Fourth-order Parabolic Equation Modeling Research Articles

Decay estimate and blow-up for a fourth order parabolic equation modeling epitaxial thin film growth

<abstract><p>In this paper, we study a fourth order parabolic equation modeling epitaxial thin film growth. By using the potential well method and some inequality techniques, we obtain the decay estimate of weak solutions. Meanwhile, the blow-up time is estimated from above and below. The blow-up rate is also derived.</p></abstract>

Lower bound of blow-up time to a fourth order parabolic equation modeling epitaxial thin film growth

In this paper, we considered a class of fourth order equation modeling epitaxial thin film growth. We establish a blow up result for the initial and boundary value problem. Furthermore, the lower bound of the blow up time and the blow up rate are derived.

Global asymptotical behavior of solutions to a class of fourth order parabolic equation modeling epitaxial growth

In this paper, we consider a class of fourth-order parabolic equation modeling epitaxial growth. For the evolution problem, we study the exponential decay of the solutions and its corresponding energy functional. For the stationary problem, we find a ground-state solution by Lagrange multiplier method. Moreover, the asymptotical behavior of the general global solution is also considered. The results of this paper extend some results got by Han (2018).

Asymptotic behavior for a fourth-order parabolic equation modeling thin film growth

The main goal of this work is to study a fourth-order parabolic equation with nonstandard growth conditions. For non-positive initial energy J(u0)≤0, by using the concavity method, we establish that weak solutions blow up in finite time. On the other hand, we also derive the blow-up time estimate and the asymptotic behavior of weak solutions.

Local and global existence of solutions to a fourth-order parabolic equation modeling kinetic roughening and coarsening in thin films

In this paper we study both the Cauchy problem and the initial boundary value problem for the equation $\partial_tu+\mbox{div}\left(\nabla\Delta u-{\bf g}(\nabla u)\right)=0$. This equation has been proposed as a continuum model for kinetic roughening and coarsening in thin films. In the Cauchy problem, we obtain that local existence of a weak solution is guaranteed as long as the vector-valued function ${\bf g}$ is continuous and the initial datum $u_0$ lies in $C^1(\mathbb{R}^N)$ with $\nabla u_0(x)$ being uniformly continuous and bounded on $\mathbb{R}^N$ and that the global existence assertion also holds true if we assume that ${\bf g}$ is locally Lipschitz and satisfies the growth condition $|{\bf g}(\xi) |\leq c|\xi|^\alpha$ for some $c>0, \alpha\in (2, 3)$, $\sup_{\mathbb{R}^N}|\nabla u_0|<\infty$, and the norm of $u_0$ in the space $L^{\frac{(\alpha-1)N}{3-\alpha}}(\mathbb{R}^N) $ is sufficiently small. This is done by exploring various properties of the biharmonic heat kernel. In the initial boundary value problem, we assume that ${\bf g}$ is continuous and satisfies the growth condition $|{\bf g}(\xi) |\leq c|\xi|^\alpha+c$ for some $c, \alpha\in (0,\infty)$. Our investigations reveal that if $\alpha\leq 1$ we have global existence of a weak solution, while if $1<\alpha<\frac{N^2+2N+4}{N^2}$ only a local existence theorem can be established. Our method here is based upon a new interpolation inequality, which may be of interest in its own right.

On the solutions of a fourth order parabolic equation modeling epitaxial thin film growth

Abstract In this paper, we study the existence and uniqueness of global weak solution, the regularity of the solutions and the existence of global attractor for a fourth order parabolic equation modeling epitaxial thin film growth with Neumann boundary conditions in two space dimensions.

Solutions of fourth-order parabolic equation modeling thin film growth

In this paper we study the well-posedness for a fourth-order parabolic equation modeling epitaxial thin film growth. Using Kato's Method [1–3] we establish existence, uniqueness and regularity of the solution to the model, in suitable spaces, namely C0([0,T];Lp(Ω)) where p=nα2−α with 1<α<2, n∈N and n≥2. We also show the global existence solution to the nonlinear parabolic equations for small initial data. Our main tools are Lp–Lq-estimates, regularization property of the linear part of e−tΔ2 and successive approximations. Furthermore, we illustrate the qualitative behavior of the approximate solution through some numerical simulations. The approximate solutions exhibit some favorable absorption properties of the model, which highlight the stabilizing effect of our specific formulation of the source term associated with the upward hopping of atoms. Consequently, the solutions describe well some experimentally observed phenomena, which characterize the growth of thin film such as grain coarsening, island formation and thickness growth.

