= Solution
For <energy stability for variable-coefficient reaction diffusion>, let $h_x=1/(M+1)$, let $D_h$ be the <Dirichlet discrete Laplacian>, and set $V_h=\operatorname{diag}(a_m)$. Use the mesh-weighted <inner product> $(v,w)_h=h_x\sum_{m=1}^Mv_mw_m$ with the corresponding discrete <L2 norm>. For zero boundary values, <summation by parts> gives
$$
(v,D_hv)_h=-\frac1{h_x}\sum_{m=0}^M(v_{m+1}-v_m)^2\leq0.
$$
For a solution, or a difference of two solutions, the <energy method> gives
$$
\frac12\frac d{dt}\|v\|_h^2=(v,D_hv)_h+(v,V_hv)_h
\leq a_+\|v\|_h^2.
$$
The <Gronwall inequality> therefore proves
$$
\boxed{\|v(t)\|_h\leq e^{a_+t}\|v(0)\|_h.}
$$
The constant is independent of $M$, so this is <stability of a numerical method> on every fixed finite time interval. The bound allows physical growth if $a_+>0$; <stability> here does not mean uniform boundedness as $t\to\infty$.
There is also a uniform maximum-<norm> bound. At a positive spatial maximum the second difference is nonpositive. Applying this observation to $e^{-a_+t}v$ and to its negative gives $\|v(t)\|_\infty\leq e^{a_+t}\|v(0)\|_\infty$. With a perturbing source $r(t)$, the <Duhamel principle> gives the corresponding initial-data bound plus $\int_0^te^{a_+(t-s)}\|r(s)\|\,ds$ in either <norm>. Thus the estimate controls accumulated residuals as well as initial perturbations.
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