Stability Results for Special Solutions of Scalar-Field Equations with Variable Coefficients

Abstract

We study the long-time behavior of general semilinear scalar-field equations on the real line with variable coefficients in the linear terms. In the first part of the dissertation, we take the coefficients to be uniformly small, but slowly decaying, perturbations of a constant-coefficient operator. We are motivated by the question of how these perturbations of the equation may change the stability properties of kink solutions (one-dimensional topological solitons). We prove existence of a stationary kink solution in our setting, and perform a detailed spectral analysis of the corresponding linearized operator, based on perturbing the linearized operator around the constant-coefficient kink. We derive a formula that allows us to check whether a discrete eigenvalue emerges from the essential spectrum under this perturbation. Known examples suggest that this extra eigenvalue may have an important influence on the long-time dynamics in a neighborhood of the kink. We also establish orbital stability of solitary-wave solutions in the variable-coefficient regime, despite the possible presence of negative eigenvalues in the linearization. In the second part, we address special solutions that are constant or perturbations of a constant state. For these solutions, we are able to prove asymptotic stability under an oddness assumption on the initial data, using a Virial argument based on defining suitable Lyapunov functionals.