Pointwise bounds and blow-up for systems of nonlinear fractional parabolic inequalities

We investigate nonnegative nonlocal solutions u( x, t) and v( x, t) of the nonlinear system of inequalities 0 <=(partial derivative(t) - Delta)(alpha)u <= upsilon(lambda) 0 <=(partial derivative(t) - Delta)(beta)upsilon <= u(sigma)} in R-n x R, n >= 1 satisfying the initial conditions u = upsilon = 0 in R-n x (-infinity, 0) where lambda, sigma, alpha and beta are positive constants. Specifically, using the definition of the fractional heat operator (partial derivative(t) - Delta)(alpha) a given in Taliaferro (2020), we obtain, when they exist, optimal pointwise upper bounds on R-n x (0,infinity) for nonnegative nonlocal solutions u and upsilon of this initial value problem with particular emphasis on these bounds as t -> 0(+) and as t -> infinity. When alpha = beta = 1 we compare our results for nonlocal solutions to those of Escobedo and Herrero (1991) for classical pointwise solutions. (C) 2020 Elsevier Ltd. All rights reserved.