When solving a $2$D Heat Equation, suppose I separate the solution into time and space, i.e., $f_1(t,\ T(t),\ T_t(t),\ ...) = f_2(x,\ y,\ Z(x,\ y),\ Z_x(x,\ y),\ Z_y(x,\ y),\ ...) = \lambda$, and then separate space into its dimensions, i.e., $f_3(x,\ X(x),\ X_x(x),\ ...) = f_4(\lambda,\ y,\ Y(y),\ Y_y(y),\ ...) = r$. The problem of this sort I worked seems to have two nontrivial paths, one in the case that $\lambda = r \neq 0$ and another in the case that $\lambda \neq r, \lambda \neq 0, r \neq 0$. Usually in other problems I have encountered only one nontrivial path.
After I have solved for $u$ in each of the paths, the former being a Fourier Series solution and the latter being a double Fourier Series solution, am I supposed to combine the answers into a single particular solution to the problem somehow, or are these separate particular solutions which I would choose between based on some physical measurement to determine whether or not $\lambda = r$?
Need boundary conditions like u = 0 or du/dn = 0 on boundary. Then the eigensolutions of del^2 u = lambda u form a complete set, so one can decompose the initial conditions U(0,x,y) into a eigenfunctions. If bdry conditions are u = 0, then all eigenvalues are negative; if boundary condition is du/dn = 0 then there is a zero eigenvalue as well. As domain goes to infinity, the eigenvalues become closer and closer packed until the fourier series "converges" to the fourier integral ...