Propagation reversal on trees in the large diffusion regime (Q6618268)

In this paper, the authors provide speed estimates of waves in the propagation reversal region of the lattice differential equation\N\[\N\dot{u}_i = d(ku_{i+1} - (k+1)u_i + u_{i-1}) + g(u_i;a), \quad i\in \mathbb{Z},\N\]\Nwith \(k>1\). The equation can be seen as the lattice reaction-advection-diffusion equation\N\[\N\dot{u}_i = d(u_{i+1} - 2u_i + u_{i-1}) + d(k-1)(u_{i+1} - u_i) + g(u_i;a), \quad i\in \mathbb{Z}.\N\]\Nbut can also describe dynamics on homogeneous infinite \(k\)-ary tree \(\mathcal{T}_k\) if \(k\in\mathbb{N}\), \(k>1\). A general class of smooth bistabilities \(g\) is considered, results are then made explicit for the classical cubic\N\[\Ng(u;a)=u(1-u)(u-a).\N\]\NPropagation reversal was discussed in [\textit{H. J. Hupkes} et al., SIAM J. Appl. Dyn. Syst. 22, No. 3, 1906--1944 (2023; Zbl 1523.34015)] via the method of sub- and super-solutions. This phenomenon describes the wave speed change for a fixed viability constant \(a\sim 1\) and increasing value of the diffusion \(d>0\). The wave solutions \(u_i(t)=U(i-ct)\) are initially pinned (\(c=0\)), then move down the tree (\(c>0\)), are pinned again (\(c=0\)), and then move up the tree (\(c<0\)).\N\NIn this paper, the authors provide fine estimates for the speed \(c\) and its sign in the thin region of the diffusion parameter \(d\) where the transition from \(c>0\) to \(c<0\) occurs for \(a\sim 1\). Using the spectral method of \textit{P. W. Bates} et al. [SIAM J. Math. Anal. 35, No. 2, 520--546 (2003; Zbl 1050.37041)]], they associate the properties of linearized operators of related partial differential equations to the above lattice equations.\N\NIn the case of the cubic nonlinearity, these estimates are made explicit and are closely tied to the critical curve\N\[\Nd(a,k)=\frac{k+1}{(k-1)^2}\left(a-\frac{1}{2}\right).\N\]