Existence and stability of equilibrium solutions of a nonlinear heat equation

The main purpose of this paper is to investigate the existence and stability of periodic and non-periodic equilibrium solutions related to the nonlinear heat equation: equation(1)ut=uxx+wu+u3+u5.ut=uxx+wu+u3+u5.The existence of periodic equilibriums with a fixed period L is deduced from the Theory of Jacobian Elliptical Functions and the Implicit Function Theorem. We show that these periodic equilibriums tend to the non-periodic positive equilibrium solution in the real line. Our stability/instability results are obtained trough the spectral study of the linear operator associated to the linearized stability problem as well as the study of a certain scalar quantity.