Breather collapse for a dispersive effective equation: Asymptotics on the well-posedness boundary

For the solution of the Cauchy problem for the equation

Initial Boundary Value Problem for a Damped Nonlinear Hyperbolic Equation

In the paper, the existence and uniqueness of the generalized global solution and the classical global solution of the initial boundary value problems for the nonlinear hyperbolic equation u_{tt} + k_1u_{xxxx} + k_2u_{xxxxt} + g(u_{xx})_{xx} = f(x, t) are proved by Galerkin method and the sufficient conditions of blow-up of solution in finite time are given.     

Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation

We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear damped wave equation

Computation of longitudinal motions of a viscoelastic bar

A Gelerkin type method, based on trigonometric functions and Crank-Nicolson discretizations of the time variable, is applied to compute solutions of the initial boundary value problem associated with the equation $$\rho _0 u_{tt} = a_1 u_{xx} + a_2 u_x^2 u_{xx} + a_3 u_{xtx} + f,x \in [0,1],t > 0.$$ Error estimates are derived and a numerical example is presented.     

A Nekhoroshev Type Theorem for the Nonlinear Wave Equation in Gevrey Space

In this paper, the authors prove a Nekhoroshev type theorem for the nonlinear wave equation u_(tt)=u_(xx)-mu-f(u),x∈[0, π] in Gevrey space.     

On Initial-boundary-value Problems for a Class of Systems of Quasi-linear Evolution Equations

In this paper the initial-boundary-value problems for pseudo-hyperbolic system of quasi-linear equations: {(-1)^Mu_{tt} + A(x, t, U, V)u_x^{2M}_{tt} = B(x, t, U, V)u_x^{2M}_{t} + C(x, t, U, V)u_x^{2M} + f(x, t, U, V) u_x^k(0,t) = ψ_{0k}(t), \quad u_x^k(l,t) = ψ_{lk}(t), \quad k = 0,1,…,M - 1 -u(x,0) = φ_0(x), \quad u_t(x,0) = φ_1(x) is studied, where U = (u_1, u_x,…,u_x^{2M - 1}) V = (u_t, u_{xt},…,u_x^{2M - 1_t}), A, B, C are m × m matrices, u, f, ψ_{0k}, ψ_{1k}, ψ_0, ψ_1 are m-dimensional vector functions. The existence and uniqueness of the generalized solution (in H² (0, T; H^{2M} (0, 1))) of the problems are proved.     

Global attractor for the weakly damped driven Schrödinger equation in $ H^2 (\mathbb{R}) $

We discuss the asymptotic behaviour of the Schrödinger equation ¶¶$ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 $ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 ¶¶ with the initial condition u(x,0) = u₀ (x) u(x,0) = u_0 (x) . We prove existence of a global attractor in\ H² (\mathbbR) H^2 (\mathbb{R}) , by using a decomposition of the semigroup in weighted Sobolev spaces to overcome the noncompactness of the classical Sobolev embeddings.     

The initial value problem for the cubic nonlinear Klein–Gordon equation

We study the initial value problem for the cubic nonlinear Klein–Gordon equation

A Nonhomogeneous Boundary Value Problem for the Boussinesq Equation on a Bounded Domain

In this paper, we study the well-posedness of an initial-boundary-value problem (IBVP) for the Boussinesq equation on a bounded domain,\begin{cases} &u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0,\quad x\in (0,1), \;\;t>0,\\ &u(x,0)=\varphi(x),\;\;\; u_t(x,0)=ψ(x),\\ &u(0,t)=h_1(t),\;\;\;u(1,t)=h_2(t),\;\;\;u_{xx}(0,t)=h_3(t),\;\;\;u_{xx}(1,t)=h_4(t).\\ \end{cases} It is shown that the IBVP is locally well-posed in the space $H^s (0,1)$ for any $s\geq 0$ with the initial data $\varphi,$ $\psi$ lie in $H^s(0,1)$ and $ H^{s-2}(0,1)$, respectively, and the naturally compatible boundary data $h_1,$ $h_2$ in the space $H_{loc}^{(s+1)/2}(\mathbb{R}^+)$, and $h_3 $, $h_4$ in the the space of $H_{loc}^{(s-1)/2}(\mathbb{R}^+)$ with optimal regularity.     

On Modified Tricomi Problem of Semi-linear Equation of Mixed Type of Second Kind

In this paper we considered the semi-linear equation of mixed type of second kind, k(x, y)u_{tt} + u_{xx} + a(x, y)u_t + P(x, y)u_t + y(x, y)u - |u|^pu = f(z,y) For the above equation, we solved the modified Tricomi problem and have proved the existence and uniqueness of strong solution in H_1.     

