Unbounded solutions for the Muskat problem

Abstract

We prove the local existence of solutions of the form $x^2+ct+g,$ with $g\in H^s(\mathbb R)$ and $s\geq 3,$ for the Muskat problem in the stable regime. We use energy methods to obtain a bound of $g$ in Sobolev spaces. In the proof we deal with the loss of the Rayleigh-Taylor condition at infinity and a new structure of the kernels in the equation. Remarkably, these solutions grow quadratically at infinity.