The Cauchy problem for a semilinear wave equation. I

L. V. Kapitanski

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 19, Tome 163 (1987), pp. 76-104

Résumé

In this paper the Cauehy problem for the semilinear wave equation on the torus $\mathbb{T}^n$, $n\geq3$: \begin{equation} \ddot{u}-\triangle u+f(u)=h,\qquad u|_{t=0}=\varphi,\qquad \dot{u}|_{t=0}=\varphi. \tag{1} \end{equation} is studied. It is supposed that the function $f:\mathbb{R}^1\longrightarrow\mathbb{R}^1$ continuous and there exist non negative constants $A_1$, $A_2$, $A_3$ and $a\geq1$ such that $$ A_1+A_2s^2+\int^s_0f(\theta)d\theta\geq0,\qquad\forall s\in\mathbb{R}^1, $$ $$ |f(s_1)-f(s_2)|\leq A_3(1+|s_1|^{a-1}+|s_2|^{a-1})|s_1-s_2|,\qquad\forall s_1,s_2\in\mathbb{R}^1. $$ The main result of the present paper is the theorem: if $1\leq a<(n+2)/(n-2)$, then for arbitrary data $\varphi\in W_2^1(\mathbb{T}^n)$, $\psi\in L_2(\mathbb{T}^n)$, $h\in L_{1,\operatorname{loc}}(\mathbb{R}^1\to L_2(\mathbb{T}^n))$ the problem (I) has the global in time solution $u$ with the following properties: $u\in C_{\operatorname{loc}}(\mathbb{R}^1\to W_2^1(\mathbb{T}^n))$, $\dot{u}\in C_{\operatorname{loc}}(\mathbb{R}^1\to L_2(\mathbb{T}^n))$ and $u\in L_{q,\operatorname{loc}}(\mathbb{R}^1\to L_p(\mathbb{T}^n))$ for all $p$, $q$, satisfying $$ \frac{n-3}{2n}<\frac1p<\frac{n-2}{2n},\qquad\frac1q=\frac{n-2}{2}-\frac np, $$ and such a solution is unique.