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 $\zeta \in (H^2(0,l)\cap H_0^1(0,l){)}^{\prime }$, the function $t\to ⟨\zeta ,u(t)⟩$ 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 $\mathbb{R}\times (0,l)$ with $u=0$ on $\mathbb{R}\times \{0,l\}$ 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 $(0,l)$.