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 expansion of a chord diagram and the Tutte polynomial

A chord diagram is a set of chords of a circle such that no pair of chords has a common endvertex. A chord diagram $E$ is called nonintersecting if $E$ contains no crossing. For a chord diagram $E$ having a crossing $S=\{x_1{x}_3,x_2{x}_4\}$, the expansion of $E$ with respect to $S$ is to replace $E$ with $E_1=(E?S)\cup \{x_2{x}_3,x_4{x}_1\}$ or $E_2=(E?S)\cup \{x_1{x}_2,x_3{x}_4\}$. For a chord diagram $E$, let $f\left(E\right)$ be the chord expansion number of $E$, which is defined as the cardinality of the multiset of all nonintersecting chord diagrams generated from $E$ with a finite sequence of expansions.In this paper, it is shown that the chord expansion number $f\left(E\right)$ equals the value of the Tutte polynomial at the point $(2,?1)$ for the interlace graph $G_E$ corresponding to $E$. The chord expansion number of a complete multipartite chord diagram is also studied. An extended abstract of the paper was published (Nakamigawa and Sakuma, 2017) [13].     

Infinite Fujita exponents for nonlinear diffusion equations with localized sources

This paper deals with Cauchy problem to nonlinear diffusion $u_t=\Delta u^m+{\lambda }_1{u}^{p_1}(x,t)+{\lambda }_2{u}^{p_2}(x^1\left(t\right),t)$ with $m\ge 1$, $p_i,{\lambda }_i\ge 0$ ($i=1,2$) and $x^1\left(t\right)$ Hölder continuous. A new phenomenon is observed that the critical Fujita exponent $p_c=+\infty$ whenever ${\lambda }_2>0$. More precisely, the solution blows up under any nontrivial and nonnegative initial data for all $p=max\{p_1,p_2\}\in (1,+\infty )$. This result is then extended to a coupled system with localized sources as well as the cases with other nonlinearities.     

Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $b(G_1,G_2,\dots ,G_k)$ is the least positive integer $b$ so that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a copy of $G_i$ in the $i$th color for some $i$. In this paper, our main focus will be to bound the following numbers: $b(C_{2t_1},C_{2t_2})$ and $b(C_{2t_1},C_{2t_2},C_{2t_3})$ for all $t_i\ge 3,$$b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4})$ for $3\le t_i\le 9,$ and $b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4},C_{2t_5})$ for $3\le t_i\le 5.$ Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.     

Soliton solutions of a coupled Ramani equation

In this work, multi-soliton solutions of the coupled Ramani equations$u_{xxxxxx}+15u_{xx}{u}_{xxx}+15u_x{u}_{xxxx}+45u_x^2{u}_{xx}-5(u_{xxxt}+3u_{xx}{u}_t+3u_x{u}_{xt})-5u_{tt}+18w_x=0\text{,}$$w_t-w_{xxx}-3w_x{u}_x-3w{u}_{xx}=0\text{,}$ are derived and expressed using Pfaffians in a compact form.     

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.