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

• Published in: Nonlinear analysis 2020-06, Vol.195, p.111744, Article 111744
• Main Author: Taliaferro, Steven D.
• Format: Article
• Language: English
• Subjects: Blow-up; Fractional heat operator; Parabolic system; Pointwise bound
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2020.111744

We investigate nonnegative nonlocal solutions u(x,t) and v(x,t) of the nonlinear system of inequalities 0≤(∂t−Δ)αu≤vλ0≤(∂t−Δ)βv≤uσin Rn×R,n≥1satisfying the initial conditions u=v=0in Rn×(−∞,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 Rn×(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.