On quasilinear parabolic evolution equations in weighted L p -spaces

Abstract

Abstract. In this paper we develop a geometric theory for quasilinear parabolic problems in weighted Lp-spaces. We prove existence and uniqueness of solutions as well as the continuous dependence on the initial data. Moreover, we make use of a regularization effect for quasilinear parabolic equations to study the ω-limit sets and the long-time behaviour of the solutions. These techniques are applied to a free boundary value problem. The results in this paper are mainly based on maximal regularity tools in (weighted) Lp-spaces.

Critical spaces for quasilinear parabolic evolution equations and applications

“…as µ c − 1/p < β < 1 − 1/p. As a consequence, the qualitative theory of Köhne, Prüss and Wilke [10] and Prüss, Simonett and Zacher [18] is available; see also Prüss and Simonett [17], Chapter 5.…”

On Critical Spaces for the Navier–Stokes Equations

“…If the friction coefficient α > 0, then σ(A N,w ) ⊂ (0, ∞), hence the equilibrium u * = 0 of (3.4) is exponentially stable in X w γ,1 , by the principle of linearized stability (see e.g. [15,21]). Choosing u 0 X w γ,µc sufficiently small, then u(t, u 0 ) is arbitrarily close to u * = 0 in B (Ω) for µ ∈ (1/p, 1/2) and (3.7), that for eachε > 0 there existsr > 0 such that for all s ∈ [δ, a] we have u(s, u 0 ) Xγ,µ ≤ε provided u 0 X w γ,µ ≤r.…”

“…Since σ(A N ) ⊂ (0, ∞) in case α > 0, we may apply the principle of linearized stability to (4.1), see e.g. [15,21].…”

Preface

“…In [24,55,74,82,90,97,105,112], the basic issues of existence, uniqueness, and regularity of solutions and their geometric properties are investigated. The approach taken makes systematic use of the theory of L p -maximal regularity.…”

“…A hallmark of this theory, which itself is rooted in deep results of vector-valued harmonic analysis, is the ensuing ability to employ basic results of nonlinear analysis, such as the inverse function theorem and the implicit function theorem, in order to establish further properties of solutions. The short paper [63], where maximal L p -regularity in the presence of time weights is investigated, turned out to be of great importance for the study of quasilinear parabolic equations [97,112,127] as it allows to encode and treat the well-known smoothing property that is typical for parabolic equations. In [90], Jan considered the situation where the set E of equilibria of a quasilinear parabolic equation forms a finite-dimensional C 1 -manifold.…”