2019/03/28 by Steven D. Taliaferro, Taliaferro, Steven
Mathematics · #35B09 #35B33 #35B44 #35B45 #35K58 #35R11 #35R45 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1903.12485
openalex publication_date 2019/03/28 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
We investigate nonnegative solutions u(x,t) and v(x,t) of the nonlinear\nsystem of inequalities \0
leq(
partialt -
Delta)^
alpha u
leq v^
lambda n0
leq (
partialt -
Delta)^
beta v
leq u^
sigma in \ℝn\n\×\ℝ, n\≥ 1, satisfying the initial conditions n u=v=0
quad
text in
mathbbRn
times(-
infty,0) where\n\λ,\σ,\α, and \β are positive constants.\n Specifically, using the definition of the fractional heat operator\n(\∂t-\Δ)^\α given in citeT, we obtain, when they exist,\noptimal pointwise upper bounds on \ℝn \×(0,\∞) for\nnonnegative solutions u and v of this initial value problem with particular\nemphasis on these bounds as t\→0+ and as t\→\∞.\n