Lower bounds for solutions of hyperbolic inequalities in unbounded regions

Abstract:

This paper considers ${C^2}$ solutions $u = u(t,x)$ of the differential inequality $|Lu| \leqq {k_1}(t,x)|u| + {k_2}(t,x)||\nabla u||$. The coefficients of the hyperbolic operator $L$ depend on both $t$ and $x$. Explicit lower bounds are given for the energy of $u$ in a region of $x$-space expanding at least as fast as wave-fronts for $L$. These bounds depend on the asymptotic behavior of ${k_1},{k_2}$, and the coefficients of $L$. They do not require boundary conditions on $u$.