Let $b\in C_b(R;R)$. Consider the following LINEAR equation on $R^2$: \begin{equation} u-\partial_{xx}^2 u + (b(x+y)-b(x)) \partial_y u=f\in C^\infty_c(R^2). \tag{1} \end{equation} Assume that $u_1$ and $u_2$ are two bounded sub-solutions to (1) and $|u_i(x)|\to 0$ as $|x|\to \infty$.
My question: Is $u=(u_1+u_2)/2$ still a sub-solution to (1)?
This point seems not easy to see from the very definition of viscosity solution. I guess some regularity assumptions on $b$ is need.
