An L_q(L_p)-theory for parabolic pseudo-differential equations: Calderón-Zygmund approach

2015/03/16 by Ildoo Kim, Kim, Ildoo, Kyeong-Hun Kim +3 · 1 citation
Mathematics · Engineering · #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.1503.04521

Abstract

In this paper we present a Calderón-Zygmund approach for a large class of parabolic equations with pseudo-differential operators A(t) of arbitrary order γ∈(0,∞). It is assumed that \cA(t) is merely measurable with respect to the time variable. The unique solvability of the equation (∂ u)/(∂ t)=\cA u-λu+f, (t,x)∈ \fR^d+1 and the L_q(\fR,L_p)-estimate ‖u_t‖_L_q(\fR,L_p)+‖(-Δ)^γ/2u‖_L_q(\fR,L_p) +λ‖u‖_L_q(\fR,L_p)≤ N‖f‖_L_q(\fR,L_p) are obtained for any λ> 0 and p,q∈ (1,∞).

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• Asymptotic behaviors of fundamental solution and its derivatives related to space-time fractional differential equations

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