Abstract
In this paper, we investigate a reaction–diffusion equation with a Caputo fractional derivative in time and with boundary conditions. According to the principle of contraction mapping, we first prove the existence and uniqueness of local solutions. Then, under some conditions of the initial data, we obtain two sufficient conditions for the blow-up of the solutions in finite time. Moreover, the existence of global solutions is studied when the initial data is small enough. Finally, the long-time behavior of bounded solutions is analyzed.
Highlights
The purpose of this paper is to study the Cauchy problem for the following time fractional reaction–diffusion equation: cCitation: Shi, L.; Cheng, W.; Mao, J.; Dtα u − d∆u = −u(1 − u), x ∈ Ω, t > 0, (1)Xu, T
Blow-Up and Global Existence of Solutions for the Time Fractional supplemented with a boundary condition: Reaction–Diffusion Equation
By Jensen’s inequality and the Kaplan’s first eigenvalue method, we obtain some sufficient conditions for a finite-time blow-up where a principal eigenvalue problem plays a crucial role
Summary
The purpose of this paper is to study the Cauchy problem for the following time fractional reaction–diffusion equation: c. Blow-Up and Global Existence of Solutions for the Time Fractional supplemented with a boundary condition: Reaction–Diffusion Equation. Fractional calculus is a generalization of ordinary differential as well as arbitrary non-integer orders In recent years, it has achieved considerable development and has been widely used for modeling in various fields of science and engineering such as in diffusion process, signal processing, porous media, economics, physics and chemistry, etc. In [24], the blow-up phenomenon and conditions of its appearance were proved by Xu. This paper is motivated by the recent work of [25], in which they proved the dissipativity of the time fractional-order sub-diffusion equation: Dtα u − d∆u + f (u) = 0, x ∈ Ω, α ∈ (0, 1),.
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