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.     

Periodic Solutions of Nonlinear Wave Equations with Dissipative Boundary Conditions

Applying Nash-Moser's implicit function theorem, the author proves the existence of periodic solution to nonlinear wave equation u_{tt} - u_{xx} + εg(t, x, u, u_t, u_x, u_{tt}, u_{tx}, u_{xx}) = 0 with a dissipative boundary condition, provided ε is sufficiently small.     

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.     

Convergence rate to traveling waves for generalized Benjamin–Bona–Mahony–Burgers equations

This paper is concerned with the large time behavior of traveling wave solutions to the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers equations

The Dirichlet problem for a mixed-type equation with characteristic degeneration in a rectangular domain

We study the first boundary-value problem for the following mixed-type equation of the second kind:

Viscosity and relaxation approximations for a hyperbolic-elliptic mixed type system

To a given system of conservation laws

we associate the system

which is of mixed type. Under certain conditions, convergence of this latter system for with is established without the need of stability criteria or hyperbolicity of the left-hand sides of the equations.

     

Some Properties of Free Boundary for Solution of Porous Medium Equation with Gravity Term in One-dimension

Consider the degenerate parabolic equation (porous medium equation with gravity term): u_t = (u^m)_{xx} + (u^n)_x, -∞ < x < ∞, t > 0, m > 1 u(x, 0) = u_0(x), -∞ < x < ∞ The main results consist of the estimation of t^∗_i called waiting time, the behavior of pressure V = \frac{m}{m-1}u^{m-1} near a vertical or a nonvertical part of ς_i(t) and a condition of that ς_i(t) is continuously differentiable.     

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

Numerical blow-up for a nonlinear heat equation

This paper concerns the study of the numerical approximation for the following initialboundary value problem

Multiple periodic solutions of a superlinear forced wave equation

We study the existence of forced vibrations of nonlinear wave equation: (*) $$\begin{array}{*{20}c} {u_{tt} - u_{xx} + g(u) = f(x,t),} & {(x,t) \in (0,\pi ) \times R,} \\ {\begin{array}{*{20}c} {u(0,t) = u(\pi ,t) = 0,} \\ {u(x,t + 2\pi ) = u(x,t),} \\ \end{array} } & {\begin{array}{*{20}c} {t \in R,} \\ {(x,t) \in (0,\pi ) \times R,} \\ \end{array} } \\ \end{array}$$ whereg(ξ)∈C(R,R)is a function with superlinear growth and f(x, t) is a function which is 2π-periodic in t. Under the suitable growth condition on g(ξ), we prove the existence of infinitely many solution of (*) for any given f(x, t).     

Solutions of minimal period for a semilinear wave equation

In questo lavoro si considera il problema

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.     

Some dichotomy results for the quenching problem

It is shown that any solution to the semilinear problem{ ut = uxx + δ(1-u)-p , (x, t) ∈ (-1 , 1) × (0 , T ), u( ±1 , t) = 0, t ∈ (0 , T ), u(x, 0) = u0(x) 1, x ∈ [ 1 , 1] either touches 1 in finite time or converges smoothly to a steady state as t →∞. Some extensions of this result to higher dimensions are also discussed.     

LIFE SPAN OF CLASSICAL SOLUTIONS TO $u_{tt}-u_{xx}=|u|^{1+\alpha}$

This paper studies the life span of classical solutions to $u_{tt}-u_{xx}=|u|^{1+\alpha}$,$t=0:u=sf(x),u_t=sg(x)$,where $\alpha$ is a positive real number,$f\in C_{0}^{2}(R)$,$g\in C_{2}^{1}(R)$,s is a small parameter.The upper and lower bounds of the same order of magnitude for the life span are obtained respectively.     

Existence of Generalized Heteroclinic Solutions of the Coupled Schr¨odinger System under a Small Perturbation

The following coupled Schrodinger system with a small perturbation
is considered, where β and ε are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and its dominant system has a heteroclinic solution. Then adjusting some appropriate constants and applying the fixed point theorem and the perturbation method yield that this heteroclinic solution deforms to a heteroclinic solution exponentially approaching the obtained periodic solution （called the generalized heteroclinic solution thereafter）.     

Nontrivial compact blow-up sets of smaller dimension

We provide examples of solutions to parabolic problems with nontrivial blow-up sets of dimension strictly smaller than the space dimension. To this end we just consider different diffusion operators in different variables, for example, or . For both equations, we prove that there exists a solution that blows up in the segment .

     

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.     

Blow up for Initial-Boundary Value Problem of Wave Equation with a Nonlinear Memory in 1-D

The present paper is devoted to studying the initial-boundary value problem of a 1-D wave equation with a nonlinear memory: u_(tt) - u_(xx) = 1/ Γ(1 - γ) ∫_0~t (t - s)~(-γ)|u(s)|~pds. The blow up result will be established when p 1 and 0 γ 1, no matter how small the initial data are, by introducing two test functions and a new functional.     

Conditions of solvability of quasilinear periodic boundary-value problems for hyperbolic equations of the second order

On the basis of properties of the Vejvoda-Shtedry operator, we obtain solvability conditions for the 2π-periodic problem

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.