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An outstanding problem in Earth science is understanding the method oftransport of magma in the Earth's mantle. Models for this process, transport ina viscously deformable porous media, give rise to scalar degenerate,dispersive, nonlinear wave equations. We establish a general localwell-posedness for a physical class of data (roughly $H^1$) via fixed pointmethods. The strategy requires positive lower bounds on the solution. This isextended to global existence for a subset of possible nonlinearities by makinguse of certain conservation laws associated with the equations. Furthermore, weconstruct a Lyapunov energy functional, which is locally convex about theuniform state, and prove (global in time) nonlinear dynamic stability of theuniform state for any choice of nonlinearity. We compare the dynamics to thatof other problems and discuss open questions concerning a larger range ofnonlinearities, for which we conjecture global existence.

Degenerate dispersive equations arising in the study of magma dynamics