A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization

The Keller–Segel–Navier–Stokes system

(?)$\{\begin{array}{ccc}\hfill n_t+u?\mathrm{?}n& \hfill =\hfill & \mathrm{\Delta }n?\chi \mathrm{?}?(n\mathrm{?}c)+\rho n?\mu n^2,\hfill \\ \hfill c_t+u?\mathrm{?}c& \hfill =\hfill & \mathrm{\Delta }c?c+n,\hfill \\ \hfill u_t+(u?\mathrm{?})u& \hfill =\hfill & \mathrm{\Delta }u+\mathrm{?}P+n\mathrm{?}?+f(x,t),\hfill \\ \hfill \mathrm{?}?u& \hfill =\hfill & 0,\hfill \end{array}$

is considered in a bounded convex domain $\mathrm{\Omega }?{\mathbb{R}}^3$ with smooth boundary, where $?\in W^{1,\infty }(\mathrm{\Omega })$ and $f\in C^1(\overline{\mathrm{\Omega }}\times [0,\infty ))$, and where $\chi >0,\rho \in \mathbb{R}$ and $\mu >0$ are given parameters.It is proved that under the assumption that ${\sup }_{t>0}?{\int }_t^{t+1}{6f(?,s)6}_{L^{\frac{6}{5}}(\mathrm{\Omega })}ds$ be finite, for any sufficiently regular initial data $(n_0,c_0,u_0)$ satisfying $n_0\ge 0$ and $c_0\ge 0$, the initial-value problem for (?) under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u possesses at least one globally defined solution in an appropriate generalized sense, and that this solution is uniformly bounded in with respect to the norm in $L^1(\mathrm{\Omega })\times L^6(\mathrm{\Omega })\times L^2(\mathrm{\Omega };{\mathbb{R}}^3)$.Moreover, under the explicit hypothesis that $\mu >\frac{\chi \sqrt{{\rho }_{+}}}{4}$, these solutions are shown to stabilize toward a spatially homogeneous state in their first two components by satisfying

$\begin{array}{c}(n(?,t),c(?,t))\to (\frac{{\rho }_{+}}{\mu },\frac{{\rho }_{+}}{\mu })\text{in }{L}^1(\mathrm{\Omega })\times L^p(\mathrm{\Omega })\hfill \\ \text{for all }p\in [1,6)\text{as }t\to \infty .\hfill \end{array}$

Finally, under an additional condition on temporal decay of f it is shown that also the third solution component equilibrates in that $u(?,t)\to 0$ in $L^2(\mathrm{\Omega };{\mathbb{R}}^3)$ as $t\to \infty$.     

Blow-up phenomena for the Liouville equation with a singular source of integer multiplicity

We are concerned with the existence of blowing-up solutions to the following boundary value problem

$?\mathrm{\Delta }u=\lambda a(x)e^u?4\pi N{\delta }_0\text{ in }\mathrm{\Omega },u=0\text{ on }?\mathrm{\Omega },$

where Ω is a smooth and bounded domain in ${\mathbb{R}}^2$ such that $0\in \mathrm{\Omega }$, $a(x)$ is a positive smooth function, N is a positive integer and $\lambda >0$ is a small parameter. Here ${\delta }_0$ defines the Dirac measure with pole at 0. We find conditions on the function a and on the domain Ω under which there exists a solution $u_{\lambda }$ blowing up at 0 and satisfying $\lambda {\int }_{\mathrm{\Omega }}a(x)e^{u_{\lambda }}\to 8\pi (N+1)$ as $\lambda \to 0^{+}$.     

