A Sixth-Order Nonlinear Parabolic Equation for Quantum Systems

The Existence of Global Attractor for a Sixth Order Parabolic Equation

This paper is concerned with a sixth-order parabolic equation which arises naturally as a continuum model for the formation of quantum dots and their faceting. Based on the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors, we prove that the sixth order parabolic equation possesses a global attractor in the H (k ≥ 0) space, whi...

Estimates on Front Propagation for Nonlinear Higher-order Parabolic Equations: an Algorithmic Approach

We present an algorithm for the derivation of lower bounds on support propagation for a certain class of nonlinear parabolic equations. We proceed by combining the ideas in some recent papers by the author with the algorithmic construction of entropies due to Jüngel and Matthes, reducing the problem to a quantifier elimination problem. Due to its complexity, the quantifier elimination problem c...

Finite element approximation of a sixth order nonlinear degenerate parabolic equation

We consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation ut = ∇.( b(u)∇∆u), where generically b(u) := |u| for any given γ ∈ (0,∞). In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d ≤ 3. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some num...

Global Nonnegative Solutions of a Nonlinear Fourth-Order Parabolic Equation for Quantum Systems

A NOTE ON "A SIXTH ORDER METHOD FOR SOLVING NONLINEAR EQUATIONS"

In this study, we modify an iterative non-optimal without memory method, in such a way that is becomes optimal. Therefore, we obtain convergence order eight with the some functional evaluations. To justify our proposed method, some numerical examples are given.