A weighted L_p-regularity theory for parabolic partial differential equations with time measurable pseudo-differential operators

2022/05/25 by Choi, Jae-Hwan, Kim, Ildoo
#35B65 #35S05 #47G30 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2205.12463

Abstract

We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problem \begincases ∂_t u(t,x)=ψ(t,-i∇)u(t,x)+f(t,x), amp;(t,x)∈(0,T)×ℝ^d,
u(0,x)=0, amp; x∈ℝ^d \endcases in (Muckenhoupt) weighted L_p-spaces with time-measurable pseudo-differential operators ψ(t,-i∇)u(t,x):=F^-1[ψ(t,⋅)F[u](t,⋅)](x). More precisely, we find sufficient conditions of the symbol ψ(t,ξ) (especially depending on the smoothness of the symbol with respect to ξ) to guarantee that equation is well-posed in (Muckenhoupt) weighted L_p-spaces. Here the symbol ψ(t,ξ) is merely measurable with respect to t, and the sufficient smoothness of ψ(t,ξ) with respect to ξ is characterized by a property of each weight. In particular, we prove the existence of a positive constant N such that for any solution u to the equation, ∫₀^T ∫_ℝ^d |(-Δ)^γ/2 u(t,x) |^p (t² + |x|²)^α/2 dxdt ≤ N∫₀^T ∫_ℝ^d |f(t,x)|^p (t² + |x|²)^α/2 dxdt and ∫₀^T (∫_ℝ^d |(-Δ)^γ/2 u(t,x) |^p |x|^α₂ dx )^q/p t^α₁dt ≤ N∫₀^T (∫_ℝ^d |f(t,x) |^p |x|^α₂ dx )^q/p t^α₁dt, where p,q∈(1,∞), -d-1<α< (d+1)(p-1), -1 < α₁ < q-1, -d <α₂< d(p-1), and γ is the order of the operator ψ(t,-i∇).

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