Aside from solutions that one obtains depending on initial and or boundary conditions (using fundamental solution or eigenfunctions depending on domain) is there a "generic solution" of the heat equation?
For example for the wave equation $U_{tt} =c^2 U_{xx}$ we have $U=f(x-ct)+g(x+ct)$ for any smooth functions $f,g$.
Are there papers discussing the solution of $U_t+a^2U_{tt}=c^2 U_{xx}$ as $a \to 0$ ?