We consider the nonlinear wave equation with a potential and study the question of global existence for radially symmetric data of non-compact support. For small data that do not decay rapidly enough, a blow-up theorem is established for potentials that are not positive at infinity. For potentials that lead to unstable growth in the linearized problem, a second blow-up theorem is derived for small data of compact support. Finally, we prove the existence of global solutions when both data and the potential are small and rapidly decaying. Our work extends the results of Strauss and Tsutaya to higher space dimensions and allows for potentials that are small in amplitude but of arbitrary sign.
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