Asymptotic behavior of solutions of hyperbolic inequalities

Abstract:

This paper discusses the asymptotic behavior of ${C^2}$ solutions $u = u(t,{x_1}, \ldots ,{x_v})$ of the inequality (1) $|Lu| \leqq {k_1}(t,x)|u| + {k_2}(t,x)||{\nabla _u}||$, in domains in $(t,x)$-space which grow unbounded in $x$ as $t \to \infty$. The operator $L$ is a second order hyperbolic operator with variable coefficients. The main results establish the maximum rate of decay of nonzero solutions of (1). This rate depends on the asymptotic behavior of ${k_1},{k_2}$, and the time derivatives of the coefficients of $L$.