Existence and blow up of small-amplitude nonlinear waves with a   sign-changing potential
  We study the nonlinear wave equation with a sign-changing potential in any space dimension. If the potential is small and rapidly decaying, then the existence of small-amplitude solutions is driven by the nonlinear term. If the potential induces growth in the linearized problem, however, solutions that start out small may blow-up in finite time. 
Paschalis Karageorgis (Northwestern University)