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We examine a degenerate, dispersive, nonlinear wave equation related to theevolution of partially molten rock in dimensions two and higher. Thissimplified model, for a scalar field capturing the melt fraction by volume, hasbeen studied by direct numerical simulation where it has been observed todevelop stable solitary waves. In this work, we prove local in timewell-posedness results for the time dependent equation, on both the whole spaceand the torus, for dimensions two and higher. We also prove the existence ofthe solitary wave solutions in dimensions two and higher.

Existence theory for magma equations in dimension two and higher