Parabolic Fourth-order Equation Research Articles

Cahn–Hilliard–Brinkman systems for tumour growth

<p style='text-indent:20px;'>A phase field model for tumour growth is introduced that is based on a Brinkman law for convective velocity fields. The model couples a convective Cahn–Hilliard equation for the evolution of the tumour to a reaction-diffusion-advection equation for a nutrient and to a Brinkman–Stokes type law for the fluid velocity. The model is derived from basic thermodynamical principles, sharp interface limits are derived by matched asymptotics and an existence theory is presented for the case of a mobility which degenerates in one phase leading to a degenerate parabolic equation of fourth order. Finally numerical results describe qualitative features of the solutions and illustrate instabilities in certain situations.</p>

Longtime convergence for epitaxial growth model under Dirichlet conditions

This paper continues our study on the initial-boundary value problem for a semilinear parabolic equation of fourth order which has been presented by Johnson-Orme-Hunt-Graff-Sudijono-Sauder-Orr [12] to describe the large-scale features of a growing crystal surface under molecular beam epitaxy. In the preceding paper [1], we already constructed a dynamical system generated by the problem and verified that the dynamical system has a finite-dimensional attractor (especially, every trajectory has nonempty $\omega$-limit set) and admits a Lyapunov function (of the form (3.1)). This paper is then devoted to showing longtime convergence of trajectory. We shall prove that every trajectory converges to some stationary solution as $t \to \infty$. As a matter of fact, we have obtained in [10] the similar result for the equation but under the Neumann like boundary conditions $\frac{\partial u}{\partial n}=\frac\partial{\partial n}\varDelta u=0$ on the unknown function $u$. In this paper, we want as in [1] to handle the Dirichlet boundary conditions $u=\frac{\partial u}{\partial n}=0$, maybe physically more natural conditions than before.

Classical solvability of the multidimensional free boundary problem for the thin film equation with quadratic mobility in the case of partial wetting

We prove locally in time the existence of the unique smooth solution (including smooth interface) to the multidimensional free boundary problem for the thin film equation with the mobility n = 2 in the case of partial wetting. We also obtain the Schauder estimates and solvability for the Dirichlet and the Neumann problem for a linear degenerate parabolic equation of fourth order.

Liouville Property for Solutions of the Linearized Degenerate Thin Film Equation of Fourth Order in a Halfspace

We consider a boundary value problem in a half-space for a linear parabolic equation of fourth order with a degeneration on the boundary of the half-space. The equation under consideration is substantially a linearized thin film equation. We prove that, if the right hand side of the equation and the boundary condition are polynomials in the tangential variables and time, the same property has any solution of a power growth. It is shown also that the specified property does not apply to the normal variable. As an application, we present a theorem of uniqueness for the problem in the class of functions of power growth.

Existence for coupled pseudomonotone–strongly monotone systems and application to a Cahn–Hilliard model with elasticity

A system of two operator equations is considered–one of pseudomonotone type and the other of strongly monotone type–both being strongly coupled. Conditions are given that allow to reduce the solvability of this system to a single operator equation of a pseudomonotone mapping. This result is applied to a coupled system consisting of a parabolic equation of fourth order in space of Cahn–Hilliard type and a nonlinear elliptic equation of second order modeling a quasi-steady mechanical equilibrium. Using an appropriate notion of weak solutions and a framework for evolution equations developed by Gröger (2001), the system is reduced to a single parabolic operator equation and the existence of solutions is shown under restrictions on the strength of the coupling.

Existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order

This paper is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order: u t + ∇ ⋅ ( | ∇ Δ u | p − 2 ∇ Δ u ) = f ( u ) in Ω ⊂ R N with boundary condition u = Δ u = 0 and initial data u 0 . The substantial difficulty is that the general maximum principle does not hold for it. The solutions are obtained for both the steady-state case and the developing case by the fixed point theorem and the semi-discretization method. Unlike the general procedures used in the previous papers on the subject, we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables, respectively. By a compactness argument with necessary estimates, we show that the two approximation sequences converge to the same limit, i.e., the solution to be determined. In addition, the decays of solutions towards the constant steady states are established via the entropy method. Finally, it is interesting to observe that the solutions just tend to the initial data u 0 as p → ∞ .

Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation

A finite element method for the Cahn-Hilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidiscrete case and in a completely discrete case based on the backward Euler method. Error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data. A detailed study of the regularity of the exact solution is included. The analysis is based on local Lipschitz conditions for the nonlinearity with respect to Sobolev norms, and the existence of a Ljapunov functional for the exact and the discretized equations is essential. A result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup (upper semicontinuity with respect to the discretization parameters) is obtained as a simple application of the nonsmooth data error estimate.

Error Estimates with Smooth and Nonsmooth Data for a Finite Element Method for the Cahn-Hilliard Equation

A finite element method for the Cahn-Hilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidiscrete case and in a completely discrete case based on the backward Euler method. Error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data. A detailed study of the regularity of the exact solution is included. The analysis is based on local Lipschitz conditions for the nonlinearity with respect to Sobolev norms, and the existence of a Ljapunov functional for the exact and the discretized equations is essential. A result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup (upper semicontinuity with respect to the discretization parameters) is obtained as a simple application of the nonsmooth data error estimate.