Oscillatory behavior of solutions of nonlinear wave equations

✍ Scribed by Hiroshi Uesaka

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 367 KB
• Volume: 30
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(97)00233-2

✦ Synopsis

We shall show that if u(z,t) is a solution of some nonlinear wave equations with the homogeneous Dirichlet boundary condition, it oscillates as time t goes on. We shall state two theorems. The first theorem is : There are al. least two points (~1, tl), (Q. tz) E c1 x R such that u(z,, tl)u(z2, tz) < 0. This holds for some nonlinear wave equation and for n spatial dimension. The second theorem is: Let z be fizzed in R c R. Zf ~(2, t) dose not identico.lly vanish for any t E R, then the sign of u(z, t) abays changes in the time interual with suituble length. This will be proved for some semilinear wave equat,ion and for 1 spatial dimension.