The asymptotic behavior of solutions of nonlinear second-order difference equations

In this paper, we study the boundedness and monotonicity properties of solutions of the difference equation

where {r_n}, {q_n}, {p_n}, and {e_n} are real sequences and α and β are ratios of odd positive integers. Examples illustrating our results are included.     

On asymptotic proximity of solutions of the Cauchy problem for second-order parabolic equations

A theorem about asymptotic (as t) proximity of weak fundamental solutions of the Cauchy problem is proved for divergent second-order parabolic equations. It is assumed that the coefficients have derivatives generalized in the Sobolev sense. A possible application of this theorem to establishing the uniform proximity of weak solutions of the Cauchy problem is also discussed.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 693–700, May, 1995.     

On numerical asymptotic behavior of finite element solutions for parabolic equations

Summary In this paper, we investigate the numerical asymptotic behavior of the finite element solutions for linear parabolic equations under some appropriate conditions. We also give some results of numerical experiments in the two dimensional problems to indicate the effectiveness of our results.     

On the asymptotic behavior of solutions of a mildly nonlinear parabolic system

In this paper we study the asymptotic behavior of solutions to the mixed initial boundary value problem for the system of nonlinear parabolic equations

$\text{?u}{}_{\mathrm{t}}\text{+Lu=f(x.t.u.v)}\text{?v}{}_{\mathrm{t}}\text{+Mv=g(x,t,u,v)}$

We show, under suitable technical assumptions, that these solutions converge to solutions of the Dirichlet problem for the corresponding limiting elliptic system, provided that the solution of the Dirichlet problem is unique.     

Pseudo-almost periodic viscosity solutions of second-order nonlinear parabolic equations

In this paper we generalize the comparison result of Bostan and Namah (2007) [8] to the second-order parabolic case and prove two properties of pseudo-almost periodic functions; then by using Perron’s method we prove the existence and uniqueness of time pseudo-almost periodic viscosity solutions of second-order parabolic equations under usual hypotheses.     

Verification of asymptotic solutions for one-dimensional nonlinear parabolic equations

The present article proves a result that is new for partial differential equations. According to this result, the solution of the Cauchy problem for a nonlinear parabolic equation with variable, slowly changing coefficients will turn into (asymptotically approach) a special asymptotic solution, either a solution or a kink, for high values of t.Translated from Matematicheskie Zametki, Vol. 52, No. 3, pp. 10–16, September, 1992.     

Limiting behavior of the solutions to linear second-order parabolic equations

We study the properties of solutions to parabolic equations in smooth cylindrical domains and establish the conditions for the existence of limiting nontangents andL ₂-limits ast 0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 1029–1038, July, 1993.     

Global existence and asymptotic behavior of solutions for nonlinear parabolic equations on unbounded domains

We study the existence and the asymptotic behavior of positive solutions for the parabolic equation on D×(0,∞), where is a some unbounded domain in and V belongs to a new parabolic class J^∞ of singular potentials generalizing the well-known parabolic Kato class at infinity P^∞ introduced recently by Zhang. We also show that the choice of this class is essentially optimal.     

On behavior of solutions of the quasilinear second-order parabolic equations in unbounded noncylindrical domains

The theorems of uniqueness of solutions are formulated in the classes of increasing functions for a mixed initial boundary value problem for the second-order degenerate quasiparabolic equations in unbounded noncylindrical domains. We presenta priori estimates of a special kind, analogous to the Saint-Venant principle. The proofs are based on the method of introducing a parameter.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 45, No. 4, pp. 492–499, April, 1993.     

Oscillation and asymptotic behavior for second-order nonlinear perturbed differential equations

In this paper, for all regular solutions of a class of second-order nonlinear perturbed differential equations, new oscillation criteria are established. Asymptotic behavior for forced equations is also discussed.     

On properties of solutions of semilinear second-order parabolic equations

We consider semilinear second-order parabolic equations whose principal parts may have either divergence or nondivergence form and whose nonlinear terms satisfy conditions of Bernstein-Dini type. We study the qualitative properties of the classical solutions of nondivergence equations and generalized solutions of equations with divergent principal parts: the behavior of solutions in various unbounded domains and near the boundaries of domains, removability of singularities of solutions, vanishing of solutions in unbounded domains, in particular solutions of compact support and uniqueness and continuous dependence on the boundary conditions for solutions of the exterior initial/boundary problem.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 70–107, 1991. Original article submitted June 18, 1987.