Time-dependent global attractor for extensible Berger equation

In this paper, we investigate the asymptotic behavior of the nonautonomous Berger equation

$\epsilon (t)u_{tt}+{\mathrm{\Delta }}^{2}u?(Q+\underset{\mathrm{\Omega }}{\int }|\mathrm{?}u{|}^{2}dx)\mathrm{\Delta }u+g(u_t)+\phi (u)=f,t>\tau ,$

on a bounded smooth domain $\mathrm{\Omega }?{\mathbb{R}}^N$ with hinged boundary condition, where $\epsilon (t)$ is a decreasing function vanishing at infinity. Under suitable assumptions, we establish an invariant time-dependent global attractor within the theory of process on time-dependent space.     

Long-time behavior for 3D MHD equations with nonlinear damping

The existence of global attractors is proved for the MHD equations with damping terms $|u{|}^{\alpha ?1}u$ and $|B{|}^{\beta ?1}B$ $(\alpha ,\beta ?1)$ on a bounded domain $\mathrm{\Omega }?{\mathbb{R}}^3$. First we establish the well-posedness of strong solutions. Then, the continuity of the corresponding semigroup is verified under the assumption $\alpha ,\beta <5$, which is guided by Gagliardo-Nirenberg inequality. Finally, the system is shown to possess an $(\mathbb{V},\mathbb{V})$-global attractor and an $(\mathbb{V},{\mathbf{\text{H}}}^2)$-global attractor.     

Existence of nontrivial solutions for modified nonlinear fourth-order elliptic equations with indefinite potential

In this paper, we investigate the following modified nonlinear fourth-order elliptic equations$\{\begin{array}{cc}{\mathrm{\Delta }}^{2}u?\mathrm{\Delta }u+V(x)u?\frac{1}{2}u\mathrm{\Delta }(u^2)=g(u),\hfill & \text{in}{\mathbb{R}}^N,\hfill \\ u\in H^2({\mathbb{R}}^N)\hfill \end{array}$ where ${\mathrm{\Delta }}^2=\mathrm{\Delta }(\mathrm{\Delta })$ is the biharmonic operator, V is an indefinite potential, g grows subcritically and satisfies the Ambrosetti-Rabinowitz type condition $g(t)t\ge \mu G(t)\ge 0$ with $\mu >3$. Using Morse theory, we obtain nontrivial solutions of the above equations. Our result complements recent results in [17], where g has to be 3-superlinear at infinity.     

Long time behavior of solutions to the 3D Hall-magneto-hydrodynamics system with one diffusion

This paper studies the asymptotic behavior of smooth solutions to the generalized Hall-magneto-hydrodynamics system (1.1) with one single diffusion on the whole space ${\mathbb{R}}^3$. We establish that, in the inviscid resistive case, the energy ${6b(t)6}_2^2$ vanishes and ${6u(t)6}_2^2$ converges to a constant as time tends to infinity provided the velocity is bounded in $W^{1?\alpha ,\frac{3}{\alpha }}({\mathbb{R}}^3)$; in the viscous non-resistive case, the energy ${6u(t)6}_2^2$ vanishes and ${6b(t)6}_2^2$ converges to a constant provided the magnetic field is bounded in $W^{1?\beta ,\infty }({\mathbb{R}}^3)$. In summary, one single diffusion, being as weak as ${(?\mathrm{\Delta })}^{\alpha }b$ or ${(?\mathrm{\Delta })}^{\beta }u$ with small enough $\alpha ,\beta$, is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system.     

