Tent space well-posedness for parabolic Cauchy problems with rough coefficients

[摘要] We study the well-posedness of Cauchy problems on the upper half space R-+(n+1) associated to higher order systems partial derivative(t)u = (-1)(m+1) div(m)A del(m)u with bounded measurable and uniformly elliptic coefficients. We address initial data lying in L-p (1 < p < infinity) and B M O (p = infinity) spaces and work with weak solutions. Our main result is the identification of a new well-posedness class, given for p is an element of (1, infinity] by distributions satisfying del(m)u is an element of T-m(p,2), where T-m(p,2) is a parabolic version of the tent space of Coifman-Meyer-Stein. In the range p is an element of [2, infinity], this holds without any further constraints on the operator and for p = infinity it provides a Carleson measure characterization of BM O with non-autonomous operators. We also prove higher order L-p well-posedness, previously only known for the case m = 1. The uniform L-p boundedness of propagators of energy solutions plays an important role in the well-posedness theory and we discover that such bounds hold for p close to 2. This is a consequence of local weak solutions being locally Holder continuous with values in spatial L-loc(p), for some p > 2, what is also new for the case m > 1. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-12-05

[关键词] Higher order parabolic equations;Non-autonomous elliptic operators;The polyharmonic operator;Well-posedness of parabolic Cauchy problems;Carleson measures;Tent spaces