Asymptotic behavior of solutions of degenerating parabolic equations

Translated from Issledovaniya po Prikladnoi Matematike, No. 17, pp. 166–173, 1990.     

Asymptotic behavior of solutions of nonlinear Volterra equations

An energy decay rate is obtained for solutions of wave type equations in a bounded region in Rⁿ whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface.     

Asymptotic behavior of solutions of functional difference equations

For linear functional difference equations, we obtain some results on the asymptotic behavior of solutions, which correspond to a Perron-type theorem for linear ordinary difference equations. We also apply our results to Volterra difference equations with infinite delay.     

Asymptotic behavior of solutions of linear delay equations

The delay differential equations of the formx′(t)=?a(t)x(t?1),t≥0 are considered, wherea(t)≥0 is locally integrable on [0,∞). The main result: Let 0<c(t)≤a(t)≤k(t) for large ∫^∞, andc(t)≤Mc(t′) fort, t′≥T,|t?t′|≤l with some constantsl>0,M>1,T≥0. Then the condition \(k(t) \leqslant \frac{3}{2} + \alpha c(t), t \geqslant T\) with some constant α>0 dependent onl, M, ensures that all solutions of (*) tend to zero ast→∞.     

Asymptotic behavior of solutions to polynomial renewal equations

Many special functions arise as “renormalized” limits of sequences of polynomials that satisfy a polynomial renewal equation. We determine the asymptotic behavior of these sequences of polynomials for both ordinary and “coefficientwise” convergence, and illustrate it with specific examples.     

Asymptotic behavior of solutions of pseudo-parabolic partial differential equations

Summary Consider a solution to a second-order pseudo-parabolic equation with sufficiently smooth time-independent coefficients in a cylindrical domain. If it vanishes on the cylindrical surface for all times and if its restriction to a fixed instant belongs toC _2+a , then its pointwise values decay exponentially as t→∞ while its Dirichlet norm grows expontially as t→−∞. Similar conclusion still hold for solutions to non-homogeneous equations under non-homogeneous boundary conditions provided the free term and the boundary data posses these asymptotic behaviors. Work of the second named author was partially supported by N.S.F. Grant No. GP-19590. Entrata in Redazione il 29 gennaio 1971.     

Asymptotic behavior of the solutions of certain parabolic equations

Some laws in physics describe the change of a flux and are represented by parabolic equations of the form (*) \documentclass{article}\pagestyle{empty}\begin{document}$$\frac{{\partial u}}{{\partial t}}=\frac{\partial}{{\partial x_j }}(\eta \frac{{\partial u}}{{ax_j}}-vju),$$\end{document} j≤m, where η and v_j are functions of both space and time. We show under quite general assumptions that the solutions of equation (*) with homogeneous Dirichlet boundary conditions and initial condition u(x, 0) = u_o(x) satisfy The decay rate d > 0 only depends on bounds for η, v and G § R^m the spatial domain, while the constant c depends additionally on which norm is considered. For the solutions of equation (*) with homogeneous Neumann boundary conditions and initial condition u₀(x) ≥ 0 we derive bounds d₁u₁ ≤ u(x, t) ≤ d₂u₂, Where d_i, i = 1, 2, depend on bounds for η, v and G, and the u_i are bounds on the initial condition u₀.     

Asymptotic behavior of solutions of linear ordinary differential equations

We present some conditions which ensure that the solution Y(x) of the ordinary differential equation Y′(x) = A(x) Y(x), Y(x₀) = I, where x₀ ? x < ∞ and A(x), Y(x) are n × n complex matrix-valued functions with A(x) continuous, has a nonsingular limit as x → ∞.     

Asymptotic behavior of solutions to nonlinear Volterra integral equations

A class of operator Riccati integral equations is associated with a factorization problem in a certain Banach algebra. Recent results concerning factorization in this algebra are used to obtain existence, uniqueness, and continuous dependence results for the Riccati equations.     

Asymptotic behavior of solutions to abstract functional differential equations

In this paper, a new approach is provided to study the asymptotic behavior of functions. A Tauberian theorem is improved and applied to describe the asymptotic behavior of abstract functional differential equations of the form

     

一类Euler方程解的渐近行为

给出了一类Ginzburg-Landau型泛函的极小元所满足的Euler方程的解的某些弱收敛性质。     

Asymptotic behavior of solutions of delay equations in Hilbert spaces

We establish unimprovable estimates of solutions of inhomogeneous delay differential-difference equations, the coefficients of which are unbounded operators and operator-functions acting in a Hilbert space. We also present results about expansions of those solutions into a sum of a (finite) linear combination of exponential solutions for the homogeneous equation and a function with a smaller power of the exponential growth. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 21, Proceedings of the Seminar on Differential and Functional Differential Equations Supervised by A. Skubachevskii (Peoples’ Friendship University of Russia), 2007.