Do the class of PDEs that lack initial conditions have a name?

Question

I am trying to think of what this kind of problem is called. I illustrate it with a telegrapher's equation with (hopefully) standard notation.

Find $u:\Omega\times \mathbb{R} \to \mathbb{R}$ such that

$$ \begin{align*} u_{tt}+u_t-\nabla^2 u &= 0& \mathrm{for}\, (x,t)\in \Omega\times \mathbb{R}\\ u&= u_{bc} \sin(\omega t)& \mathrm{for}\ (x,t)\in \partial\Omega \times \mathbb{R} \end{align*} $$

The main point is that I am interested in solving the PDE for all time, or perhaps more precisely I am interested in the limit cycle behavior of the PDE, (after any transients associated to the initial condition have died down).

Mathematically, I guess a valid problem would be the following. Fine $u: \Omega \times \mathbb{R}\to \mathbb{R}$ such that $$ \begin{align*} u_{tt} +u_t - \nabla^2 u &= 0 &\mathrm{for} (x,t)\in \Omega \times \mathbb{R}\\ u &= u_{bc} \sin(\omega t)& \mathrm{for} (x,t)\in \partial\Omega \times \mathbb{R}\\ u(x,t+2\pi/\omega) &= u(x,t)& \mathrm{for} (x,t)\in \Omega \times \mathbb{R} \end{align*} $$

Does this problem have a name?

Answer 1

Such problems (sometimes called lateral Cauchy problems) are in general not well-posed (meaning they either lack a solution, or there are infinitely many of them, or the solution is unstable under perturbations of the boundary conditions).

For parabolic (or dissipative) equations, it makes sense to study the stationary limit (simply omit the term $u_t$ in the PDE together with the initial condition), but I don't think that's the case for wave-like hyperbolic problems, where information is transported without loss (meaning the influence of the initial conditions never disappears, unless you have absorbing boundary conditions, in which case the limit solution is the trivial solution $u\equiv 0$).

Your second problem doesn't really make sense mathematically, because the solution $u$ is either completely determined by the periodicity and boundary conditions (if they are compatible), or no solution exists (if they are not).