Global estimates for quasilinear parabolic equations on Reifenberg flat domains and its applications to Riccati type parabolic equations with distributional data

Abstract

In this paper, we prove global weighted Lorentz and Lorentz-Morrey estimates for gradients of solutions to the quasilinear parabolic equations:, under minimal regularity assumptions on the boundary of domain and on nonlinearity A. Then results yields existence of a solution to the Riccati type parabolic equations:where q > 1 and µ is a bounded Radon measure.MSC: primary 35K59; secondary 42B37

Good-λ type bounds of quasilinear elliptic equations for the singular case

“…For the proofs of these above theorems 1.1 and 1.2 in the present paper, it is possible to apply some results developed for quasilinear equations with given measure data, or linear/nonlinear potential and Calderón-Zygmund theories (see in [3,11,12,13,14,23,20,26,25]), to some new comparison estimates in the singular case 3n−2 2n−1 < p ≤ 2 − 1 n . The outline of paper is organized as follows.…”

Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

“…As mentioned earlier, this technique is not applicable to our settings due to the lack of homogeneity, and in the particular case p = 2, we recover the results in [34] for the unweighted Lorentz estimates by a different approach.…”

“…In most of papers on the global Lorentz estimates for the gradient of the weak solutions to elliptic and parabolic equations, the HardyLittlewood maximal function technique is often employed, see for example [8,34,30,31]. In particular case of the parabolic equation in the form (1) and some of its specific cases, the maximal function technique works well only in the case p = 2, see for example [8,34]. However, the case p = 2 rules out the homogeneity of the parabolic equations.…”

“…See Section 2.2. (c) The global weighted Lorentz estimates for the gradient of the weak solution to the problem (1) corresponding to p = 2 was obtained in [34] by using the maximal function technique.…”

Porous medium equation with nonlocal pressure in a bounded domain

“…We refer to [32,33] for L p estimates for the semi-group e t∆ . Let H(t, x, y) be the Heat kernel in Ω × (0, ∞).…”