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

### Alain Haraux

Université Pierre et Marie Curie, Paris, France

## Abstract

Let $l$ be any positive number. For any non-negative potential $p∈L_{∞}(0,l)$, we show that for any solution $u$ of $u_{tt}+u_{xxxx}+p(x)u=0$ in $R×(0,l)$ with $u=u_{xx}=0$ on $R×{0,l}$ , and for any form $ζ∈(H_{2}(0,l)∩H_{0}(0,l))_{′}$, the function $t→⟨ζ,u(t)⟩$ has a zero in each closed interval $I$ of $R$ with length $∣I∣≥3π l_{2}$. 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 $R×(0,l)$ with $u=0$ on $R×{0,l}$ 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 $(0,l)$.

## Cite this article

Alain Haraux, On the strong oscillatory behavior of all solutions to some second order evolution equations. Port. Math. 72 (2015), no. 2/3, pp. 193–206

DOI 10.4171/PM/1964