On the strong oscillatory behavior of all solutions to some second order evolution equations

Abstract

Let $l$ be any positive number. For any non-negative potential $p\in L^\infty (0, l)$, we show that for any solution $u$ of $u_{tt} +u_{xxxx}+ p(x) u = 0$ in $\mathbb R\times (0, l)$ with $u = u_{xx} = 0$ on $\mathbb R\times \{0, l\}$ , and for any form $\zeta \in (H^2(0, l) \cap H^1_0(0, l))'$, the function $t\rightarrow \langle \zeta, u(t)\rangle$ has a zero in each closed interval $I$ of $\mathbb R$ with length $|I|\ge \frac{\pi} {3} l^2$. A similar result of uniform oscillation property on each interval of length at least equal to $2l$ is established for all weak solutions of the equation $u_{tt} - u_{xx}+ a(t) u = 0$ in $\R\times (0, l)$ with $u = 0$ on $\R\times \{0, l\}$ where $a$ is a nonnegative essentially bounded coefficient. These results apply in particular to any finite linear combination of evaluations of the solution $u$ at arbitrary points of $(0, l)$.