This paper is devoted to estimates on weighted $L^q$-norms of the nonstationary 3D Navier-Stokes flow in an exterior domain. By multiplying the Navier-Stokes equation with a well selected vector field, an integral equation is derived, from which, we establish the weighted estimate $∥|x|^αu(t)∥_q ≤ C(1+t^{\frac{α}{2} +ε} )t^{-\frac{3}{2}(1-\frac{1}{q})} ,$$t>0,$ where $ 0<α≤1$ and $\frac{3}{2}<q<∞,$ or $1<α<2$ and $\frac{3}{3−α }<q<∞,$ $0<ε<1$ is arbitrary, and $u_0∈L^3_σ(Ω),$ $|x|^αu_0 ∈L^1(Ω)$ with $∥u_0∥_3$ sufficiently small. With the aid of the representation of the flow, we also prove that if in addition $u_0∈D^{1−1/b,b} _a$  for some $\frac{6}{5} \le a<\frac{3}{2}$ and $1<b<2$ with $\frac{3}{a}+\frac{2}{b} =4,$ then the optimal estimate $∥|x|^αu(t)∥_q≤C(1+t^{\frac{\alpha}{2}})  t^{-\frac{3}{2}(1-\frac{1}{q})},$ $t>0$ holds, where $α>0$ and $1<q<∞.$ Compared with the literature, here no extra restriction is laid on the range of the exponents $α$ and $q.$

In this paper, we study the asymptotic behavior of solutions to a quasilinear two-species chemotaxis system with nonlinear sensitivity and nonlinear signal production

under homogeneous Neumann boundary conditions in a smooth bounded domain $Ω⊂\mathbb{R}^n(n≥2)$ where the parameter $\mu_1,\mu_2,γ_1,γ_2$ are positive constants, $τ ∈ \{0,1\}.$ The diffusion coefficients $D_i,Si ∈ C^2([0,∞))$ satisfy $D_i(s)≥a_0(s+1)^{−m_i},$ $0≤S_i(s)≤ b_0s(s+1)^{β_i−1},$ $s≥0,$ $m_i,β_i ∈\mathbb{R},$ $a_0,b_0 >0,$ $i=1,2.$ Under the assumption of properly initial data regularity, we can find appropriate $\mu_i$ such that the globally bounded solution of this system satisfies the following relationship.
(I) If $a_1,a_2∈(0,1)$ and $\mu_1$ and $\mu_2$ are sufficiently large, then any globally bounded solution exponentially converges to $(\frac{1-a_1}{1-a_1a_2}, (\frac{1-a_2}{1-a_1a_2})^{\gamma_1}, \frac{1-a_2}{1-a_1a_2}, (\frac{1-a_1}{1-a_1a_2})^{\gamma_2})$  as $t→∞;$
(II) If $a_1 >1>a_2 >0$ and $\mu_2$ is sufficiently large, then any globally bounded solution exponentially converges to (0,1,1,0) as $t→∞;$
(III) If $a_1=1>a_2 >0$ and $\mu_2$ is sufficiently large, then any globally bounded solution algebraically converges to (0,1,1,0) as $t→∞.$

In this short note, we consider the Dirichlet problem associated to an even order elliptic system with antisymmetric first order potential. Given any continuous boundary data, we show that weak solutions are continuous up to boundary.

In this paper, we consider the Cauchy problem of $d$-dimensional $(d=2,3)$  tropical climate model without thermal diffusion and construct global smooth solutions by choosing a class of special initial data $(u_0,v_0,θ_0)$ whose $L^2$ norm can be arbitrarily large, and improve the previous results in [1,2].

This paper considers initial boundary value to a pseudo-parabolic equation with singular potential $\frac{u_t}{|x|^s}-\Delta u_t-\Delta u =|u|^{p-2}u$ with $2<p<\frac{2N}{N−2},$ which was studied in [1] by Lian et al. They dealt with the global existence, asymptotic behavior with low initial level $J(u_0)≤d$ and got the blow-up conditions of solutions with low and high initial level. In this paper, we give a new blow-up result which independent of the initial Nehari functional $I(u_0)$, and estimate the lower bound for blow-up time under some conditions. Finally, the precise exponential decay estimate is obtained for global solution with some conditions.

Recently, qualitative analysis of peaked solutions of Lane-Emden problem in dimension two has been widely considered. In particular, the Morse index of concentrated solutions with a single peak or multi peaks has been computed in [15,16] separately. In this paper, we continue to consider the qualitative properties of the eigenvalues and eigenfunctions of the linearized Lane-Emden problem associated to peak solutions. Here we establish the fine behaviors of the first $m$ eigenvalues and eigenfunctions of the linearized Lane-Emden problem in dimension two, and correspondingly the number of concentrated points of the first $m$ eigenfunctions are given.

In this paper, we are concerned about a food chain model with a protection zone for the prey species. Dynamical behavior, nonexistence and existence of positive steady states are obtained and there exist several critical values determined by the parameters and the protection zone for the growth rate of the prey species. The results reveal that the protection zone is effective for the survival of the prey species and beneficial for the coexistence of multiple species. Moreover, different properties of positive steady states from those of the two-species models are shown. The introduction of the prey or the top predator can be either favorable or unfavorable for the coexistence of multiple species.