# Wave equation: ensuring a minimal value by setting a constraint on initial conditions

Assume the wave equation:

$$u_{tt} = u_{xx}$$ for $$x \in [0,1]$$

with either zero-Neumann or -Dirichlet boundary conditions on $$x = 1$$ and $$x = 0$$.

Next, I would like to have the space of initial conditions $$u(x,0)$$ and $$u_t(x,0)$$ such that:

$$u_x(x,t) > -1$$ $$\forall x \in [0,1]$$ and $$\forall t \in [0,\infty]$$. (1)

one may presume $$u(x,0) < \infty$$ is Lipschitz-continuous and $$u_t(x,0) < \infty$$ is not.

I know that $$u_x(x,t)$$ do have a global maximum and minimum as the resulting motion is periodic in $$t$$. Also, the PDE and boundary conditions are homogeneous and without any parametric excitation, so I am confident that the global minimum can be defined by the initial conditions. However, I am not sure how to translate (1) into a constraint on initial conditions.

Any help would be appreciated. Thank you!