On the Cauchy problem for a nonlinear Sobolev type equation

We study the large-time asymptotics of the solution of the Cauchy problem for a nonlinear Sobolev type equation. We show that if initial data are sufficiently small in some norm, then the leading term of the asymptotic expansion is determined by the linear part of the equation and has exponential character, while the nonlinearity affects the decay order of the remainder.     

Periodic problem for an evolution equation with a quadratic and a cubic nonlinearity

We study conditions for the existence of a solution of a periodic problem for a model nonlinear equation in the spatially multidimensional case and consider various types of large time asymptotics (exponential and oscillating) for such solutions. The generalized Kolmogorov-Petrovskii-Piskunov equation, the nonlinear Schrödinger equation, and some other partial differential equations are special cases of this equation. We analyze the solution smoothing phenomenon under certain conditions on the linear part of the equation and study the case of nonsmall initial data for a nonlinearity of special form. The leading asymptotic term is presented, and the remainder in the asymptotics of the solution is estimated in a spatially uniform metric.     

Periodic boundary value problem for nonlinear Sobolev-type equations

The large-time asymptotic behavior of solutions to the periodic boundary value problem for a nonlinear Sobolev-type equation is studied. In particular, the case where the initial perturbations are not small is considered. In this case, the large-time behavior of solutions is dichotomous.     

Large-Time Asymptotics for Periodic Solutions of the Modified Burgers Equation

In this paper, we construct large-time asymptotic solution of the modified Burgers equation with sinusoidal initial conditions by using a balancing argument. These asymptotics are validated by a careful numerical study.     

Large-Time Asymptotics for Periodic Solutions of Some Generalized Burgers Equations

In this paper, we construct large-time asymptotic solutions of some generalized Burgers equations with periodic initial conditions by using a balancing argument. These asymptotics are validated by a careful numerical study. We also show that our asymptotic results agree with the approximate solutions obtained by Parker [1] in certain limits.     

Perturbation theorems for Hele-Shaw flows and their applications

In this work, we give a perturbation theorem for strong polynomial solutions to the zero surface tension Hele-Shaw equation driven by injection or suction, the so called Polubarinova–Galin equation. This theorem enables us to explore properties of solutions with initial functions close to polynomials. Applications of this theorem are given in the suction and injection cases. In the former case, we show that if the initial domain is close to a disk, most of the fluid will be sucked before the strong solution blows up. In the latter case, we obtain precise large-time rescaling behaviors for large data to Hele-Shaw flows in terms of invariant Richardson complex moments. This rescaling behavior result generalizes a recent result regarding large-time rescaling behavior for small data in terms of moments. As a byproduct of a theorem in this paper, a short proof of existence and uniqueness of strong solutions to the Polubarinova–Galin equation is given.     

A Periodic Problem for the Landau--Ginzburg Equation

In this paper, we consider a periodic problem for the n-dimensional complex Landau--Ginzburg equation. It is shown that in the case of small initial data there exists a unique classical solution of this problem, and an asymptotics of this solution uniform in the space variable is given. The leading term of the asymptotics is exponentially decreasing in time.     

A constructive theory for shock-free,isentropic flow

We construct by finite differences solutions of the Cauchy problem for the nonlinear wave equation in one space dimension. We make certain monotonicity assumptions about the initial data, and we show that the resulting solution is Lipschitz continuous for positive times. In addition, we prove the uniqueness of the solution in a certain class, and we characterize its large-time behavior in terms of the equilibrium state for a corresponding Riemann problem. Finally, we show how our results can be extended to more general 2 × 2 systems of hyperbolic conservation laws which are genuinely nonlinear.     

Asymptotic solutions of the one-dimensional linearized Korteweg–de Vries equation with localized initial data

The Cauchy problem with localized initial data for the linearized Korteweg–de Vries equation is considered. In the case of constant coefficients, exact solutions for the initial function in the form of the Gaussian exponential are constructed. For a fairly arbitrary localized initial function, an asymptotic (with respect to the small localization parameter) solution is constructed as the combination of the Airy function and its derivative. In the limit as the parameter tends to zero, this solution becomes the exactGreen function for the Cauchy problem. Such an asymptotics is also applicable to the case of a discontinuous initial function. For an equation with variable coefficients, the asymptotic solution in a neighborhood of focal points is expressed using special functions. The leading front of the wave and its asymptotics are constructed.     

