  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.
