Asymptotic profiles for a traveling front solution of a biological equation (Q2890983)

The existence of depolarization waves in the human brain is considered. For the modeling of the studied biological process, the equation NEWLINE\[NEWLINE \frac{{\partial u}} {{\partial \tau}} = \Delta u + f(u)I_\Omega - \alpha uI_{\mathbb{R}^N \backslash \Omega } ,\quad \tau \in \mathbb{R},\;x \in \mathbb{R}^N NEWLINE\]NEWLINE is used, where \(f(u) = \lambda u(u - a)(1 - u)\) , \(a \in (0,1/2)\) and \(\lambda > 0\), \(\alpha > 0\) are numerical parameters; \(I_\Omega \) is the characteristic function of the domain \(\Omega \). It is assumed that \(\Omega \) is a straight cylinder of radius r. The influence of \(\Omega \) on the geometry of the propagation of the waves, the existence, the stability and the energy of nontrivial asymptotic profiles of the possible traveling fronts is studied. Applying dynamical systems' techniques and graphic criteria based on Sturm-Liouville, trhee different types of behavior for the solution \(u(t, x)\) are established depending on the thickness of the grey matter. Some biological effects are demonstrated as results of numerical experiments.