## Infinite dimensional PDE

Is there any established theory concerning infinite dimensional PDE?

 Do you mean that the function has infinitely many variables, or that it is an infinite dimensional function of a finite number of variables?
 Infinitely many variables. For example a quantum mechanical real Klein-Gordon field, if I have understood correctly, can be pretty much described by the infinite dimensional non-homogenous heat equation (the Shrodinger's equation, with certain constants and with the harmonic potential). Something like this $$i\partial_t \Psi(t,\phi) = \sum_{k\in\mathbb{R}^3} \Big(-\alpha \partial^2_{k} + \beta |k|^2\Big)\Psi(t, \phi)$$ where $$\Psi:\mathbb{R}\times\mathbb{R}^{\mathbb{R}^3}\to\mathbb{C}.$$ It can be solved by a separation attempt $$\Psi(t,\phi) = \prod_{k\in\mathbb{R}^3} \Phi_k(t) \Psi_k (\phi(k)),$$ where $$\Phi_k,\;\Psi_k:\mathbb{R}\to\mathbb{C}$$ This is total honest pseudo mathematics, motivated by physics, don't complain about it! In fact his is a very vague example with uncountable set of variables. There could be more rigor examples with only countably many variables.

## Infinite dimensional PDE

It could be these are supposed to be called functional differential equations, but I'm not sure. Some quick google hits were slightly confusing.