Persistence of travelling wave solutions of a fourth order diffusion system (Q1763662)

The paper considers an equation \[ u_t=-\delta u_{xxxx} + u_{xx} -\alpha uu_x -\beta u(u-1) (u-\gamma), \] which is a generalization of the so-called Burgers-Huxley equation, which corresponds to \(\delta = 0\). The paper offers a rigorous proof of the fact that, under certain constraints imposed on initial conditions, the solution to the fourth-order equation with periodic boundary conditions (in a finite domain) converges to the constant solution, \(u=1\), uniformly in \(x\). Then, using the geometric singular perturbation theory, it is shown that the traveling shock-wave solution, which is known in the equation with \(\delta =0\) in an exact analytical form, is stable in the generalized equation with small \(\delta\).