Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant

This paper deals with a two-competing-species chemotaxis system with consumption of chemoattractant

$\{\begin{array}{cc}{u}_t=d_1\mathrm{\Delta }u?\mathrm{?}?(u{\chi }_1(w)\mathrm{?}w)+{\mu }_{1}u(1?u?a_{1}v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=d_2\mathrm{\Delta }v?\mathrm{?}?(v{\chi }_2(w)\mathrm{?}w)+{\mu }_{2}v(1?a_{2}u?v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=d_3\mathrm{\Delta }w?(\alpha u+\beta v)w,\hfill & x\in \mathrm{\Omega },t>0\hfill \end{array}$

under homogeneous Neumann boundary conditions in a bounded domain $\mathrm{\Omega }?{\mathbb{R}}^n$ ($n\ge 1$) with smooth boundary, where the initial data $(u_0,v_0)\in {(C^0(\stackrel{￣}{\mathrm{\Omega }}))}^2$ and $w_0\in W^{1,\infty }(\mathrm{\Omega })$ are non-negative and the parameters $d_1,d_2,d_3>0$, ${\mu }_1,{\mu }_2>0$, $a_1,a_2>0$ and $\alpha ,\beta >0$. The chemotactic function ${\chi }_i(w)$ ($i=1,2$) is smooth and satisfying some conditions. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution if one of the following cases hold: for $i=1,2$,(i) ${\chi }_i(w)={\chi }_{0,i}>0$ and

${6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}<\frac{\pi \sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}?\frac{2\sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}\arctan ?\frac{d_i?d_3}{2}\sqrt{\frac{n+1}{d_i{d}_3}};$

(ii) $0<{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}\le \frac{d_3}{3(n+1){6{\chi }_{i}6}_{L^{\infty }[0,{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}]}}\min ?\{\frac{2d_i}{d_i+d_3},1\}$.Moreover, we prove asymptotic stabilization of solutions in the sense that:? If $a_1,a_2\in (0,1)$ and $u_0\ne 0\ne v_0$, then any global bounded solution exponentially converge to $(\frac{1?a_1}{1?a_1{a}_2},\frac{1?a_2}{1?a_1{a}_2},0)$ as $t\to \infty$;? If $a_1>1>a_2>0$ and $v_0\ne 0$, then any global bounded solution exponentially converge to $(0,1,0)$ as $t\to \infty$;? If $a_1=1>a_2>0$ and $v_0\ne 0$, then any global bounded solution algebraically converge to $(0,1,0)$ as $t\to \infty$.     

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.     

Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus

Let $L=?/?t+{\sum }_{j=1}^N(a_j+i{b}_j)(t)?/?x_j$ be a vector field defined on the torus ${\mathbb{T}}^{N+1}?{\mathbb{R}}^{N+1}/2\pi {\mathbb{Z}}^{N+1}$, where $a_j$, $b_j$ are real-valued functions and belonging to the Gevrey class $G^s({\mathbb{T}}^1)$, $s>1$, for $j=1,\dots ,N$. We present a complete characterization for the s-global solvability and s-global hypoellipticity of L. Our results are linked to Diophantine properties of the coefficients and, also, connectedness of certain sublevel sets.     

A scheme for interpolation by Hankel translates of a basis function

This work discusses interpolation of complex-valued functions defined on the positive real axis $I$ by certain special subspaces, in a variational setting that follows the approach of Light and Wayne [W. Light, H. Wayne, Spaces of distributions, interpolation by translates of a basis function and error estimates, Numer. Math. 81 (1999) 415–450]. The set of interpolation points will be a subset $\{a_1,\dots ,a_n\}$ of $I$ and the interpolants will take the form $u\left(x\right)=\sum _{i=1}^n{\alpha }_i\left({\tau }_{a_i}?\right)\left(x\right)+\sum _{j=0}^{m?1}{\beta }_j{p}_{\mu ,j}\left(x\right)(x\in I)\text{,}$ where $\mu \ge ?1/2,?$ is a complex function defined on $I$ (the so-called basis function), $p_{\mu ,j}\left(x\right)=x^{2j+\mu +1/2}(j\in {\mathbb{Z}}_{+},0\le j\le m?1)$ is a Müntz monomial, ${\tau }_z(z\in I)$ denotes the Hankel translation operator of order $\mu$, and ${\alpha }_i,{\beta }_j(i,j\in {\mathbb{Z}}_{+},1\le i\le n,0\le j\le m?1)$ are complex coefficients. An estimate for the pointwise error of these interpolants is given. Some numerical examples are included.     

Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption

The chemotaxis system

$\{\begin{array}{c}{u}_t=\mathrm{\Delta }u?\mathrm{?}?\left(\frac{u}{v}\mathrm{?}v\right),\hfill \\ v_t=\mathrm{\Delta }v?uv,\hfill \end{array}(?)$

is considered under homogeneous Neumann boundary conditions in the ball $\mathrm{\Omega }=B_R(0)?{\mathbb{R}}^n$, where $R>0$ and $n\ge 2$.Despite its great relevance as a model for the spontaneous emergence of spatial structures in populations of primitive bacteria, since its introduction by Keller and Segel in 1971 this system has been lacking a satisfactory theory even at the level of the basic questions from the context of well-posedness; global existence results in the literature are restricted to spatially one- or two-dimensional cases so far, or alternatively require certain smallness hypotheses on the initial data.For all suitably regular and radially symmetric initial data $(u_0,v_0)$ satisfying $u_0\ge 0$ and $v_0>0$, the present paper establishes the existence of a globally defined pair $(u,v)$ of radially symmetric functions which are continuous in $(\overline{\mathrm{\Omega }}?\{0\})\times [0,\infty )$ and smooth in $(\overline{\mathrm{\Omega }}?\{0\})\times (0,\infty )$, and which solve the corresponding initial-boundary value problem for (?) with $(u(?,0),v(?,0))=(u_0,v_0)$ in an appropriate generalized sense. To the best of our knowledge, this in particular provides the first result on global existence for the three-dimensional version of (?) involving arbitrarily large initial data.     

Exponential stability for the wave model with localized memory in a past history framework

In this paper we discuss the asymptotic stability as well as the well-posedness of the damped wave equation posed on a bounded domain Ω of ${\mathbb{R}}^n,n\ge 2$,

$\rho (x)u_{tt}?\mathrm{\Delta }u+\underset{0}{\overset{\infty }{\int }}g(s)\text{div}[a(x)\mathrm{?}u(?,t?s)]ds+b(x)u_t=0,$

subject to a locally distributed viscoelastic effect driven by a nonnegative function $a(x)$ and supplemented with a frictional damping $b(x)\ge 0$ acting on a region A of Ω, where $a=0$ in A. Assuming that $\rho (x)$ is constant, considering that the well-known geometric control condition $(\omega ,T_0)$ holds and supposing that the relaxation function g is bounded by a function that decays exponentially to zero, we prove that the solutions to the corresponding partial viscoelastic model decay exponentially to zero, even in the absence of the frictional dissipative effect. In addition, in some suitable cases where the material density $\rho (x)$ is not constant, it is also possible to remove the frictional damping term $b(x)u_t$, that is, the localized viscoelastic damping is strong enough to assure that the system is exponentially stable. The semi-linear case is also considered.     

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 $\text{Rank}(D_y^{2}K)(y_0)\ge n?k+1$.     

On a Liouville-type comparison principle for solutions of semilinear elliptic partial differential inequalities

This Note is devoted to the study of a Liouville-type comparison principle for entire weak solutions of semilinear elliptic partial differential inequalities of the form $Lu+{|u|}^{q?1}u?Lv+{|v|}^{q?1}v$, where $q>0$ is a given number and L is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergent form given formally by the relation

$L=\sum _{i,j=1}^n\frac{?}{?x_i}[a_{ij}(x)\frac{?}{?x_j}].$

We assume that $n?2$, that the coefficients $a_{ij}(x)$, $i,j=1,\dots ,n$, are measurable bounded functions on ${\mathbb{R}}^n$ such that $a_{ij}(x)=a_{ji}(x)$, and that the corresponding quadratic form is non-negative. The results obtained in this work complete similar results on solutions of quasilinear elliptic partial differential inequalities announced in Kurta [C. R. Acad. Sci. Paris, Ser. I 336 (11) (2003) 897–900]. To cite this article: V.V. Kurta, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Ramsey numbers of odd cycles versus larger even wheels

The generalized Ramsey number $R(G_1,G_2)$ is the smallest positive integer $N$ such that any red–blue coloring of the edges of the complete graph $K_N$ either contains a red copy of $G_1$ or a blue copy of $G_2$. Let $C_m$ denote a cycle of length $m$ and $W_n$ denote a wheel with $n+1$ vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers $R(C_{2k+1},W_n)$ of odd cycles versus larger wheels, leaving open the particular case where $n=2j$ is even and $k<j<3k∕2$. They conjectured that for these values of $j$ and $k$, $R(C_{2k+1},W_{2j})=4j+1$. In 2015, Sanhueza-Matamala confirmed this conjecture asymptotically, showing that $R(C_{2k+1},W_{2j})\le 4j+334$. In this paper, we prove the conjecture of Zhang, Zhang and Chen for almost all of the remaining cases. In particular, we prove that $R(C_{2k+1},W_{2j})=4j+1$ if $j?k\ge 251$, $k<j<3k∕2$, and $j\ge 212299$.