Well posedness of nonlinear parabolic systems beyond duality

Abstract

We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial-boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem accessible to our technique is the following system \[ \partial_tu-\mathrm{div} ( ν(|\nabla u|) \nabla u )= -\mathrm{div} f \] with a given {strictly} positive bounded function $\nu$, {such that ${\lim }_{k\to \infty }\nu (k)={\nu }_{\infty }$} and $f\in L^q$ with $q\in (1,\infty )$. The {existence, uniqueness and regularity} results for $q\ge 2$ are by now standard. However, even if a priori estimates are available, the existence in case $q\in (1,2)$ was essentially missing. We overcome the related crucial difficulty, namely the lack of a standard duality pairing, by resorting to proper weighted spaces and consequently provide existence, uniqueness and optimal regularity in the entire range $q\in (1,\infty )$.