Consider the following uniformly parabolic lattice differential equation

$ \begin{array}{ccc} \dot{u}_{n,m} & = & \alpha_{n,m}(u_{n+1,m} - u_{n,m}) + \beta_{n,m}(u_{n-1,m}-u_{n,m}) \\ & & \quad + \gamma_{n,m}(u_{n,m+1}-u_{n,m}) + \delta_{n,m}(u_{n,m-1}-u_{n,m}). \end{array} \quad (1) $

Here the coefficients $\alpha$, $\beta$, $\gamma$ and $\delta$ are bounded uniformly in $n$ and $m$ both above and below by positive constants, thus (1) enjoys a maximum principle.

Equation (1) is related to diffusion on a bidirectional graph as well as a semi-discretization of a parabolic PDE. In my own work it arose in the study of stability of planar fronts in lattice differential equations. I don't understand the behavior of solutions to (1) very well. I am curious to know whether or not their qualitative asymptotic behavior is well-known and/or somewhat trivial or if there is some deepish mathematics lurking here. I suspect that there is deep mathematics in the quantitative asymptotics because heat kernels for graphs have been a topic of study for a long time and to the best of my knowledge there are no results on decay estimates for heat kernels in the non-self-adjoint case.

In the case that the coefficients $\alpha$, $\beta$, $\gamma$ and $\delta$ are independent of space ($n$ and $m$), solutions to (1) can be obtained via Fourier transform. One can show for example that $u \to 0$ (e.g. in $\ell^2$ for initial data in $\ell^2$).

In the case that the right hand side is self-adjoint ($\alpha_{n,m} = \beta_{n,m}$ and $\gamma_{n,m} = \delta_{n,m}$) an energy estimate together with a Nash inequality gives algebraic decay in e.g. $\ell^2$.

In the case that the right hand side is neither self-adjoint nor constant-coefficient, $\ell^1$ norm need not be preserved and $\ell^2$ norm need not be non-increasing. Arguments involving the Nash inequality and/or concentration compactness seem not to have traction.

Question: What can be said about the asymptotic behavior of solutions to (1). Do solutions to (1) with initial data in $\ell^\infty$ necessarily approach a constant? Do solutions with initial data in $\ell^1$ necessarily approach zero? Can anything be said about the rate of decay?