Wave train solutions for general reaction-diffusion systems (Q1355823)

For the system \[ p_t= D\nabla^2p+ F_1(p,q),\quad q_t= F_2(p,q),\tag{1} \] where \(p(x,t)\in \mathbb{R}^n\), \(q(x,t)\in\mathbb{R}^m\) (\(n\), \(m\) positive integers); \(\nabla^2\) is the Laplacian operator with respect to the spatial variable \(x\in\mathbb{R}^k\) (\(k\) a positive integer); \(D\) is a diagonal \(n\times n\) matrix with diagonal elements \(d_i>0\); the functions \(F_1: \mathbb{R}^{n+m}\to\mathbb{R}\), \(F_2: \mathbb{R}^{n+m}\to\mathbb{R}^m\) are smooth enough, the existence of wave train solutions is proved. A wave train solution for the system (1) is a periodic solution in the variable \(ct-\langle\vec\sigma, x\rangle\), where \(c>0\) is the wave velocity, \(\sigma>0\) is the wave number, \(\vec\sigma= (\sigma,\sigma,\dots,\sigma)\in \mathbb{R}^k\), and \(\langle\cdot\rangle\) is the usual inner product in \(\mathbb{R}^k\). System (1) can be seen as a reaction diffusion system in which some diffusivities are equal to zero.