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

Abstract

Keywords:Parabolic equation of fourth order Existence of solutions Semi-discretization Asymptotic behavior Large-time behavior Entropy method This paper is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order: u t + ∇ · (|∇ u| p−2 ∇ u) = f (u) in Ω ⊂ R N with boundary condition u = 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 → ∞.

Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity

“…Thirdly, it is easily seen that Ψ(u) is weakly lower semi-continuous on B 1 by the weakly lower semi-continuous of a semi-modular ρ. So, following the lines of Lemma 2.1 in [5], we conclude that there exists a u * ∈ B 1 such that…”

“…For example, this model may describe the surface tension driven evolution of the height u(x, t) of a thin liquid film on a solid surface in lubrication approximation [1,2]. Especially p ≡ 2, Problem (1.1) becomes the classical Cahn-Hilliard problem, and there have been many results about the existence, uniqueness, and some other properties of the solutions, the readers may refer to the bibliography given in [3][4][5].…”

Weak solutions for p-Laplacian equation

“…It is easy to obtain the following estimate from the above results kjrw .n/ j p 2 rw .n/ k L p 0 .Q T / Ä C: 11]). The second kind of approximate solution is defined as follows: Proof.…”

Existence of Solutions to a Nonlinear Parabolic Equation of Fourth-Order in Variable Exponent Spaces

“…The paper [3] has studied the existence of the Cahn-Hilliard equation and the reader may refer to [4] to obtain its physical background. For the constant exponent case of (1), the paper [5] has given the existence and uniqueness of weak solutions. For the problems in variable exponent spaces, the papers [6][7][8] have studied the existence of some fourth-order parabolic equations with a variable exponent, and [9] has given the Fujita type conditions for fast diffusion equation.…”