Fermi-Dirac-Fokker-Planck equation: Well-posedness & long-time asymptotics

A Fokker-Planck type equation for interacting particles with exclusion principle is analyzed. The nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function. Assuming enough initial moments are finite, we can show the global existence of weak solutions for this problem. The natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy. As a consequence, long-time asymptotics in L¹ are characterized by the Fermi-Dirac equilibrium with the same initial mass. This result is achieved without rate for any constructed global solution and with exponential rate due to entropy/entropy-dissipation arguments for initial data controlled by Fermi-Dirac distributions. Finally, initial data below radial solutions with suitable decay at infinity lead to solutions for which the relative entropy towards the Fermi-Dirac equilibrium is shown to converge to zero without decay rate.     

Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity

We study the asymptotic behavior in time of solutions to the initial value problem of the nonlinear Schrödinger equation with a subcritical dissipative nonlinearity λ|u|^p−1u, where 1<p<1+2/n, n is the space dimension and λ is a complex constant satisfying Imλ<0. We show the time decay estimates and the large-time asymptotics of the solution, when the space dimension n?3, p is sufficiently close to 1+2/n and the initial data is sufficiently small.     

Properties of solutions of a nonlinear system of equations

The objective of this paper is to study a highly general model system of nonlinear equations that describe propagation of nerve impulses in a cell. We prove time-local existence of a classical solution, derive sufficient conditions for the existence of a time-global solution, and consider the question of eventual smoothing of discontinuous initial values. The large-time asymptotic expansion of the solutions of the Cauchy problem is constructed. The coefficients of the principal term of the asymptotic expression for the solution of the Cauchy problem are computed in terms of the Fourier transform of the initial values.The study has been partially supported by the Russian Foundation of Basic Research (93-011-131).Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 3, pp. 64–93, 1993.     

Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term

The paper studies the global existence, asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term. It proves that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as t→+∞, respectively, in the states of large initial data and small initial energy. In particular, in the case of space dimension N=1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. And an example is given.     

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE SYSTEM OF ONE-DIMENSIONAL NONLINEAR THERMOVISCOELASTICITY

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The Cauchy problem for a singularly perturbed integro-differential Fredholm equation

An initial problem is considered for an ordinary singularly perturbed integro-differential equation with a nonlinear integral Fredholm operator. The case when the reduced equation has a smooth solution is investigated, and the solution to the reduced equation with a corner point is analyzed. The asymptotics of the solution to the Cauchy problem is constructed by the method of boundary functions. The asymptotics is validated by the asymptotic method of differential inequalities developed for a new class of problems.     

大初值的抛物型守恒律方程解的大时间性态(英文)

In this paper,we study the large-time behavior of periodic solutions for parabolic conservation laws.There is no smallness assumption on the initial data.We firstly get the local existence of the solution by the iterative scheme,then we get the exponential decay estimates for the solution by energy method and maximum principle,and obtain the global solution in the same time.     

带梯度吸收项的快扩散方程的自相似奇性解

研究一类带有非线性梯度吸收项的快速扩散方程的自相似奇性解．通过自相似变换,该自相似奇性解满足一个非线性常微分方程的边值问题,再利用打靶法技巧研究该常微分方程初值问题解的存在唯一性并根据初值的取值范围对其解进行了分类．通过对这些解类的性质的分析研究,得出了自相似强奇性解存在唯一性的充分必要条件,此时自相似奇性解就是强奇性解．     

Optimal homotopy analysis and control of error for solutions to the non-local Whitham equation

The Whitham equation is a non-local model for nonlinear dispersive water waves. Since this equation is both nonlinear and non-local, exact or analytical solutions are rare except for in a few special cases. As such, an analytical method which results in minimal error is highly desirable for general forms of the Whitham equation. We obtain approximate analytical solutions to the non-local Whitham equation for general initial data by way of the optimal homotopy analysis method, through the use of a partial differential auxiliary linear operator. A method to control the residual error of these approximate solutions, through the use of the embedded convergence control parameter, is discussed in the context of optimal homotopy analysis. We obtain residual error minimizing solutions, using relatively few terms in the solution series, in the case of several different kernels and associated initial data. Interestingly, we find that for a specific class of initial data, there exists an exact solution given by the first term in the homotopy expansion. A specific example of initial data which satisfies the condition producing an exact solution is included. These results demonstrate the applicability of optimal homotopy analysis to equations which are simultaneously nonlinear and non-local.     

Existence and Asymptotic Behavior of the Wave Equation with Dynamic Boundary Conditions

The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t>0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time.     

Bifurcations in the Generalized Korteweg–de Vries Equation

We study the generalized Korteweg–de Vries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) equation with periodic in the spatial variable boundary conditions. For various values of parameters, in a sufficiently small neighborhood of the zero equilibrium state we construct asymptotics of periodic solutions and invariant tori. Separately we consider the case when the stability spectrum of the zero solution contains a countable number of roots of the characteristic equation. In this case we state a special nonlinear boundary-value problem which plays the role of a normal form and determines the dynamics of the initial problem.