2017/10/02 by Steven D. Taliaferro, Taliaferro, Steven D.
Computer Science · Mathematics · #35B09 #35B33 #35B44 #35B45 #35K10 #35K58 #35R09 #35R45 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1710.00896
openalex publication_date 2017/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the behavior as t\→ 0+ of nonnegative functions\n\
label0.1 u
in C2,1 (
mathbbRn
times (0,1))
cap\nL^
lambda (
mathbbRn
times (0,1)),
quad n
ge 1, satisfying\nthe parabolic Choquard-Pekar type inequalities \
label0.2\n 0
leq ut-
Delta u
leq(
Phi
alpha/n*u^
lambda )u^
sigma
quad
text in\nB1 (0)
times (0,1) where \α\∈(0,n+2), \λ>0, and\n\σ\≥0 are constants, \Φ is the heat kernel, and * is the\nconvolution operation in \ℝn\× (0,1). We provide optimal\nconditions on \α,\λ, and \σ such that nonnegative solutions\nu satisfy pointwise bounds in compact subsets of B1(0) as t\→ 0+. We\nobtain similar results for nonnegative solutions when \Φ\α/n is\nreplaced with the fundamental solution \Φ_\α of the fractional heat\noperator (\(\∂)/(\∂ t)-\Δ)\α/2.\n