Irina Astashova

On power and non-power asymptotic behavior of positive solutions to Emden-Fowler type higher-order equations

AbstractFor the equation y(n)=yk,k>1,n=12,13,14, the existence of positive solutions with non-power asymptotic behavior is proved, namely y=(x∗−x)−αh(log(x∗−x)),α=nk−1,x<x∗, where x∗ is an arbitrary point, h is a positive periodic non-constant function on R.To prove this result, the Hopf bifurcation theorem is used.

On Asymptotic Classification of Solutions to Fourth-Order Differential Equations with Singular Power Nonlinearity

The asymptotic behavior of all solutions to the fourth-order Emden– Fowler type differential equation with singular nonlinearity is investigated. The equation is transformed into a system on the three-dimensional sphere. By investigation of the asymptotic behavior of all possible trajectories of this system an asymptotic classification of all solutions to the equation is obtained.

Uniform Estimates and Existence of Solutions with Prescribed Domain to Nonlinear Third-Order Differential Equation

For differential equation of the third order with power nonlinearity, uniform estimates of solutions with the same domain are obtained. The existence of solutions with prescribed domain is proved.

On Asymptotic Behavior of Blow-Up Solutions to Higher-Order Differential Equations with General Nonlinearity

New results are proved on the asymptotic behavior of blow-up solutions to a higher-order equation with general potential are proved. Several author’s results are presented concerning both positive and oscillatory solutions to equations with regular and singular nonlinearities. Some applications of the results obtained are proposed.

On asymptotic equivalence of nth order nonlinear differential equations

Abstract This paper is devoted to the problem of asymptotic equivalence of nth order differential equations with exponentially equivalent right-hand sides. With the help of this result asymptotic behaviour of solutions to nonhomogeneous differential equations is described