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Perturbation of second order nonlinear equation by delta-like potential
T. R. Gadyl'shina,
F. Kh. Mukminovb a Ufa State Aviation Technical University,
Karl Marx str. 12,
450008, Ufa, Russia
b Institute of Mathematics,
Ufa Federal Research Center, RAS,
Chernyshevsky str. 112,
450008, Ufa, Russia
Abstract:
We consider boundary value problems for
one-dimensional second order quasilinear equation on
bounded and unbounded intervals
$I$ of the real axis. The equation perturbed by the delta-shaped potential
$\varepsilon^{-1}Q\left(\varepsilon^{-1}x\right)$, where
$Q(\xi)$
is a compactly supported function,
$0<\varepsilon\ll1$. The mean value of
$\left<Q\right>$ can be
negative, but it is assumed to be bounded from below
$\left<Q\right>\ge-m_0$. The number
$m_0$ is defined in terms of
coefficients of the equation. We study the convergence rate of
the solution of the perturbed problem
$ u^\varepsilon $ to
the solution of the limit problem
$ u_0 $ as the
parameter
$ \varepsilon $ tends to zero. In
the case of a bounded interval
$I$, the estimate of the form
$|u^\varepsilon(x)-u_0(x)|<C\varepsilon$ is established.
As the interval
$I$ is unbounded, we prove a weaker estimate
$|u^\varepsilon(x)-u_0(x) / <C\varepsilon^{1/2}$.
The estimates are proved by using original
cut-off functions as trial functions. For simplicity, the proof of the existence of solutions to
perturbed and limiting problems are made by the method of contracting mappings. The disadvantage of this approach, as it is known, is the smallness of the nonlinearities in the equation.
We consider the cases of the Dirichlet, Neumann and Robin condition.
Keywords:
second order nonlinear equation, delta-like potential, small parameter.
UDC:
517.927.2:517.928
MSC: 34E15 Received: 16.09.2017
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