88 research outputs found
A waiting time phenomenon for thin film equations
We prove the occurrence of a waiting time phenomenon for solutions to fourth order degenerate parabolic differential equations which model the evolution of thin films of viscous fluids. In space dimension less or equal to three, we identify a general criterion on the growth of initial data near the free boundary which guarantees that for sufficiently small times the support of strong solutions locally does not increase. It turns out that this condition only depends on the smoothness of the diffusion coefficient in its point of degeneracy. Our approach combines a new version of Stampacchia's iteration lemma with weighted energy or entropy estimates. On account of numerical experiments, we conjecture that the
aforementioned growth criterion is optimal
Existence of solutions for a higher order non-local equation appearing in crack dynamics
In this paper, we prove the existence of non-negative solutions for a
non-local higher order degenerate parabolic equation arising in the modeling 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
difficulty associated to higher order equations (e.g. lack of maximum
principle). However, there are important differences with, for instance, the
thin film equation: First, our equation is nonlocal; Also the natural energy
estimate is not as good as in the case of the thin film equation, and does not
yields, for instance, boundedness and continuity of the solutions (our case is
critical in dimension in that respect)
Refined asymptotics for the infinite heat equation with homogeneous Dirichlet boundary conditions
The nonnegative viscosity solutions to the infinite heat equation with
homogeneous Dirichlet boundary conditions are shown to converge as time
increases to infinity to a uniquely determined limit after a suitable time
rescaling. The proof relies on the half-relaxed limits technique as well as
interior positivity estimates and boundary estimates. The expansion of the
support is also studied
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