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Asymptotic profiles for a traveling front solution of a biological equation. (English) Zbl 1252.35061
The existence of depolarization waves in the human brain is considered. For the modeling of the studied biological process, the equation $\frac{{\partial u}} {{\partial \tau}} = \Delta u + f(u)I_\Omega - \alpha uI_{\mathbb{R}^N \backslash \Omega } ,\quad \tau \in \mathbb{R},\;x \in \mathbb{R}^N$ 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.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K15 Initial value problems for second-order parabolic equations 35K58 Semilinear parabolic equations 35C07 Traveling wave solutions 35B35 Stability in context of PDEs 35K57 Reaction-diffusion equations 92C20 Neural biology
##### Keywords:
spreading depression; Sturm-Liouville theory
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##### References:
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