I have the partial differential equation $$\partial_t S(x,v,t) = v \partial_x S - (v + a ) \partial_v S + \partial_v^2 S$$ subject to the following boundary conditions: $$S(x,v,0)=1,$$ $$S(0+,|v|,t)=S(0+,-|v|,t)$$ $$S(H,-|v|,t)=0.$$ These conditions represent reflection and absorption at the boundaries $x=0$ and $x=H$ respectively. The valid spatial domain is $0 \leq x \leq H$ while the valid velocity domain is $-\infty < v < \infty$.
At long times I expect the solution to have an asymptotic form like $$ S(x,v,t) \sim \exp[-\lambda(x,v) t] $$ for physical reasons, at least for small $a$. I would like to attempt to derive this form approximately.
One approach I have tried is simply plugging the asymptotic form in the PDE with intent to find a $\lambda(x,v)$ which works. I obtain: $$ 0 \approx \lambda - v \lambda_x t + (v+a) \lambda_v t + D \lambda_v^2 t^2 - D \lambda_{vv} t.$$ Since by assumption $t \rightarrow \infty$, this seems to imply $$ \lambda_v^2 \approx 0,$$ or $\lambda = f(x),$ which violates the original assumption that $\lambda = \lambda(x,v)$.
Is there any means by which I might solve this equation perturbatively to obtain the large $t$ solution?