A Degenerate Fourth-Order Parabolic Equation Modeling Bose-Einstein Condensation Part II: Finite-Time Blow-Up

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model can be motivated from the spatially homogeneous and isotropic Boltzmann-Nordheim equation by a formal Taylor expansion of the collision integral. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of local-in-time weak solutions satisfying a certain entropy inequality is proven. The main result asserts that if a weighted L 1 norm of the initial data is sufficiently large and the initial data satisfies some integrability conditions, the solution blows up with respect to the L ∞ norm in finite time. Furthermore, the set of all such blow-up enforcing initial functions is shown to be dense in the set of all admissible initial data. The proofs are based on approximation arguments and interpolation inequalities in weighted Sobolev spaces. By exploiting the entropy inequality, a nonlinear integral inequality is proved which implies the finite-time blow-up property.

A Degenerate Fourth-Order Parabolic Equation Modeling Bose–Einstein Condensation. Part I: Local Existence of Solutions

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model is a Fokker-Planck-type approximation of the Boltzmann-Nordheim equation, only keeping the leading order term. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of a local-in-time nonnegative continuous weak solution is proven. If the solution is not global, it blows up with respect to the $L^\infty$ norm in finite time. The proof is based on approximation arguments, interpolation inequalities in weighted Sobolev spaces, and suitable a priori estimates for a weighted gradient $L^2$ norm.

Optimal control of a fourth-order parabolic equation modeling epitaxial thin film growth

In this paper, for the fourth-order parabolic equation modeling epitaxial thin film growth, the optimal control under boundary condition is given, and the existence of optimal solution to the equation is proved.

Global solutions in higher dimensions to a fourth-order parabolic equation modeling epitaxial thin-film growth

The initial-value problem for $$u_t=-\Delta^2 u - \mu\Delta u - \lambda \Delta |\nabla u|^2 + f(x)\qquad \qquad (\star)$$ is studied under the conditions \({{\frac{\partial}{\partial\nu}} u={\frac{\partial}{\partial\nu}} \Delta u=0}\) on the boundary of a bounded convex domain \({\Omega \subset {\mathbb{R}}^n}\) with smooth boundary. This problem arises in the modeling of the evolution of a thin surface when exposed to molecular beam epitaxy. Correspondingly the physically most relevant spatial setting is obtained when n = 2, but previous mathematical results appear to concentrate on the case n = 1. In this work, it is proved that when n ≤ 3, μ ≥ 0, λ > 0 and \({f \in L^\infty(\Omega)}\) satisfies \({{\int_\Omega} f \ge 0}\), for each prescribed initial distribution \({u_0 \in L^\infty(\Omega)}\) fulfilling \({{\int_\Omega} u_0 \ge 0}\), there exists at least one global weak solution \({u \in L^2_{loc}([0,\infty); W^{1,2}(\Omega))}\) satisfying \({{\int_\Omega} u(\cdot,t) \ge 0}\) for a.e. t > 0, and moreover, it is shown that this solution can be obtained through a Rothe-type approximation scheme. Furthermore, under an additional smallness condition on μ and \({\|f\|_{L^\infty(\Omega)}}\), it is shown that there exists a bounded set \({S\subset L^1(\Omega)}\) which is absorbing for \({(\star)}\) in the sense that for any such solution, we can pick T > 0 such that \({e^{2\lambda u(\cdot,t)}\in S}\) for all t > T, provided that Ω is a ball and u0 and f are radially symmetric with respect to x = 0. This partially extends similar absorption results known in the spatially one-dimensional case. The techniques applied to derive appropriate compactness properties via a priori estimates include straightforward testing procedures which lead to integral inequalities involving, for instance, the functional \({{\int_\Omega} e^{2\lambda u}dx}\), but also the use of a maximum principle for second-order elliptic equations.

A fourth-order parabolic equation modeling epitaxial thin film growth

We study the continuum model for epitaxial thin film growth from Phys. D 132 (1999) 520–542, which is known to simulate experimentally observed dynamics very well. We show existence, uniqueness and regularity of solutions in an appropriate function space, and we characterize the existence of nontrivial equilibria in terms of the size of the underlying domain. In an investigation of asymptotical behavior, we give a weak assumption under which the ω-limit set of the dynamical system consists only of steady states. In the one-dimensional setting we can characterize the set of steady states and determine its unique asymptotically stable element. The article closes with some illustrative numerical examples.