Polynomial bounds on the Sobolev norms of the solutions of the nonlinear wave equation with time dependent potential

Archiv der Mathematik - We consider the Cauchy problem for the nonlinear wave equation $$u_{tt} - \Delta _x u +q(t, x) u + u^3 = 0$$ with smooth potential $$q(t, x) \ge 0$$ having compact support...