A local theory for a fractional reaction-diffusion equation

Abstract

In this paper, we study the local well-posedness for the Cauchy problem of a semilinear fractional diffusion equation where the perturbations behave like [Formula: see text] and [Formula: see text], and [Formula: see text] is the characteristic function of a ball [Formula: see text]. Here, we are interested in the solvability of the problem when singular initial data [Formula: see text] are taken in [Formula: see text]. Eventually, we give sufficient conditions to the nonexistence of positive global solutions.

Existence, symmetries, and asymptotic properties of global solutions for a fractional diffusion equation with gradient nonlinearity

“…aN (see [11,23]) while it is q F ¼ 1 þ 2 N (the same as the heat equation), for o a t u ¼ Du þ juðt; xÞj qÀ1 uðt; xÞ; ð1:6Þ…”

Some well‐posed results on the time‐fractional Rayleigh–Stokes problem with polynomial and gradient nonlinearities

“…More precisely, in another study, 22 he and his co‐authors consider some problems that appear in the science of viscoelastic materials. A time‐fractional biology model for chemotaxis introduced by Keller and Segel was studied by Cuevas et al 25 Additionally, there are also many other typical authors, namely, Bruno de Andrade, 23,26,27 Arlúcio Viana, 26–28 etc.…”

Existence and regularity results of a backward problem for fractional diffusion equations

“…More recently, Andrade and Viana 16 have developed the global well-posedness and spatiotemporal asymptotic behavior in the case of semilinear situation. Viana 17 studied the local well-posedness of such equations with perturbations behavior.…”