I have the following PDE:
$$\frac{\partial u}{\partial t} = \frac{1}{x^2} \frac{\partial }{\partial x}\left(x^2 \frac{\partial u}{\partial x}\right) + \frac{1}{x^2} \frac{\partial }{\partial y}\left(\frac{1}{\sin y}\frac{\partial}{\partial y}\left(u\sin y\right)\right)$$
defined on $\{(x,y):0\leq x \leq 1, 0\leq y \leq \pi\}$ and with BCs $u(0,y) = u(x,0) = u(x,\pi) = 0$ and $u(1,y) = \sin(y)$.
Are there any standard existence theorems that apply to this class of pde? I've made a few assumptions (starting with the Navier-Stokes equations) to get here, and if this PDE has a solution then I can construct a solution to the N-S equations for my problem. I can solve it numerically, and the numerical solution looks fine and everything makes physical sense, but I'd sleep better at night if I knew that this pde has a solution. I'm also curious as to how powerful the existence theorems for PDEs are :)
Thanks