Fourth-order Parabolic Equation Research Articles

Exponential time decay of solutions to a nonlinear fourth-order parabolic equation

In this paper we investigate the large-time behavior of weak solutions to the nonlinear fourth-order parabolic equation $n_t = -(n(\log n)_{xx})_{xx}$ modeling interface fluctuations in spin systems. We study here the case $x\in \Omega =(0,1)$ , with n = 1, n x = 0 on $\partial \Omega$ . In particular, we prove the exponential decay of u(x,t) towards the constant steady state $n_\infty =1$ in the L 1 norm for long times and we give the explicit rate of decay. The result is based on classical entropy estimates, and on detailed lower bounds for the entropy production.

On an Averaged Model for the 2-fluid Immiscible Flow with Surface Tension in a Thin Cylindrical Tube

We consider the two-fluid incompressible immiscible flow, with surface tension, in a thin cylindrical tube. Using the method of asymptotic expansions with respect to the thickness of the domain, effective 1D equations are derived. Supposing that the fluids are strictly separated at t=0 and that only one phase is in contact with the rigid wall, we show that the derived equations equal a nonlinear degenerate fourth order parabolic equation for the saturation. If we set the surface tension to be zero, it reduces to the Buckley–Leverett equation recently derived by Mikelic and Paoli using the same technique. In the general case we obtain the generalized Darcy's equations with extra-diagonal terms and the relative permeabilities are calculated explicitly. The capillary pressure results only from te surface tension between two fluid phases.

Stable patterns for fourth-order parabolic equations

We consider fourth-order parabolic equations of gradient type. For the sake of simplicity, the analysis is carried out for the specific equation $u\sb t=-\gamma\ u\sb {xxxx}+\beta u\sb {xx}-F\sp \prime(u)$ with $(t,x)\in (0,\infty)\times(0, L)$ and $\gamma,\beta>0$ and where $F(u)$ is a bistable potential. We study its stable equilibria as a function of the ratio $\gamma/beta\sp 2$. As the ratio $\gamma/beta\sp 2$ crosses an explicit threshold value, the number of stable patterns grows to infinity as $L\to \infty$. The construction of the stable patterns is based on a variational gluing method that does not require any genericity conditions to be satisfied.

Positive entropic schemes for a nonlinear fourth-order parabolic equation

A finite-difference scheme with positivity-preserving and entropy-decreasing properties for a nonlinear fourth-order parabolic equation arising in quantum systems and interface fluctuations is derived. Initial-boundary value problems, the Cauchy problem and a rescaled equation are discussed. Based on this scheme we elucidate properties of the long-time asymptotics for this equation.

NULL BOUNDARY CONTROLLABILITY FOR A FOURTH ORDER PARABOLIC EQUATION

We study the null boundary controllability for a one-dimensional fourth order parabolic equation. We show that if the initial data is continuous then the fourth order parabolic equation is controllable.

Exact solutions for variable coefficients fourth-order parabolic partial differential equations in higher-dimensional spaces

In this paper analytic solution of variable coefficient fourth-order parabolic partial differential equation in two and three space variables are developed. The calculations are accelerated by using the noise terms phenomenon for nonhomogeneous problems. Numerical examples are investigated to illustrate the pertinent features of the proposed algorithm.

Asymptotic Behavior of Solutions to the Cahn-Hilliard Equation in Spherically Symmetric Domains

The Cahn-Hilliard equation is a fourth-order parabolic partial differential equation that is one of the leading models for the study of phase separation in isothermal, isotropic, binary mixtures, such as molten alloys. The asymptotic behavior of solutions to the Cahn-Hilliard equation with Dirichlet boundary conditions and the associated stationary problem have been studied. In particular, it is proved that the only possible stable equilibrium solutions in spherically symmetric domains are spherically symmetric and monotone in the radial direction.

A Renormalization Method for Modulational Stability of Quasi-Steady Patterns in Dispersive Systems

We employ global quasi-steady manifolds to rigorously reduce forced, linearly damped dispersive partial differential equations to finite dimensional flows. The manifolds we consider are not invariant, but through a renormalization group method we capture the long-time evolution of the full system as a flow on the manifold. For the parametric nonlinear Schrödinger equation we consider a manifold describing N well-separated pulses and derive an explicit system of ordinary differential equations for the flow on the manifold which captures the leading order pulse motion through the tail-tail interactions. We also outline a rigorous connection between the slow evolution in the hyperbolic PNLS and the fourth-order parabolic phase sensitive amplification equation for fiber optic systems.

