Quasilinear parabolic equations with superlinear nonlinearities in critical spaces

TL;DR

This paper establishes well-posedness for certain quasilinear parabolic equations in critical spaces, demonstrating the existence of solutions and semiflows using regularizing effects, with applications to atmospheric flow models and equations with scaling invariance.

Contribution

It introduces new well-posedness results for quasilinear parabolic equations in critical spaces, expanding understanding of solution behavior in these settings.

Findings

Well-posedness in critical intermediate spaces is proven for specific quasilinear equations.

The solution map generates a semiflow in these critical spaces.

Applications include models for atmospheric flows and equations with scaling invariance.

Abstract

Well-posedness in time-weighted spaces for quasilinear (and semilinear) parabolic evolution equations $u'=A(u)u+f(u)$ is established in a certain critical case of strict inclusion $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ for the domains of the (superlinear) function $u\mapsto f(u)$ and the quasilinear part $u\mapsto A(u)$. Based upon regularizing effects of parabolic equations, it is proven that the solution map generates a semiflow in a critical intermediate space. The applicability of the abstract results is demonstrated by several examples including a model for atmospheric flows and semilinear and quasilinear evolution equations with scaling invariance for which well-posedness in the critical scaling invariant intermediate spaces is shown.