Local uniqueness of multi-bump solutions for singularly perturbed Kirchhoff problems

2022 ◽  
Vol 124 ◽  
pp. 107684
Author(s):  
Mingzhu Yu ◽  
Hongxia Shi
Keyword(s):  
Singularly Perturbed ◽  
Local Uniqueness

Local uniqueness of the solutions for a singularly perturbed nonlinear nonautonomous transmission problem

2020 ◽  
Vol 191 ◽  
pp. 111645
Author(s):  
Matteo Dalla Riva ◽  
Riccardo Molinarolo ◽  
Paolo Musolino
Keyword(s):  
Transmission Problem ◽  
Singularly Perturbed ◽  
Local Uniqueness

scholarly journals Boundary layer solutions to singularly perturbed quasilinear systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Valentin Butuzov ◽  
Nikolay Nefedov ◽  
Oleh Omel'chenko ◽  
Lutz Recke
Keyword(s):  
Boundary Layer ◽  
Convergence Rates ◽  
Singularly Perturbed ◽  
Uniqueness Condition ◽  
Scalar Case ◽  
Monotonicity Condition ◽  
Asymptotic Convergence ◽  
Local Uniqueness ◽  
Quasilinear Systems ◽  
Ode Systems

<p style='text-indent:20px;'>We consider weak boundary layer solutions to the singularly perturbed ODE systems of the type <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon^2\left(A(x, u(x), \varepsilon)u'(x)\right)' = f(x, u(x), \varepsilon) $\end{document}</tex-math></inline-formula>. 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.</p>

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