Let $\Omega := [0,\infty)\times [0,\infty)$ be a domain and $u, w \in C^\infty (\Omega)$. Consider the system
\begin{align} \frac{\partial u}{\partial t} &= D_u \frac{\partial^2 u}{\partial x ^2} - \lambda u + \beta w \\ \frac{\partial w}{\partial t} &= D_w \frac{\partial^2 w}{\partial x ^2} + \alpha u - v \frac{\partial w}{\partial x} - \beta w. \end{align}
with $D_u, D_w, \lambda, \beta, \alpha, v \in \mathbb{R}$ constants and conditions
\begin{align} \lim_{x\to \infty} u (x,t) &= 0, \qquad \lim_{x\to \infty} w (x,t) = 0 \\ u(0, t) &= f_u(t), \ \ \ w(0, t) = g_w(t) \\ u(x, 0) &= 0, \qquad w(x, 0) = 0 \end{align}
and
\begin{align} \int_{0}^\infty u (x,t) + w (x,t) dx &= 1 \end{align}
What I have done so far
I have numerically solved the problem using the method of lines and found the steady states analytically solving. The solution seems to be a negative exponential.
Now I want to find an explicit solution. I tried to solve the first equation for $w$ and then replace it on the second. But the substitution yields a $4^{th}$ order PDE that I do not know how to solve.
Does anyone know how to find the explicit solution for this system of PDE?
Many thanks.