Existence of solutions of a non-linear eigenvalue problem with a variable weight

We study the non-linear minimization problem on $H_0^1(\mathrm{\Omega })?L^q$ with $q=\frac{2n}{n?2}$, $\alpha >0$ and $n\ge 4$:

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }a(x,u)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2$

where $a(x,s)$ presents a global minimum α at $(x_0,0)$ with $x_0\in \mathrm{\Omega }$. In order to describe the concentration of $u(x)$ around $x_0$, one needs to calibrate the behavior of $a(x,s)$ with respect to s. The model case is

$\underset{\begin{array}{c}\hfill u\in H_0^1(\mathrm{\Omega })\hfill \\ \hfill {6u6}_{L^q}=1\hfill \end{array}}{\inf }?\underset{\mathrm{\Omega }}{\int }(\alpha +|x{|}^{\beta }|u{|}^k)|\mathrm{?}u{|}^2?\lambda \underset{\mathrm{\Omega }}{\int }|u{|}^2.$

In a previous paper dedicated to the same problem with $\lambda =0$, we showed that minimizers exist only in the range $\beta <kn/q$, which corresponds to a dominant non-linear term. On the contrary, the linear influence for $\beta \ge kn/q$ prevented their existence. The goal of this present paper is to show that for $0<\lambda \le \alpha {\lambda }_1(\mathrm{\Omega })$, $0\le k\le q?2$ and $\beta >kn/q+2$, minimizers do exist.     

Large time behavior of solutions to a fully parabolic chemotaxis–haptotaxis model in N dimensions

This paper deals with the Neumann problem for a fully parabolic chemotaxis–haptotaxis model of cancer invasion given by

$\{\begin{array}{c}{u}_t=\mathrm{\Delta }u?\chi \mathrm{?}?(u\mathrm{?}v)?\xi \mathrm{?}?(u\mathrm{?}w)+u(a?\mu u^{r?1}?\lambda w),x\in \mathrm{\Omega },t>0,\hfill \\ \tau v_t=\mathrm{\Delta }v?v+u,x\in \mathrm{\Omega },t>0,\hfill \\ w_t=?vw,x\in \mathrm{\Omega },t>0.\hfill \end{array}$

Here, $\mathrm{\Omega }?{\mathbb{R}}^N(N\ge 1)$ is a bounded domain with smooth boundary and $\tau >0,r>1,\lambda \ge 0$, $a\in \mathbb{R}$, $\mu ,\xi$ and χ are positive constants. It is shown that the corresponding initial–boundary value problem possesses a unique global bounded classical solution in the cases $r>2$ or $r=2$, with $\mu >{\mu }^{?}=\frac{{(N?2)}_{+}}{N}(\chi +C_{\beta })C_{\frac{N}{2}+1}^{\frac{1}{\frac{N}{2}+1}}$ for some positive constants $C_{\beta }$ and $C_{\frac{N}{2}+1}$. Furthermore, the large time behavior of solutions to the problem is also investigated. Specially speaking, when a is appropriately large, the corresponding solution of the system exponentially decays to $({(\frac{a}{\mu })}^{\frac{1}{r?1}},{(\frac{a}{\mu })}^{\frac{1}{r?1}},0)$ if μ is large enough. This result improves or extends previous results of several authors.     

Global well-posedness and asymptotic stabilization for chemotaxis system with signal-dependent sensitivity

A fully parabolic chemotaxis system

$u_t=\mathrm{\Delta }u?\mathrm{?}?(u\chi (v)\mathrm{?}v),v_t=\mathrm{\Delta }v?v+u,$

in a smooth bounded domain $\mathrm{\Omega }?{\mathbb{R}}^N$, $N\ge 2$ with homogeneous Neumann boundary conditions is considered, where the non-negative chemotactic sensitivity function χ satisfies $\chi (v)\le \mu {(a+v)}^{?k}$, for some $a\ge 0$ and $k\ge 1$. It is shown that a novel type of weight function can be applied to a weighted energy estimate for $k>1$. Consequently, the range of μ for the global existence and uniform boundedness of classical solutions established by Mizukami and Yokota [23] is enlarged. Moreover, under a convexity assumption on Ω, an asymptotic Lyapunov functional is obtained and used to establish the asymptotic stability of spatially homogeneous equilibrium solutions for $k\ge 1$ under a smallness assumption on μ. In particular, when $\chi (v)=\mu /v$ and $N<8$, it is shown that the spatially homogeneous steady state is a global attractor whenever $\mu \le 1/2$.     

