Existence of solutions for a higher order non-local equation appearing in crack dynamics

Abstract

In this paper, we prove the existence of non-negative solutions for a non-local higher order degenerate parabolic equation arising in the modelling of hydraulic fractures. The equation is similar to the well-known thin-film equation, but the Laplace operator is replaced by a Dirichlet-to-Neumann operator, corresponding to the square root of the Laplace operator on a bounded domain with Neumann boundary conditions (which can also be defined using the periodic Hilbert transform). In our study, we have to deal with the usual difficulties associated with higher order equations (e.g. a lack of maximum principle). However, there are important differences to, for instance, the thin-film equation. Firstly, our equation is non-local. Secondly the natural energy estimate is not as good as that of the thin-film equation, and does not yield, for instance, boundedness and continuity of the solutions (our equation is critical in dimension 1 in that respect).