Semilinear wave equation with time dependent potential

摘要

We consider the following semilinear wave equation:squareu + V(t,x)u = +/-uu(lambda-1)u(0) = rhof, partial derivative(t)u(0) = rhogfor (t,X) is an element of R-t X 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)less than or equal toC(1 + t)(-sigma0)(1 + x)(-(2+sigma0)) for a.e. (t,x)is an element ofR(t) x R-x(3)where C, sigma(0) > 0 are real constants, then for any real number lambda that satisfies 1 + root2 < lambda < 3 there exists a real number rho(f,g,lambda)>0 such that the equation has a global solution provided that 0less than or equal top(f,g,lambda). Copyright (C) 2004 John Wiley Sons, Ltd.