Abstract
In this work we consider a nonlinear parabolic higher order partial differential equation that has been proposed as a model for epitaxial growth. This equation possesses both global-in-time solutions and solutions that blow up in finite time, for which this blow-up is mediated by its Hessian nonlinearity. Herein, we further analyze its blow-up behaviour by means of the construction of explicit solutions in the square, the disc, and the plane. Some of these solutions show complete blow-up in either finite or infinite time. Finally, we refine a blow-up criterium that was proved for this evolution equation. Still, existent blow-up criteria based on a priori estimates do not completely reflect the singular character of these explicit blowing up solutions.
Highlights
In this work we consider both the initial and initial-boundary value problems for the partial differential equation ut = det(D2u) − 2u, (1)posed either on R2 or on a bounded subset of the plane
Equation (1) is a higher order parabolic equation provided with a Hessian nonlinearity and, because of that, it possesses no second order analogue; in this respect, it is an interesting model to be analyzed
It has been studied in the context of condensed matter physics as a model for epitaxial growth [3,9]
Summary
Our approach is two-fold: first, we compute explicit solutions to model (1) that belong to three different classes; some blow up in finite time, some blow up in infinite time, and others do not blow up at all This program is carried out in Sect. In all the situations considered at least in the cases exhaustively analyzed, solutions that blow up in infinite time are present. Overall, this shows that Eq (1) still presents some features that are not completely captured by the theoretical developments built so far.
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