Travelling fronts in non-local evolution equations (Q1912916)

The following non-local evolution equation is considered: \[ {\partial m\over \partial t}= - m+ \text{tanh} \{\beta[J* m+ h]\},\tag{1} \] where \(m= m(x, t)\) denotes the magnetization density in \(x\in \mathbb{R}\) at time \(t\in \mathbb{R}_+\); \(\beta> 0\) the inverse temperature of the Ising system; \(J\) a non-negative, even function on \(\mathbb{R}\), \(h\geq 0\) an external magnetic field; and \(J* m\) the convolution between \(J\) and \(m\). The authors prove the existence of travelling fronts and their uniqueness modulo translations in the context of (1) and discuss the linear stability and local stability for the related problem of equation (1).