Brownian-time processes: The PDE Connection and the half-derivative generator

We introduce a class of interesting stochastic processes based on Brownian-time processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of Brownian motion. They generalize the iterated Brownian motion (IBM) of Burdzy and the Markov snake of Le Gall, and they introduce new interesting examples. After defining Brownian-time processes, we relate them to fourth order parabolic partial differential equations (PDE’s). We then study their exit problem as they exit nice domains in $\mathbb{R}^d$ , and connect it to elliptic PDE’s. We show that these processes have the peculiar property that they solve fourth order parabolic PDE’s, but their exit distribution—at least in the standard Brownian time process case—solves the usual second order Dirichlet problem. We recover fourth order PDE’s in the elliptic setting by encoding the iterative nature of the Brownian-time process, through its exit time, in a standard Brownian motion. We also show that it is possible to assign a formal generator to these non-Markovian processes by giving such a generator in the half-derivative sense.

Two generalisations of the thin film equation

This paper is concerned with two generalisations of the widely studied thin film equation u t = −( u nu xxx ) x . Both are degenerate fourth-order parabolic equations in conservation form; the first is ▪ which shares the scaling properties of the thin film equation (with special cases arising in applications and in earlier analyses), while the second is a doubly nonlinear equation ▪ which is relevant to capillary driven flows of thin films of power-law fluids. We focus on giving a characterisation of nonnegative mass preserving compactly supported solutions, exploiting local analyses about the edge of the support and special closed form solutions; however, other properties are also noted and open questions raised.

Spatial chaos in a fourth-order nonlinear parabolic equation

Abstract Bounded stationary solutions for a fourth-order extended Fisher–Kolmogorov equation with cubic-like piecewise linear nonlinearity are given. Those solutions which have a finite number of zeros are characterized by means of a set of integers associated with the distance between the zeros. The value of the spatial entropy is estimated and the existence of spatial chaos is shown.

Analytic treatment for variable coefficient fourth-order parabolic partial differential equations

The variable coefficient fourth-order parabolic partial differential equation in one space variable that arise in the study of the transverse vibrations of a uniform flexible beam is examined. The analytic solution is obtained by using Adomian decomposition method. The nonhomogeneous problem is solved by using the noise terms phenomenon. The method is tested on three problems which have appeared in physical applications.

Travelling Waves for Fourth Order Parabolic Equations

We study travelling wave solutions for a class of fourth order parabolic equations. Travelling wave fronts of the form u(x,t)=U(x+ct), connecting homogeneous states, are proven to exist in various cases: connections between two stable states, as well as connections between an unstable and a stable state, are considered.

Singular limit of a fourth-order problem arising in the microphase separation of diblock copolymers

We study the limiting behavior as $\varepsilon$ tends to zero of the solution of a system arising in the microphase separation of diblock copolymers. This system involves a fourth-order parabolic equation. We consider the case of spherical symmetry, and we show the convergence to a free-boundary Hele--Shaw-type problem.

Approximate Self-Similarity in Models of Geological Folding

We propose a model for the folding of rock under the compression of tectonic plates. This models an elastic rock layer imbedded in a viscous foundation by a fourth-order parabolic equation with a nonlinear constraint. The large-time behavior of solutions of this problem is examined and found to be of approximately self-similar form.

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

The existence of nonnegative weak solutions globally in time of a nonlinear fourth-order parabolic equation in one space dimension is shown. This equation arises in the study of interface fluctuations in spin systems and in quantum semiconductor modeling. The problem is considered on a bounded interval subject to initial and Dirichlet and Neumann boundary conditions. Further, the initial datum is assumed only to be nonnegative and to satisfy a weak integrability condition. The main difficulty of the existence proof is to ensure that the solutions stay nonnegative and exist globally in time. The first property is obtained by an exponential transformation of variables. Moreover, entropy-type estimates allow for the proof of the second property. Results concerning the regularity and long-time behavior are given. Finally, numerical experiments underlining the preservation of positivity are presented.

Implicit time splitting for fourth-order parabolic equations

A coordinate-splitting economic difference scheme is proposed for generalized parabolic equations (GPE) containing fourth-order diffusion operators and the algorithm for its implementation is developed. The performance of the scheme is demonstrated for different cases, e.g. for treating bifurcation phenomena. The technique is applied to the numerical solution of Swift-Hohenberg equation describing the Rayleigh-Bénard convection and results are obtained for very large system sizes and for very long times on small computational platform.

Homogeneous reonomic contact transformations and path reparametrization in functional integrals with respect to quasimeasures

Homogeneous reonomic contact transformations and path reparametrizations in functional integrals with respect to quasimeasures are considered for linear fourth-order parabolic differential equations with coefficients explicitly depending on time. An integral relation between the Green’s functions is found for fourth-order differential equations that correspond to each other. In a special case, the differential operator of the transformed equation coincides with the Bol conformally covariant operator, which is well known in exactly integrable and conformal models.

On the Cahn–Hilliard Equation with Degenerate Mobility

An existence result for the Cahn–Hilliard equation with a concentration dependent diffusional mobility is presented. In particular, the mobility is allowed to vanish when the scaled concentration takes the values $ \pm 1$, and it is shown that the solution is bounded by 1 in magnitude. Finally, applications of our method to other degenerate fourth-order parabolic equations are discussed.

Long time asymptotics of fourth order parabolic equations

Long time asymptotics of fourth order parabolic equations