2023/10/13 by Jae-Hoon Kang, Kang, Jaehoon, Daehan Park +1
Computer Science · Mathematics · #35B65 #47G20 #60G51 #60J35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2310.08871
openalex publication_date 2023/10/13 · openalex created_date 2023/10/17 · openalex updated_date 2026/07/28
In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators ∂tu(t,x) = Lau(t,x) + f(t,x), tgt;0 in Lq(Lp) spaces. Our spatial operator La is an integro-differential operator of the form ∫ℝd ( u(x+y)-u(x) -∇ u(x) ⋅ y 1|y|≤ 1 ) a(t,y) jd(|y|)dy. Here, a(t,y) is a merely bounded measurable coefficient, and we employed the theory of additive process to handle it. We investigate conditions on jd(r) which yield Lq(Lp)-regularity of solutions. Our assumptions on jd are general so that jd(r) may be comparable to r-dℓ(r-1) for a function ℓ which is slowly varying at infinity. For example, we can take ℓ(r)=log(1+rα) or ℓ(r) = min\rα,1\ (α∈(0,2)). Indeed, our result covers the operators whose Fourier multiplier ψ(ξ) does not have any scaling condition for |ξ|≥ 1. Furthermore, we give some examples of operators, which cannot be covered by previous results where smoothness or scaling conditions on ψ are considered.