2020/05/12 by Steven D. Taliaferro, Taliaferro, Steven D.
Mathematics · #35B09 #35B33 #35K58 #35R11 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2005.06029
openalex publication_date 2020/05/12 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We study the existence of nontrivial nonlocal nonnegative solutions u(x,t)\nof the nonlinear initial value problems \(
partialt -
Delta)^
alpha u
geqν^
lambda
quad
textin
mathbbRn
times
mathbbR,
,n
geq 1 \u=0\n
quad
textin
mathbbRn
times(-
infty,0) and \C1 u^
lambda\n
leq(
partialt -
Delta)^
alpha u
leq C2 u^
lambda
quad
textin \n
mathbbRn
times
mathbbR,
,n
geq1 \u=0
quad
textin
mathbbRn\n
times(-
infty,0), where \λ,\α,C1, and C2 are positive\nconstants with C1 <C2. We use the definition of the fractional heat\noperator (\∂t -\Δ)^\α given in [Taliaferro, 2020] and compare\nour results in the classical case \α=1 to known results.\n