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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!

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