# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021226
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## Boundary layer solutions to singularly perturbed quasilinear systems

 1 Department of Mathematics, Faculty of Physics, Moscow State University, Vorob'jovy Gory, 19899 Moscow, Russia 2 Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam, Germany 3 Insitute of Mathematics, Humboldt University of Berlin, Rudower Chaussee 25, 12489 Berlin, Germany

* Corresponding author

Received  November 2020 Revised  August 2021 Early access September 2021

We consider weak boundary layer solutions to the singularly perturbed ODE systems of the type $\varepsilon^2\left(A(x, u(x), \varepsilon)u'(x)\right)' = f(x, u(x), \varepsilon)$. The new features are that we do not consider one scalar equation, but systems, that the systems are allowed to be quasilinear, and that the systems are spatially non-smooth. Although the results about existence, asymptotic behavior, local uniqueness and stability of boundary layer solutions are similar to those known for semilinear, scalar and smooth problems, there are at least three essential differences. First, the asymptotic convergence rates valid for smooth problems are not true anymore, in general, in the non-smooth case. Second, a specific local uniqueness condition from the scalar case is not sufficient anymore in the vectorial case. And third, the monotonicity condition, which is sufficient for stability of boundary layers in the scalar case, must be adjusted to the vectorial case.

Citation: Valentin Butuzov, Nikolay Nefedov, Oleh Omel'chenko, Lutz Recke. Boundary layer solutions to singularly perturbed quasilinear systems. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021226
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The homoclinic solution to problem (7.2)
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