Abstract
This paper studies a fourth-order parabolic equation ut + ε(unuxxx)x − δ|uxx|muxx = 0 with the boundary conditions uxx = 0, u = l and the initial condition u(x, 0) = u0(x). The existence of solutions is obtained from the semidiscretization method. When the initial function is close to a constant steady state solution, the uniqueness of the bounded solutions is obtained. Finally, by the iteration technique from its semi-discrete problem, the solution exponentially converges to a constant steady state solution as the time tends to infinity.
Highlights
In recent years, the research of nonlinear fourth-order parabolic equations has become a hot topic
Zheng and Garcke in the paper [11, 12] have studied this equation with a linear and a degenerate mobility respectively
The existence in the distributional sense and the long time decay were studied by Bertozzi and Pugh [5] for the thin film equation with a second-order diffusion term in the one-dimensional space
Summary
The research of nonlinear fourth-order parabolic equations has become a hot topic. The surface tension driven thin film flows can be modeled by using the following fourth order degenerate parabolic equation:. The existence in the distributional sense and the long time decay were studied by Bertozzi and Pugh [5] for the thin film equation with a second-order diffusion term in the one-dimensional space. For the semi-discretization method, the paper [10] has developed a unifying method to prove the existence and uniqueness of weak solutions for a nonuniformly parabolic equation. We can obtain some accurate estimates by applying the semi-discretization method and we will show the effect of the second order nonlinear diffusion term for the fourth order parabolic equation. We will apply the entropy functional method to show that the solutions of (1.1)-(1.3) decay exponentially to the constant steady state l in H1-norm as t → ∞.
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