ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR GENERALIZED BURGERS EQUATION

This paper is concerned with the stability of the rarefaction wave for the Burgers equationwhere 0 ≤ a < 1/4p (q is determined by (2.2)). Roughly speaking, under the assumption that u_ < u , the authors prove the existence of the global smooth solution to the Cauchy problem (I), also find the solution u(x, t) to the Cauchy problem (I) satisfying sup |u(x, t) -uR(x/t)| → 0 as t → ∞, where uR(x/t) is the rarefaction wave of the non-viscous Burgersequation ut f(u)x = 0 with Riemann initial data u(x, 0) =     

Positive solution of fourth order ordinary differential equation with four-point boundary conditions

In this work, the authors consider the fourth order nonlinear ordinary differential equation$u^{\left(4\right)}\left(t\right)=f(t,u\left(t\right))\text{,}0<t<1\text{,}$ with the four-point boundary conditions $u\left(0\right)=u\left(1\right)=0\text{,}$$a{u}^{″}\left({\xi }_1\right)-b{u}^{‴}\left({\xi }_1\right)=0\text{,}c{u}^{″}\left({\xi }_2\right)+d{u}^{‴}\left({\xi }_2\right)=0\text{,}$ where $0\le {\xi }_1<{\xi }_2\le 1$. By means of the upper and lower solution method and fixed point theorems, some results on the existence of positive solutions to the above four-point boundary value problem are obtained.     

Singular discrete and continuous mixed boundary value problems

For each $n\in \mathbb{N}$, $n\ge 2$, we prove the existence of a solution $(u_0,\dots ,u_n)\in {\mathbb{R}}^{n+1}$ of the singular discrete problem $\frac{1}{h^2}{\Delta }^2{u}_{k?1}+f(t_k,u_k)=0\text{,}k=1,\dots ,n?1\text{,}$$\Delta u_0=0\text{,}{u}_n=0\text{,}$ where $u_k>0$ for $k=0,\dots ,n?1$. Here $T\in (0,\infty )$, $h=\frac{T}{n}$, $t_k=hk$, $f(t,x):[0,T]\times (0,\infty )\to \mathbb{R}$ is continuous and has a singularity at $x=0$. We prove that for $n\to \infty$ the sequence of solutions of the above discrete problems converges to a solution $y$ of the corresponding continuous boundary value problem $y^{″}\left(t\right)+f(t,y\left(t\right))=0\text{,}$$y^{\prime }\left(0\right)=0\text{,}y\left(T\right)=0\text{.}$     

Homogeneous initial–boundary value problem of the Rosenau equation posed on a finite interval

This paper considers the IBVP of the Rosenau equation $\{\begin{array}{c}{\partial }_{t}u+{\partial }_t{\partial }_x^{4}u+{\partial }_{x}u+u{\partial }_{x}u=0\text{,}x\in (0,1),t>0\text{,}\\ u(0,x)=u_0\left(x\right)\\ u(0,t)={\partial }_x^{2}u(0,t)=0\text{,}u(1,t)={\partial }_x^{2}u(1,t)=0\text{.}\end{array}$ It is proved that this IBVP has a unique global distributional solution $u\in C([0,T];H^s(0,1))$ as initial data $u_0\in H^s(0,1)$ with $s\in [0,4]$. This is a new global well-posedness result on IBVP of the Rosenau equation with Dirichlet boundary conditions.     

On a nonlinear wave equation involving the term −∂∂x(μ(x,t,u,6ux62)ux): Linear approximation and asymptotic expansion of solution in many small parameters

In this paper, we consider the following nonlinear Kirchhoff wave equation (1)$\{\begin{array}{c}{u}_{tt}?\frac{?}{?x}\left(\mu (x,t,u,{6u_{x}6}^2)u_x\right)=f(x,t,u,u_x,u_t)\text{,}0<x<1,0<t<T\text{,}\hfill \\ u(0,t)=g_0\left(t\right)\text{,}u(1,t)=g_1\left(t\right)\text{,}\hfill \\ u(x,0)={\stackrel{?}{u}}_0\left(x\right)\text{,}{u}_t(x,0)={\stackrel{?}{u}}_1\left(x\right)\text{,}\hfill \end{array}$ where ${\stackrel{?}{u}}_0$, ${\stackrel{?}{u}}_1$, $\mu$, $f$, $g_0$, $g_1$ are given functions and ${6u_{x}6}^2={\int }_0^1{u}_x^2(x,t)dx$. First, combining the linearization method for nonlinear term, the Faedo–Galerkin method and the weak compact method, a unique weak solution of problem (1) is obtained. Next, by using Taylor’s expansion of the function $\mu (x,t,y,z)$ around the point $(x,t,y_0,z_0)$ up to order $N+1$, we establish an asymptotic expansion of high order in many small parameters of solution.     

The existence of solutions for boundary value problem of fractional hybrid differential equations

In this paper, we study the existence of solutions for the boundary value problem of fractional hybrid differential equations$D_{0^{+}}^{\alpha }\left[\frac{x(t)}{f(t\text{,}x(t))}\right]+g(t\text{,}x(t))=0\text{,}0<t<1\text{,}$$x(0)=x(1)=0\text{,}$where $1<\alpha ?2$ is a real number, $D_{0^{+}}^{\alpha }$ is the Riemann–Liouville fractional derivative. By a fixed point theorem in Banach algebra due to Dhage, an existence theorem for fractional hybrid differential equations is proved under mixed Lipschitz and Carathéodory conditions. As an application, examples are presented to illustrate the main results.