Pointwise bounds and blow-up for systems of nonlinear fractional parabolic inequalities. (June 2020) - Taliaferro, Steven D.
Abstract: We investigate nonnegative nonlocal solutions u ( x, t ) and v ( x, t ) of the nonlinear system of inequalities 0 ≤ ( ∂ t − Δ ) α u ≤ v λ 0 ≤ ( ∂ t − Δ ) β v ≤ u σ in R n × R, n ≥ 1 satisfying the initial conditions u = v = 0 in R n × ( − ∞, 0 ) where λ, σ, α, and β are positive constants. Specifically, using the definition of the fractional heat operator ( ∂ t − Δ ) α given in Taliaferro (2020), we obtain, when they exist, optimal pointwise upper bounds on R n × ( 0, ∞ ) for nonnegative nonlocal solutions u and v of this initial value problem with particular emphasis on these bounds as t → 0 + and as t → ∞ . When α = β = 1 we compare our results for nonlocal solutions to those of Escobedo and Herrero (1991) for classical pointwise solutions.