Study of Solutions to a Fourth Order Parabolic Equation∗

Abstract

This paper studies a fourth-order parabolic equation u_t + ε(uⁿu_xxx)_x − δ|u_xx|^mu_xx = 0 with the boundary conditions u_xx = 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.