ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35 Akisato Kubo, Fujita Health University, Japan Global existence of a classical solution to □u − μu + au^{m} = 0 We consider the following boundary value problem of the non-linear wave equation (B. V.)

\{ \begin{matrix}u_{tt}(t, x)Δu(t, x) − μu(t, x) + au^{m}(t, x) = 0, (t, x) ∈ [0, ∞) × Ω (1.1) \\ u(t, x)|_{∂Ω} = 0  (1.2) \end{matrix}

where a is a real number, m = 2,3, .. . and Ω ⊂ R^{n} is a bounded domain with a smooth boundary ∂Ω. We will show that for a constant μ > 0, one can obtain a time global solution for (1.1) and (1.2) with exponential decay property under some additional condition on μ and m.