Semilinear wave equation with time dependent potential

We consider the following semilinear wave equation:for ( t , x ) ∈ ℝ t × ℝ   x 3 . We prove that if the potential V ( t , x ) is a measurable function that satisfies the following decay assumption: ∣ V ( t , x )∣⩽ C (1+ t )   −σ   0(1+∣ x ∣)   −(2+σ   0 )for a.e. ( t , x ) ∈ ℝ t × ℝ   x 3where C , σ 0 >0 are real constants, then for any real number λ that satisfies ${1+\sqrt{2}< \lambda <3}$ there exists a real number ρ( f , g ,λ)>0 such that the equation has a global solution provided that 0<ρ⩽ρ( f , g ,λ). Copyright © 2004 John Wiley & Sons, Ltd.