The semi-discrete diffusion convection equation with decay (Q6540928)

The authors study the lattice (semi-discrete) reaction-advection-diffusion equation \N\[\N\begin{cases} u_t(t, n)-\alpha \Delta_d u(t, n)+c \nabla_d u(t, n)+\lambda u(t, n)=F(t, u(t, n)),\quad t\geq 0,\quad n\in\mathbb{Z},\\\Nu(0,n)=\varphi(n),\quad n\in\mathbb{Z}, \end{cases}\N\]\Nwhere \(\Delta_d u(t,n) = u(t,n+1)-2u(t,n)+u(t,n)\) is the (spatial) lattice discrete Laplacian operator, \(\nabla_d u(t,n)=u(t,n)-u(t,n-1)\) is the backward (nabla) difference in space. The diffusion parameter \(\alpha>0\) is positive while the advection and decay parameters \(c,\lambda\in\mathbb{R}\) are arbitrary.\N\NThe authors provide three results. First, they find explicit solution of this problem with \(F\equiv 0\)\N\[\Nu(t, n)=\sum_{m \in \mathbb{Z}}\left(1+\frac{c}{\alpha}\right)^{(n-m) / 2} e^{-(2 \alpha+c+\lambda) t} I_{n-m}(2 t \sqrt{\alpha(\alpha+c)}) \varphi(m), \N\]\Nwhere \(I_k\) is the modified Bessel function of the first kind.\N\NNext, they provide a set of parameters \((\alpha,c,\lambda)\) in which the solutions are stable. Finally, they show that if \(\alpha+c>0\) and \(\lambda\geq 0\) then the fundamental solution constitutes a uniformly continuous semigroup in \(\ell^p(\mathbb{Z})\).