The Semilinear Cahn-Hilliard-Gurtin System in Critical Spaces

Abstract

In this thesis, the semilinear Cahn-Hilliard-Gurtin equation is studied using the method of Maximal Regularity. In 2012, Wilke developed a linear theory in $L^p$-spaces, and achieved a local and global well-posedness result for large $p$. In 2013, Denk and Kaip developed a linear theory in mixed integrability $L^pL^q$-spaces, using the method of Newton polygons. In this thesis, we connected the recent weighted anisotropic Mikhlin multiplier theorem of Lorist with the method of Newton polygons, leading to a linear theory in time-weighted $L^pw_\alpha L^q$-spaces, which is novel. By a postulation that the linear theory also holds in domains, we are able to treat the local well-posedness in the recently developed critical space setting of Prüss et al. This approach draws upon recent advances in interpolation theory in the setting of fractional Sobolev spaces with power weights in time, such as exhibited in the work of Agresti and Veraar.By adapting the global well-posedness result of Wilke, we are able to treat the semilinear equation in less regular spaces, i.e. smaller integrability parameters $p$ and $q$, and with rough initial data.