Generalized rainbow Turán numbers of odd cycles

Given graphs F and H, the generalized rainbow Turán number $\mathrm{ex}(n,F,\mathrm{rainbow-}H)$ is the maximum number of copies of F in an n-vertex graph with a proper edge-coloring that contains no rainbow copy of H. B. Janzer determined the order of magnitude of $\mathrm{ex}(n,C_s,\mathrm{rainbow-}{C}_t)$ for all $s\ge 4$ and $t\ge 3$, and a recent result of O. Janzer implied that $\mathrm{ex}(n,C_3,\mathrm{rainbow-}{C}_{2k})=O(n^{1+1/k})$. We prove the corresponding upper bound for the remaining cases, showing that $\mathrm{ex}(n,C_3,\mathrm{rainbow-}{C}_{2k+1})=O(n^{1+1/k})$. This matches the known lower bound for k even and is conjectured to be tight for k odd.     

Arithmetic invariants from Sato–Tate moments

We give some arithmetic-geometric interpretations of the moments ${\mathrm{M}}_2[a_1]$, ${\mathrm{M}}_1[a_2]$, and ${\mathrm{M}}_1[s_2]$ of the Sato–Tate group of an abelian variety A defined over a number field by relating them to the ranks of the endomorphism ring and Néron–Severi group of A.     

Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation

The system$\{\begin{array}{c}{u}_t=\mathrm{\Delta }u?\chi \mathrm{?}?\left(\frac{u}{v}\mathrm{?}v\right)?uv+B_1(x,t),\hfill \\ v_t=\mathrm{\Delta }v+uv?v+B_2(x,t),\hfill \end{array}(?)$ is considered in a disk $\mathrm{\Omega }?{\mathbb{R}}^2$, with a positive parameter χ and given nonnegative and suitably regular functions $B_1$ and $B_2$ defined on $\mathrm{\Omega }\times (0,\infty )$. In the particular version obtained when $\chi =2$, (?) was proposed in [31] as a model for crime propagation in urban regions.Within a suitable generalized framework, it is shown that under mild assumptions on the parameter functions and the initial data the no-flux initial-boundary value problem for (?) possesses at least one global solution in the case when all model ingredients are radially symmetric with respect to the center of Ω. Moreover, under an additional hypothesis on stabilization of the given external source terms in both equations, these solutions are shown to approach the solution of an elliptic boundary value problem in an appropriate sense.The analysis is based on deriving a priori estimates for a family of approximate problems, in a first step achieving some spatially global but weak initial regularity information which in a series of spatially localized arguments is thereafter successively improved.To the best of our knowledge, this is the first result on global existence of solutions to the two-dimensional version of the full original system (?) for arbitrarily large values of χ.     

On the concentration phenomenon of L2-subcritical constrained minimizers for a class of Kirchhoff equations with potentials

In this paper, we study the existence and concentration behavior of minimizers for $i_V(c)=\underset{u\in S_c}{\inf }?I_V(u)$, here $S_c=\{u\in H^1({\mathbb{R}}^N)|{\int }_{{\mathbb{R}}^N}V(x)|u{|}^2<+\infty ,|u{|}_2=c>0\}$ and

$I_V(u)=\frac{1}{2}\underset{{\mathbb{R}}^N}{\int }(a|\mathrm{?}u{|}^2+V(x)|u{|}^2)+\frac{b}{4}{(\underset{{\mathbb{R}}^N}{\int }|\mathrm{?}u{|}^2)}^2?\frac{1}{p}\underset{{\mathbb{R}}^N}{\int }|u{|}^p,$

where $N=1,2,3$ and $a,b>0$ are constants. By the Gagliardo–Nirenberg inequality, we get the sharp existence of global constraint minimizers of $i_V(c)$ for $2<p<2^{?}$ when $V(x)\ge 0$, $V(x)\in {L^{\infty }}_{loc}({\mathbb{R}}^N)$ and $\underset{|x|\to +\infty }{\lim }?V(x)=+\infty$. For the case $p\in (2,\frac{2N+8}{N})\backslash \{4\}$, we prove that the global constraint minimizers $u_c$ of $i_V(c)$ behave like

