I have the following parabolic partial differential equation:
\begin{equation} \frac{\partial^2 \phi}{\partial x^2} - \alpha \sin{x} \frac{\partial \phi}{\partial t} + \beta(\cos{x} - \gamma) \phi = 0 \end{equation}
where $\phi(x,t)$ is a scalar function defined in $ \Omega \equiv [0,L_x]\times [0,L_t]$, $\alpha$,$\beta$ and $\gamma$ and complex numbers. The idea is to apply as boundary conditions: \begin{equation} \phi(x,0) = \phi(x,L_t) = 0 \quad \text{and} \quad \frac{\partial \phi}{\partial x}(0,t) = \frac{\partial \phi}{\partial x}(0,t) = 0. \end{equation} Actually, I'm not fully convinced whether there exists a solution consistent with these boundaries, so do you have any idea how to check if they are ok? Is there a systematic way to prove such things, and if so, how would you do that? I also tried to solve it analytically and numerically, but I had some issues in determining the correct boundary conditions.
Thanks in advance!