On bifurcation of eigenvalues along convex symplectic paths

We consider a continuously differentiable curve $t?\gamma (t)$ in the space of $2n\times 2n$ real symplectic matrices, which is the solution of the following ODE:

$\frac{\mathrm{d}\gamma }{\mathrm{d}t}(t)=J_{2n}A(t)\gamma (t),\gamma (0)\in \mathrm{Sp}(2n,\mathbb{R}),$

where $J=J_{2n}\stackrel{\text{def}}{=}\left[\begin{array}{cc}\hfill 0\hfill & \hfill {\mathrm{Id}}_n\hfill \\ \hfill ?{\mathrm{Id}}_n\hfill & \hfill 0\hfill \end{array}\right]$ and $A:t?A(t)$ is a continuous path in the space of $2n\times 2n$ real matrices which are symmetric. Under a certain convexity assumption (which includes the particular case that $A(t)$ is strictly positive definite for all $t\in \mathbb{R}$), we investigate the dynamics of the eigenvalues of $\gamma (t)$ when t varies, which are closely related to the stability of such Hamiltonian dynamical systems. We rigorously prove the qualitative behavior of the branching of eigenvalues and explicitly give the first order asymptotics of the eigenvalues. This generalizes classical Krein–Lyubarskii theorem on the analytic bifurcation of the Floquet multipliers under a linear perturbation of the Hamiltonian. As a corollary, we give a rigorous proof of the following statement of Ekeland: $\{t\in \mathbb{R}:\gamma (t)\text{ has a Krein indefinite eigenvalue of modulus }1\}$ is a discrete set.     

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.     

Positive solutions for a class of singular quasilinear Schrödinger equations with critical Sobolev exponent

We prove the existence of positive solutions of the following singular quasilinear Schrödinger equations at critical growth

$?\mathrm{\Delta }u?\lambda c(x)u?\kappa \alpha (\mathrm{\Delta }(|u{|}^{2\alpha }))|u{|}^{2\alpha ?2}u=|u{|}^{q?2}u+|u{|}^{2^{?}?2}u,u\in D^{1,2}({\mathbb{R}}^N),$

via variational methods, where $\lambda \ge 0$, $c:{\mathbb{R}}^N\to {\mathbb{R}}^{+}$, $\kappa >0$, $0<\alpha <1/2$, $2<q<2^{?}$. It is interesting that we do not need to add a weight function to control $|u{|}^{q?2}u$.     

Stabilization in three-dimensional chemotaxis-growth model with indirect attractant production

This paper deals with the chemotaxis-growth system: $u_t=\mathrm{\Delta }u?\mathrm{?}?(u\mathrm{?}v)+\mu u(1?u)$, $v_t=\mathrm{\Delta }v?v+w$, $\tau w_t+\delta w=u$ in a smooth bounded domain $\mathrm{\Omega }?{\mathbb{R}}^3$ with zero-flux boundary conditions, where μ, δ, and τ are given positive parameters. It is shown that the solution $(u,v,w)$ exponentially stabilizes to the constant stationary solution $(1,\frac{1}{\delta },\frac{1}{\delta })$ in the norm of $L^{\infty }(\mathrm{\Omega })$ as $t\to \infty$ provided that $\mu >0$ and any given nonnegative and suitably smooth initial data $(u_0,v_0,w_0)$ fulfills $u_0?0$, which extends the condition $\mu >\frac{1}{8{\delta }^2}$ in [8].     

Entropy solutions for stochastic porous media equations

We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well-posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators $\mathrm{\Delta }(|u{|}^{m?1}u)$ for all $m\in (1,\infty )$, and Hölder continuous diffusion nonlinearity with exponent 1/2.     

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 $t\to \infty$.     

Well posedness of nonlinear parabolic systems beyond duality

We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial–boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem accessible to our technique is the following system${?}_{t}u?\mathrm{div}(\nu (|\mathrm{?}u|)\mathrm{?}u)=?\mathrm{div}f$ with a given strictly positive bounded function ν, such that ${\lim }_{k\to \infty }?\nu (k)={\nu }_{\infty }$ and $f\in L^q$ with $q\in (1,\infty )$. The existence, uniqueness and regularity results for $q\ge 2$ are by now standard. However, even if a priori estimates are available, the existence in case $q\in (1,2)$ was essentially missing. We overcome the related crucial difficulty, namely the lack of a standard duality pairing, by resorting to proper weighted spaces and consequently provide existence, uniqueness and optimal regularity in the entire range $q\in (1,\infty )$.Furthermore, our paper includes several new results that may be of independent interest and serve as the starting point for further analysis of more complicated problems. They include a parabolic Lipschitz approximation method in weighted spaces with fine control of the time derivative and a theory for linear parabolic systems with right hand sides belonging to Muckenhoupt weighted $L^q$ spaces.     

Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension

This paper deals with positive solutions of the fully parabolic system

$\{\begin{array}{cc}{u}_t=\mathrm{\Delta }u?\chi \mathrm{?}?(u\mathrm{?}v)\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty ),\hfill \\ {\tau }_1{v}_t=\mathrm{\Delta }v?v+w\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty ),\hfill \\ {\tau }_2{w}_t=\mathrm{\Delta }w?w+u\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty )\hfill \end{array}$

under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain $\mathrm{\Omega }?{\mathbb{R}}^4$ with positive parameters ${\tau }_1,{\tau }_2,\chi >0$ and nonnegative smooth initial data $(u_0,v_0,w_0)$.Global existence and boundedness of solutions were shown if ${6u_{0}6}_{L^1(\mathrm{\Omega })}<{(8\pi )}^2/\chi$ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ${6u_{0}6}_{L^1(\mathrm{\Omega })}>{(8\pi )}^2/\chi$. This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has $8\pi /\chi$-dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in ${\mathbb{R}}^4$.     

SLE intersecting with random hulls

Lawler, Schramm, and Werner gave in 2003 an explicit formula of the probability that $\mathrm{SLE}(8/3)$ does not intersect a deterministic hull. For general $\mathrm{SLE}(\kappa )$ with $\kappa \ne 8/3$, no such explicit formula has been obtained so far. In this paper, we shall consider a random hull generated by an independent chordal conformal restriction measure and obtain an explicit formula for the probability that $\mathrm{SLE}(\kappa )$ does not intersect this random hull for any $\kappa \in (0,8)$. As a corollary, we will give a new proof of Werner's result on conformal restriction measures.     

Compactness of sign-changing solutions to scalar curvature-type equations with bounded negative part

We consider the equation ${\mathrm{\Delta }}_{g}u+hu=|u{|}^{2^{?}?2}u$ in a closed Riemannian manifold $(M,g)$, where $h\in C^{0,\theta }(M)$, $\theta \in (0,1)$ and $2^{?}=\frac{2n}{n?2}$, $n:=\dim ?(M)\ge 3$. We obtain a sharp compactness result on the sets of sign-changing solutions whose negative part is a priori bounded. We obtain this result under the conditions that $n\ge 7$ and $h<\frac{n?2}{4(n?1)}{\text{Scal}}_g$ in M, where ${\text{Scal}}_g$ is the Scalar curvature of the manifold. We show that these conditions are optimal by constructing examples of blowing-up solutions, with arbitrarily large energy, in the case of the round sphere with a constant potential function h.     

Point partition numbers: Decomposable and indecomposable critical graphs

Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number ${\chi }_t(G)$ is the least integer k for which G admits a coloring with k colors such that each color class induces a $(t?1)$-degenerate subgraph of G. So ${\chi }_1$ is the chromatic number and ${\chi }_2$ is the point arboricity. The point partition number ${\chi }_t$ with $t\ge 1$ was introduced by Lick and White. A graph G is called ${\chi }_t$-critical if every proper subgraph H of G satisfies ${\chi }_t(H)<{\chi }_t(G)$. In this paper we prove that if G is a ${\chi }_t$-critical graph whose order satisfies $|G|\le 2{\chi }_t(G)?2$, then G can be obtained from two non-empty disjoint subgraphs $G_1$ and $G_2$ by adding t edges between any pair $u,v$ of vertices with $u\in V(G_1)$ and $v\in V(G_2)$. Based on this result we establish the minimum number of edges possible in a ${\chi }_t$-critical graph G of order n and with ${\chi }_t(G)=k$, provided that $n\le 2k?1$ and t is even. For $t=1$ the corresponding two results were obtained in 1963 by Tibor Gallai.