method and presented an estimate for the life span of blowing-up solutions under suitable conditions.…”