The cauchy problem for a semilinear wave equation. I

Abstract

The Cauchy problem for a semilinear wave equation on the torus Tⁿ, n≥3:

$$\ddot u - \Delta u + f(u) = h, u\left| {_{t = 0} = \phi ,} \right. \dot u\left| {_{t = 0} = \psi .} \right.$$

((1))

is investigated. It is assumed that the function f: ℝ¹→ℝ¹ is continuous and there exist nonnegative constants A₁, A₂, A₃, and a≥1 such that

$$\begin{gathered} A_1 + A_2 s^2 + \int_0^s {f(\theta )d\theta } \geqslant 0, \forall s \in R^1 , \hfill \\ \left| {f(s_1 ) - f(s_2 )} \right| \leqslant A_3 (1 + \left| {s_1 } \right|^{a - 1} + \left| {s_2 } \right|^{a - 1} )\left| {(s_1 - s_2 )} \right|, \forall s_1 ,s_2 \in R^1 . \hfill \\ \end{gathered} $$

. It is proved that if the parameter a lies in the interval 1 ≤ a < (n+2)/ (n−2), then for any ϕε W₂ ¹{T}ⁿ), ψ ∃ L₂(Tⁿ), h ∃ L_{1, loc}(ℝ¹→L₂(Tⁿ)) the problem (1) has a unique solution u, global with respect to time, such that u ∃ C_loc(ℝ¹→ W₂ ¹(Tⁿ)), ů ∃ C_loc(ℝ¹ → L₂(Tⁿ)) and u ∃ L_{q, loc}(ℝ¹ → L_p(T_n)) for all p, q satisfying the relations

$$\frac{{n - 3}}{{2n}}< \frac{1}{p}< \frac{{n - 2}}{{2n}}, \frac{1}{q} = \frac{{n - 2}}{2} - \frac{n}{p}.$$