Nonlinear problems with biological and physical insights

We address the study of regularity and/or well-posedness for nonlinear problems with applications to biology and physics. In particular, this work is divided into three parts. In the first part, we present two non-isothermal Cahn-Hilliard models. The first one is a 2D PDE system describing the phase separation behaviour of a two-component fluid in a bounded domain. On the other hand, the second one is a 3D non-isothermal Cahn-Hilliard system describing tumor growth. The second part is devoted to study the mechanism of breathing. In particular, we consider the lungs as a viscoelastic deformable porous medium and breathing as an isothermal periodic process, which also takes into account the phenomenon of hysteresis. Eventually, in the third part, we focus on higher differentiability properties of solutions to obstacle problems with nonstandard growth conditions.