It is well-known that the backward differentation formulae (BDF) of order 1, 2 and 3 are gradient stable. This means that when such a method is used for the time discretization of a gradient flow, the associated discrete dynamical system exhibit properties similar to the continuous case, such as the existence of a Lyapunov functional. By means of a Lojasiewicz-Simon inequality, we prove convergence to equilibrium for the 3-step BDF scheme applied to the Allen-Cahn equation with an analytic nonlinearity. By introducing a notion of quadraticstability, we also show that the BDF methods of order 4 and 5 are gradient stable, and that the k-step BDF schemes are not gradient stable for k ≥ 7. Some numerical simulations illustrate the theoretical results.
This work proposes a novel nonlinear parabolic equation with p(x)-growth
conditions for image restoration and enhancement. Based on the
generalized Lebesgue and Sobolev spaces with variable exponent, we
demonstrate the well-posedness of the proposed model. As a first result,
we prove the existence of a weak solution to our model when the reaction
term is bounded by a suitable function. Secondly, we use the
approximations method to establish the existence of a nonnegative weak
SOLA solution (Solution Obtained as Limit of Approximations) to the
proposed model. Finally, numerical experiments illustrate that the
proposed model performs better for image enhancement and denoising.
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