Oscillatory phenomena associated to semilinear wave equations in one spatial dimension

Abstract:

Let $g$ be a nonincreasing, odd ${C^1}$ function and $l > 0$. We establish that for any solution $u \in C({\mathbf {R}};H_0^1(0,l))$ of the equation ${u_{tt}} - {u_{xx}} + g(u) = 0$ and any ${x_0} \in ]0,l[$, the function $t \mapsto u(t,{x_0})$ satisfies the following alternative: either $u(t,{x_0}) = 0,\forall t \in {\mathbf {R}}$, or $\forall a \in {\mathbf {R}}$, there exist ${t_1}$ and ${t_2}$ in $[a,a + 2l]$ such that $u({t_1},{x_0}) > 0$ and $u({t_2},{x_0}) < 0$. We study the structure of the set of points satisfying the first possibility. We give analogous results for ${u_x}$ and for some other homogeneous boundary conditions.