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We study a nonlinear fourth-order extension of Richards' equation thatdescribes infiltration processes in unsaturated soils. We prove thewell-posedness of the fourth-order equation by first applying Kirchhoff'stransformation to linearize the higher-order terms. The transformed equation isthen discretized in time and space and a set of a priori estimates isestablished. These allow, by means of compactness theorems, extracting a uniqueweak solution. Finally, we use the inverse of Kirchhoff's transformation toprove the well-posedness of the original equation.

On the Well-posedness of a Nonlinear Fourth-Order Extension of Richards' Equation