$u_c(x)\approx \frac{c}{|Q_p{|}_2}{\left(\frac{m_c}{c}\right)}^{\frac{N}{2}}{Q}_p(\frac{m_c}{c}x?z_c),$

for some $z_c\in {\mathbb{R}}^N$ when c is large, where $Q_p$ is, up to translations, the unique positive solution of $?\frac{N(p?2)}{4}\mathrm{\Delta }{Q}_p+\frac{2N?p(N?2)}{4}{Q}_p=|Q_p{|}^{p?2}{Q}_p$ in ${\mathbb{R}}^N$ and $m_c={(\frac{\sqrt{a^2{D}_1^2?4b{D}_2{i}_0(c)}+a{D}_1}{2b{D}_2})}^{\frac{1}{2}}$, $D_1=\frac{Np?2N?4}{2N(p?2)}$ and $D_2=\frac{2N+8?Np}{4N(p?2)}$.     

Concentrating standing waves for the Gross–Pitaevskii equation in trapped dipolar quantum gases

We are concerned with the following singularly perturbed Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases:

$\{\begin{array}{cc}\hfill & ?{\epsilon }^2\mathrm{\Delta }u+V(x)u+{\lambda }_1|u{|}^{2}u+{\lambda }_2(K?|u{|}^2)u=0\text{ in }{\mathbb{R}}^3,\hfill \\ \hfill & u>0,u\in H^1({\mathbb{R}}^3),\hfill \end{array}$

where ε is a small positive parameter, ${\lambda }_1,{\lambda }_2\in \mathbb{R}$, ? denotes the convolution, $K(x)=\frac{1?3{\cos }^2?\theta }{|x{|}^3}$ and $\theta =\theta (x)$ is the angle between the dipole axis determined by $(0,0,1)$ and the vector x. Under certain assumptions on $({\lambda }_1,{\lambda }_2)\in {\mathbb{R}}^2$, we construct a family of positive solutions $u_{\epsilon }\in H^1({\mathbb{R}}^3)$ which concentrates around the local minima of V as $\epsilon \to 0$. Our main results extend the results in J. Byeon and L. Jeanjean (2007) [6], which dealt with singularly perturbed Schrödinger equations with a local nonlinearity, to the nonlocal Gross–Pitaevskii type equation.     

On the existence of weak solutions of semilinear elliptic equations and systems with Hardy potentials

Let $\mathrm{\Omega }?{\mathbb{R}}^N$ ($N\ge 3$) be a bounded $C^2$ domain and $\delta (x)=\mathrm{dist}(x,?\mathrm{\Omega })$. Put $L_{\mu }=\mathrm{\Delta }+\frac{\mu }{{\delta }^2}$ with $\mu >0$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to

$?L_{\mu }u=u^p+\tau \text{in }\mathrm{\Omega },u=\nu \text{on }?\mathrm{\Omega },$

where $\mu >0$, $p>0$, τ and ν are measures on Ω and ?Ω respectively. We then establish existence results for the system

$\{\begin{array}{cc}\hfill & ?L_{\mu }u=?v^p+\tau \text{in }\mathrm{\Omega },\hfill \\ \hfill & ?L_{\mu }v=?u^{\stackrel{?}{p}}+\stackrel{?}{\tau }\text{in }\mathrm{\Omega },\hfill \\ \hfill & u=\nu ,v=\stackrel{?}{\nu }\text{on }?\mathrm{\Omega },\hfill \end{array}$

where $?=\pm 1$, $p>0$, $\stackrel{?}{p}>0$, τ and $\stackrel{?}{\tau }$ are measures on Ω, ν and $\stackrel{?}{\nu }$ are measures on ?Ω. We also deal with elliptic systems where the nonlinearities are more general.