Usually second-order linear PDE's are classified as elliptic, parabolic, or hyperbolic (or ultrahyperbolic) depending on the eigenvalues of the coefficient matrix. The three cases correspond to the three most famous second-order PDE's:
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Elliptic – Laplace's equation $\nabla^2 u = 0$.
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Parabolic – the heat equation $u_t = \nabla^2 u$.
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Hyperbolic – the wave equation $u_{tt} = \nabla^2 u$.
In the general study of such equations, it is common to refer to one of the coordinates as time in the parabolic and hyperbolic case, but in the elliptic case all of the coordinates are usually thought of as spatial (at least in the treatments I have seen).
My question is — is there a good theoretical reason for this? Or is this just a tradition, based on the fact that the main application of Laplace's equation in physics are spatial? Is the equation
$$
u_{tt} = -\nabla^2 u
$$
useful for modeling any physical situations?
Best Answer
The reason for this is that in physics we usually have to deal not only with the equation, but also with initial-boundary problem. And some of the initial and boundary conditions lead to a well posed problem, whereas others do not. When we talk about the "time" variable, it usually means for mathematicians that natural initial conditions can be set. However, if you consider your example for the "Laplace" equation with a "time" variable, your initial condition+equation will not constitute a well-posed problem.
For example for the Laplace equation the initial value problem is not well-posed, since small perturbation in the "initial" conditions leads to very large difference for finite t. You should google Hadamard's example of an ill posed problem. In general, Laplace equation does not go good with "initial-value" problems.
About comment by @mkl314:
This question is right to the money and very deep mathematically. Note that distinction between "initial" and "boundary" comes from the physical interpretation of our equations. Nothing prohibits to set the initial conditions for an arbitrary manifold in the space. However, for some initial conditions the problems will be well-posed, for others --- ill posed. It is a very complicated question in general: How to set the initial conditions for a given system of PDE of order $n$ to guarantee the existence, uniqueness, and continuous dependence on the initial conditions. The well known examples of heat, wave, and Laplace equations how we treat them in undergraduate courses come from physical intuition, not from carefully chosen mathematical conditions.