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.     

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.     

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) =     

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.     

Existence and convexity of local solutions to degenerate Hessian equations

In this work, we prove the existence of convex solutions to the following k-Hessian equation

$S_k[u]=K(y)g(y,u,Du)$

in the neighborhood of a point $(y_0,u_0,p_0)\in {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^n$, where $g\in C^{\infty },g(y_0,u_0,p_0)>0$, $K\in C^{\infty }$ is nonnegative near $y_0$, $K(y_0)=0$ and $\text{Rank}(D_y^{2}K)(y_0)\ge n?k+1$.     

Blow-up for coupled nonlinear wave equations with damping and source

We consider the system of nonlinear wave equations $\{\begin{array}{cc}{u}_{tt}+u_t+{\left|u_t\right|}^{m?1}{u}_t=\mathrm{div}\left({\rho }_1\left({\left|?u\right|}^2\right)?u\right)+f_1(u,v)\text{,}\hfill & (x,t)\in \Omega \times (0,T)\text{,}\hfill \\ v_{tt}+v_t+{\left|v_t\right|}^{r?1}{v}_t=\mathrm{div}\left({\rho }_2\left({\left|?v\right|}^2\right)?v\right)+f_2(u,v)\text{,}\hfill & (x,t)\in \Omega \times (0,T)\text{,}\hfill \end{array}$ with initial and Dirichlet boundary conditions. Under some suitable assumptions on the functions$f_1$, $f_2$, ${\rho }_1$, ${\rho }_2$, parameters $r,m$ and the initial data, the result on blow-up of solutions and upper bound of blow-up time are given.     

Dynamics of epidemics in homogeneous/heterogeneous populations and the spreading of multiple inter-related infectious diseases: Constant-sign periodic solutions for the discrete model

We consider the following model that describes the dynamics of epidemics in homogeneous/heterogeneous populations as well as the spreading of multiple inter-related infectious diseases:$u_i(k)=\sum _{\ell =k-{\tau }_i}^{k-1}{g}_i(k,\ell )f_i(\ell ,u_1(\ell ),u_2(\ell ),\dots ,u_n(\ell )),k\in \mathbf{Z},1⩽i⩽n\text{.}$Our aim is to establish criteria such that the above system has one or multiple constant-sign periodic solutions $(u_1,u_2,\dots ,u_n)$, i.e., for each $1⩽i⩽n$, $u_i$ is periodic and ${\theta }_i{u}_i⩾0$ where ${\theta }_i\in \{1,-1\}$ is fixed. Examples are also included to illustrate the results obtained.