Asymptotic behavior of the first boundary-value problem for elliptic equations in domains with a thin coating

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 6, pp. 74–84, November–December, 1988.     

Asymptotic analysis of a boundary-value problem with nonlinear multiphase boundary interactions in a perforated domain

We consider a boundary-value problem for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ω _ε that is ε-periodically perforated by small holes. The holes are split into two ε-periodic sets depending on the boundary interaction via their boundary surfaces. Therefore, two different nonlinear boundary conditions σ _ε (u _ε ) + εκ _m (u _ε ) = εg _ε ^(m) , m = 1, 2, are given on the corresponding boundaries of the small holes. The asymptotic analysis of this problem is performed as ε → 0, namely, the convergence theorem for both the solution and the energy integral is proved without using an extension operator, asymptotic approximations for the solution and the energy integral are constructed, and the corresponding approximation error estimates are obtained.     

Asymptotic behavior of the eigenvalues of a boundary-value problem for a second-order elliptic operator-differential equation

We study the asymptotic behavior of the eigenvalues of a boundary-value problem with spectral parameter in the boundary conditions for a second-order elliptic operator-differential equation. The asymptotic formulas for the eigenvalues are obtained. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 8, pp. 1146–1152, August, 2006.     

Approximate stabilization for a nonlinear parabolic boundary-value problem

For the problem of optimal stabilization of solutions of a nonlinear parabolic boundary-value problem with small parameter in the nonlinear term, we substantiate the form of approximate regulator on the basis of the formula of optimal synthesis of the corresponding linear-quadratic problem.     

Asymptotic behavior of solutions to a nonlinear Stefan problem with different moving parameters

This paper deals with a nonlinear diffusion equation with double free boundaries possessing different moving parameters. We present the spreading–vanishing dichotomy and threshold between spreading and vanishing. Moreover, when spreading happens, using the zero number argument we provide sharp estimates of spreading speeds of expanding fronts, and describe how the solution approaches the semi-wave.