I want to know how to deal with the following non-linear parabolic pde $$\begin{cases} W_t(t,x)+W+W_x-W_{xx}-\mathrm{e}^xW_x^{-1}W_{xx}-\mathrm{e}^x=0, \quad (t,x)\in (0,T]\times(0,\infty)\\ W(0,x)=\mathrm{e}^{\beta x}(\beta>1). \end{cases} $$ The pde is strange since $\beta\neq1$. When $\beta=1$, it is easy to see that $f(t)\mathrm{e}^x$ is a solution for some $f(\cdot)$.
I hope to know is there some techiniques in pde theory to deal with such an equation.