Existence of Global Solution of a Nonlinear Wave Equation with Short-range Potential

The existence of a global solution for the Cauchy problem for the nonlinear wave equation (1) (∂tt −∆)u = F (u′, u′′) , t ≥ 0, x ∈ R , is established by S. Klainerman [6] under a suitable algebraic condition called the null condition on the quadratic nonlinearity in F (u′, u′′). The global or “almost global” solution of the corresponding mixed problem has been studied by Y. Shibata and Y. Tsutsumi [10], P. Datti [1] and P. Godin [2]. The main goal of this work is to study the nonlinear wave equation involving a linear perturbation of short-range type, i.e. we shall consider the Cauchy problem for the perturbed wave equation (2) (∂tt −∆)u+ q(x)x = F (x, u′, u′′) , where q(x) is a smooth potential, which is nonnegative and decays sufficiently rapidly at infinity. More precisely, we assume that (H1) q(x) ≥ 0 , ∂rq(x) ≤ 0 , |q(x)| ≤ C(1 + |x|2)−2 , where ∂r is the radial derivative ∂r = ∑3 j=1(xj/r)∂j , r = |x|. Even in the simple case q ≡ 0, the results of F. John [5] show that the solution blows up for some choices of the nonlinearity F . That is why we assume that (3) F = Q(x, u′, u′′) + C(u′, u′′) , where C(v, w) is a smooth function such that C(v, w) = O((|v| + |w|)) near (c, w) = (0, 0), while Q has the form Q(x, u′, u′′) = ∑