Stability and Instability of Traveling Wave Solutions to Nonlinear Wave Equations

In this paper, we study the stability and instability of plane wave solutions to semilinear systems of wave equations satisfying the null condition. We identify a condition that allows us to prove the global nonlinear asymptotic stability of the plane wave. The proof of global stability requires us to analyze the geometry of the interaction between the background plane wave and the perturbation. When this condition is not met, we are able to prove linear instability assuming an additional genericity condition. The linear instability is shown using a geometric optics ansatz.

Non-well-ordered lower and upper solutions for semilinear systems of PDEs

We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.