Abstract
A time-fractional space-nonlocal reaction-diffusion equation in a bounded domain is considered. First, the existence of a unique local mild solution is proved. Applying Poincaré inequality it is obtained the existence and boundedness of global classical solution for small initial data. Under some conditions on the initial data, we show that solutions may experience blow-up in a finite time.
Highlights
The purpose of this paper is to study Cauchy problem for the time-fractional space-nonlocal reaction-diffusion equation
Let us mention that with the change of variable v := 1 − u, (1.1) is transformed to the Fisher equation, if α = 1, ε = 0
Differential equations with modified arguments are equations in which the unknown function and its derivatives are evaluated with modifications of time or space variables; such equations are called, in general, functional differential equations
Summary
Similar studies for time-fractional reaction-diffusion equations were considered in [3, 9]. Existence of global classical solution Let |ε| < 1 and u0(x) ∈ C([0, 1]) satisfy the estimates 0 ≤ u0(x) ≤ 1. The global classical solution 0 ≤ u ≤ 1 of nonlocal reaction-diffusion problem (1.1)-(1.3) satisfies the following estimate. If 1 + (1 − ε)π2 ≤ 2 u0(x) sin πxdx = F0, the classical solution of problem (1.1)-(1.3) blows-up in a finite time Preliminaries Let us give basic definitions of fractional differentiation and integration of the Riemann–Liouville and Caputo types. Proposition 1.5 [10] If y0 > 0, the solution of problem (1.5) blows-up in a finite time. Poincaré inequality for the differential operator with involution We consider the following eigenvalue problem.
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