The asymptotic behavior of solutions of a certain nonlinear Volterra equation

We study the asymptotic behavior of bounded and unbounded solutions to the Volterra-Hammerstein equation. We obtain conditions for the admissibility of a pair of spaces consisting of the sum of a quasipolynomial and the Taylor expansion at infinity.     

On the asymptotic spectra of the bounded solutions of a nonlinear Volterra equation

We consider the asymptotic behavior of the bounded solutions of a nonlinear Volterra integrodifferential equation with a positive definite convolution kernel. Our main result states that (under appropriate assumptions) the asymptotic spectra of the solutions are contained in the set where the real part of the Fourier transform of the kernel vanishes. We also give a new asymptotic stability theorem, and present a new proof of a known result on the asymptotic behavior of the bounded solutions of a nonlinear, nondifferentiated Volterra equation.     

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.     

On existence and asymptotic stability of solutions of a nonlinear integral equation

We prove an existence theorem for a nonlinear integral equation being a Volterra counterpart of an integral equation arising in the traffic theory. The method used in the proof allows us to obtain additional characterization in terms of asymptotic stability of solutions of an equation in question.     

On the asymptotic behavior of solutions of nonlinear differential systems

We obtain constructive conditions for the unique solvability of the singular problem dx/dt = f(t, x), x _∞ = 0, where f ∈ C ^(0,1)([0, ∞) × ? ⁿ , ? ⁿ ).     

Existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order

This paper is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order: ut+∇⋅(|∇Δu|^p−2∇Δu)=f(u) in Ω⊂RN with boundary condition u=Δu=0 and initial data u₀. The substantial difficulty is that the general maximum principle does not hold for it. The solutions are obtained for both the steady-state case and the developing case by the fixed point theorem and the semi-discretization method. Unlike the general procedures used in the previous papers on the subject, we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables, respectively. By a compactness argument with necessary estimates, we show that the two approximation sequences converge to the same limit, i.e., the solution to be determined. In addition, the decays of solutions towards the constant steady states are established via the entropy method. Finally, it is interesting to observe that the solutions just tend to the initial data u₀ as p→∞.     

On the asymptotic behavior of solutions of nonlinear second-order parabolic equations

We study the asymptotic behavior as t → +∞ of solutions to a semilinear second-order parabolic equation in a cylindrical domain bounded in the spatial variable. We find the leading term of the asymptotic expansion of a solution as t → +∞ and show that each solution of the problem under consideration is asymptotically equivalent to a solution of some nonlinear ordinary differential equation.     

On asymptotic solutions of the renewal equation

Consider the renewal equation in the form (¹) $\text{u(t) = g(t) + ∝}{}_{\mathrm{o}}{}^{\mathrm{t}}\text{u(t ? τ) ?(τ) dτ}$, where $\text{?(t)}$ is a probability density on [0, ∞) and lim_{t → ∞}g(t) = g₀. Asymptotic solutions of (¹) are given in the case when f(t) has no expectation, i.e., $\text{∝}{}_0{}^{\mathrm{\infty }}\text{t?(t)dt = ∞}$. These results complement the classical theorem of Feller under the assumption that f(t) possesses finite expectation.     

Self-similar asymptotic behavior for the solutions of a linear coagulation equation

In this paper we consider the long-time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving in a random distribution of fixed particles. The volumes v of these particles are independently distributed according to a probability distribution which decays asymptotically as a power law $v^{?\sigma }$. The validity of the equation has been rigorously proved in [22] taking as a starting point a particle model and for values of the exponent $\sigma >3$, but the model can be expected to be valid, on heuristic grounds, for $\sigma >\frac{5}{3}$. The resulting equation is a non-local linear degenerate parabolic equation. The solutions of this equation display a rich structure of different asymptotic behaviors according to the different values of the exponent σ. Here we show that for $\frac{5}{3}<\sigma <2$ the linear Smoluchowski equation is well-posed and that there exists a unique self-similar profile which is asymptotically stable.     

Global asymptotic behavior of solutions of a semilinear parabolic equation

We study the large time behavior of solutions for the semilinear parabolic equation . Under a general and natural condition on and the initial value , we show that global positive solutions of the parabolic equation converge pointwise to positive solutions of the corresponding elliptic equation. As a corollary of this, we recapture the global existence results on semilinear elliptic equations obtained by Kenig and Ni and by F.H. Lin and Z. Zhao. Our method depends on newly found global bounds for fundamental solutions of certain linear parabolic equations.

     

On a semilinear wave equation: The Cauchy problem and the asymptotic behavior of solutions

The semilinear wave equation

$\text{□u + m}{}^2\text{u + ¦u¦}{}^{\mathrm{p\; ?\; 2}}\text{u(V1 ¦u¦}{}^{\mathrm{p}}\text{) = 0}$

in $\text{Ω= R}{}^3\text{, ?∞ < t < ∞}$, is studied where □ denotes the d'Alembertian operator and 1 means spatial convolution. Under mild assumptions on the real-valued function V and 2 ? p ? 3 the well-posedness of the Cauchy problem is proved. Furthermore, some properties of the solutions of the equation are analyzed such as the asymptotic behavior of local energy as $\text{¦t¦ → + ∞}$ in the case of zero mass. Our results extend that of Perla Menzala and Strauss, where case p = 2 was studied.     

On the asymptotic behavior of the solutions of a class of second order nonlinear differential equations

In this paper, we investigate properties of the solutions of a class of second-order nonlinear differential equation such as [p(t)f(x(t))x′(t)]′ + q(t)g(x′(t))e(x(t)) = r(t)c(x(t)). We prove the theorems of monotonicity, nonoscillation and continuation of the solutions of the equation, the sufficient and necessary conditions that the solutions of the equation are bounded, and the asymptotic behavior of the solutions of the equation when t → ∞ on condition that the solutions are bounded. Also we provide the asymptotic relationship between the solutions of this equation and those of the following second-order linear differential equation: [p(t)u′(t)]′ = r(t)u(t)