Global existence and blow-up of solutions of the time-fractional space-involution reaction-diffusion equation

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.

Solvability of mixed problems for the wave equation with reflection of the argument

“…The questions of solvability of direct, inverse problems for differential equations with involution in the case of constant coefficients are discussed in the works of many authors. 11,12,[14][15][16][17][18][31][32][33][34][35][36] For example, in Turmetov and Kadirkulov, 31 solvability of a boundary value problem for a nonlocal analog of a mixed parabolic-hyperbolic equation of fractional order with involution is studied. Previous works [13][14][15][16][17][18]32 are devoted to the study of inverse problems for a fractional parabolic equation with involution.…”

“…Inverse problems with involution perturbations have been considered in previous works. [11][12][13][14][15][16][17][18] Mathematical models describing the phenomena of population dynamics, ecology, physiology, and other applications are considered in monographs. 1,3 Definition 2.…”

A variant of quasi‐reversibility method for a class of heat equations with involution perturbation

“…To our best knowledge, the mathematical literature devoted to the regularization methods and their numerical approximation, dealing with inverse problems and ill‐posed problems governed by PDEs with deviating variables, is characterized by the scarcity of the works. Concerning inverse problems with involution perturbations, we can quote the recent works 9,11,23‐28 …”

Remarks on the Initial and Terminal Value Problem for Time and Space Fractional Diffusion Equation

“…Jin et al [27] applied two semidiscrete schemes of Galerkin FEM method in order to approximate the solution of Problems (1) and (2). In [28], the authors investigated a reaction-diffusion equation with a Caputo fractional derivative in time. In [29], the authors established the existence and uniqueness of the weak solution and the regularity of the solution for coupled fractional diffusion system.…”