Initial value problem for a class of nonlinear pseudo-hyperbolic equations

The existence of global weak solutions to the periodic boundary problem or the initial value problem for the nonlinear Pseudo-hyperbolic equation u_(tt)-[a_1+a_2(u_x)~(2m)]u_(xx)-a_3u_(xxt)=f(x,t,u,u_x) is proved by the method of the vanishing of the additional diffusion terms, Leray-Schauder's fixedpoint argument and Sobolev's estimates,where m≥1 is a natural number and a_i>0(i=1,2,3)are constants.     

Numerical Approximation of a Reaction-Diffusion System with Fast Reversible Reaction

The authors consider the finite volume approximation of a reaction-diffusion system with fast reversible reaction.It is deduced from a priori estimates that the approximate solution converges to the weak solution of the reaction-diffusion problem and satisfies estimates which do not depend on the kinetic rate.It follows that the solution converges to the solution of a nonlinear diffusion problem,as the size of the volume elements and the time steps converge to zero while the kinetic rate tends to infinity.     

Travelling Wave Front Solutions for Reaction-diffusion Systems

In this paper by using upper-lower solution method, under appropriate assumptions on f and g the existence of travelling wave front solutions for the following reaction-diffusion system is proved: {u_t - u_{xx}, = f(u,v) v_t - v_{xx} = g(u, v) As an application, the necessary and sufficient condition of the existence of monotone solutions for the boundary value problem {u" + cu' + u(1 - u- rv) = 0 v" + cv' - buv = 0 u(-∞) = v(+∞) = 0 u(+∞) = v(-∞) = 1 where 0 < r < 1, 0 < b < \frac{1 - r}{r} are known constants and c is unknown constant to be obtained.     

Nonlinear wavelet methods for high-dimensional backward heat equation

The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation ut (x, y, t) = u xx (x, y, t) + uyy (x, y, t), x ∈ R, y ∈ R, 0 ≤ t 1, u(x, y, 1) = (x, y), x ∈ R, y ∈ R. Motivated by Regińska's work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent.     

Solutions of minimal period for a semilinear wave equation

In questo lavoro si considera il problema

New explicit periodic wave solutions and their limits for modified form of Camassa-holm equation

In this paper, we consider the following nonlinear equation ut＋2kux-uxxt＋au^2ux=2uxuxx＋uuxxx,which is a modified form of the Camassa-Holm equation. We construct four new explicit periodic wave solutions by bifurcation method of dynamical systems. We also obtain two explicit solitary wave solutions via the limits of the explicit periodic wave solutions. One of the two solitary wave solutions is new.     

一类组合方程的势对称分类

首先给出一类含有任意函数的变系数波动方程uxx=H(x)utt的古典对称及其势对称的完全分类,然后借助于这个波动方程的对称分类,系统讨论了含有两个任意函数的一类组合方程的势对称分类,所得结果确实扩充了原方程的对称.在计算过程中,采用微分形式的吴方法,微分特征列的程序包起到了重要作用.     

Regularity of Degenerate Monge–Ampère and Prescribed Gaussian Curvature Equations in Two Dimensions

We use a priori inequalities for quasilinear equations to obtain a regularity theorem for the Dirichlet problem for the Monge–Ampère equation,

Asymptotic behavior relative to a large parameter of the solution of the Fok-Klein-Gordon equation in the case of a discontinuous initial condition

One considers the problem of the asymptotic behavior for K→+∞ of the solution of the Cauchy problem $$u_{tt} - u_{xx} + \kappa ^2 u = 0; u|_{t = 0} = \theta (x), u_t |_{t = 0} = 0 (t > 0 - fixed)$$ Hereθ(x) is the Heaviside function. In the neighborhood of the characteristics x=±t function u(x,t)_?2 oscillates exceptionally fast (the wavelength is of order k^?2). Near the t axis the asymptotics of u(x,t) contains the Fresnel integral.     

On the new critical exponent for the nonlinear Schrödinger equations

We consider the Cauchy problem for the pth order nonlinear Schrödinger equation in one space dimension $$\left\{\begin{array}{ll}iu_{t} + \frac{1}{2} u_{xx} = |u|^{p}, x \in {\bf R}, \, t > 0, \\ \qquad u(0, x) = u_{0} (x), \; x \in {\bf R},\end{array}\right.$$ where \({p > p_{s} = \frac{3 + \sqrt{17}}{2}}\) . We reveal that p = 4 is a new critical exponent with respect to the large time asymptotic behavior of solutions. We prove that if p _s < p < 4, then the large time asymptotics of solutions essentially differs from that for the linear case, whereas it has a quasilinear character for the case of p > 4.