Oscillatory behavior of solutions of nonlinear wave equations Original Research Article

We shall show that if u(x,t) is a solution of some nonlinear wave equations with the ho- mogeneous Dirichlet boundary condition, it oscillates as time t goes on. We shall state two theorems. The first theorem is : There are ai Least two points (x1,t1), (x2,t2) ε Ω × R such that u (x1, t1)u (x2, t2) < 0. This holds for some nonlinear wave equation and for n spatial dimension. The second theorem is: Let x be fixed in Ω ⊂ R. If u(x,t) dose not identically vanish for any t ε R, then the sign of u(x,t) always changes in the time interval with suitable length. This will be proved for some semilinear wave equation and for 1 spatial dimension.