Constructing solutions of an ill-posed nonlinear singularly perturbed problem for an equation of elliptic typeстатья
Информация о цитировании статьи получена из
Web of Science
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 27 мая 2015 г.
Аннотация:Consider the following singularly perturbed problem
ε 2 Δu=f(u,x),x=(x 1 ,x 2 )∈Ω⊂ℝ 2 ,(1)
where ε>0, with the boundary conditions
u| ∂Ω =0or∂u ∂n| ∂Ω =0·(2)
Regarding f(u,x) as the right-hand side of the Poisson equation, one can write out the equation of the second kind for u(x,ε) in terms of the source function G(x,y) with known properties
-∫ Ω G(x,y)f(u(y),y)dτ y =ε 2 u(x,ε)·(3)
The limiting case, that is
-∫ Ω G(x,y)f(u(y),y)dy=0(4)
is ill-posed. Depending on the roots of the equation f(u,x)=0, the author studies asymptotics of bounded solutions (3) as ε→0.