Unbounded solutions for the Muskat problem

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Abstract We prove the local existence of solutions of the form x 2 + c t + g , with g ∈ H s ( R ) and s ⩾ 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.