In this paper, we investigate the global existence of solutions to the following consumptive Keller-Segel model with singular sensitivity and porous medium diffusion $$\begin{align*} \left\{ \begin{aligned} &u_t=\Delta u^m-\chi\nabla\cdot(\frac{u}{v^\beta}\nabla v), \\ &v_t=\Delta v-vu^{\alpha}. \end{aligned}\right. \end{align*}$$ In the two dimensional space, it is shown that for any $m>1$, $\beta<\frac{11+8\sqrt 2}{28}(\approx 0.797)$, $\alpha<m+3(m-1)$, there exists a locally bounded global weak solution for any positive initial datum, furthermore, the solution is uniformly bounded in the sense of $L^p$-norm for any $p>1$. In the three dimensional space, it is shown that for any $m>\frac{10}9$, $\beta<\frac{3+\sqrt 3}{6}(\approx 0.789)$, $\alpha<\min\{\frac{32}5(m-1), m+3( m-\frac{10}9)\}$, there exists a locally bounded global weak solution, and the weak solution is uniformly bounded in the sense of $L^p$-norm for any $1<p<9(m-1)$. In addition, for any such solution, we prove that $v$ goes to zero uniformly as $t\to\infty$. It is worth noting that the global existence conclusion of the solution in this paper does not require any smallness restrictions on the initial values and parameters, thus expanding the scope of applicability of existing studies that rely on small initial values or small parameters.

In this paper, we formulate a free boundary model which is based on a chemotaxis system, and then prove the local existence of weak solutions for this model using a semigroup argument. By studying the properties of the moving boundary, we show the finite-time blowup and chemotactic collapse for this system.

This article mainly investigates the initial boundary value problem of a chemotaxis-convection model pertaining to the growth of capillary sprouts during tumor angiogenesis $$\begin{equation*} \begin{cases} \frac{\partial u}{\partial t}=\Delta u-\chi_1\nabla\cdot(u\nabla v)+\chi_2\nabla\cdot(u\nabla w), &(x,t)\in\Omega\times(0, \infty),\\ \frac{\partial v}{\partial t}=\Delta v+\xi_1\nabla\cdot(v\nabla w) - v + u, &(x,t)\in\Omega\times(0, \infty),\\ \frac{\partial w}{\partial t}=\Delta w - w + u, &(x,t)\in\Omega\times(0, \infty), \end{cases} \end{equation*}$$ where $\Omega\subset\mathbb{R}^N(N\ge1)$ is a smoothly bounded domain with Neumman boundary condition. The parameters $\chi_1$, $\chi_2$ and $\xi_1>0$. We demonstrate that this model admits a globally bounded classical solution under the smallness condition on $\chi_1$ and $\chi_2$. Moreover, this solution will converge exponentially to its constant steady state $(\bar{u}_0,\bar{u}_0,\bar{u}_0)$ under the further smallness condition on $\xi_1$ and $\chi_2$, where $\bar{u}_0=\frac{1}{|\Omega|}\int_\Omega u \mathrm{d}x$.

In this paper, we extend the Bahri-Lions theorem (1988) to a class of semilinear problems associated with Dirichlet forms. By introducing a new min-max scheme based on the notion of relative genus, we construct novel critical point structures and establish corresponding estimates for the Morse index of the obtained solutions. The results provide a unified framework for treating variational problems arising from degenerate and non-uniformly elliptic equations, and are expected to have further applications in geometric analysis and the study of elliptic and degenerate elliptic partial differential equations.

This paper is mainly concerned with the time-asymptotic stability of stationary solutions to the outflow problem of the non-isentropic compressible Navier-Stokes-Allen-Cahn equations in the half space $\mathbb{R}^+$. The models can be used to describe the motion of a mixture of macroscopically immiscible two viscous compressible fluids. When the adiabatic exponent $\gamma$ is sufficiently close to $1$, we prove that the one-dimensional non-isentropic compressible Navier-Stokes-Allen-Cahn equations admits a unique global solution, which tends to the non-degenerate stationary solution as time goes to infinity. In this paper, the initial perturbations of temperature function and the phase field variable, and the strength of the stationary solution are required to be sufficiently small, but the initial perturbations of the density and velocity functions can be arbitrarily large. Our analysis is based on the basic $L^2$-energy method and some new techniques, which take into account the effect the phase field variable and the stationary solution.

This paper investigates the Dirichlet eigenvalue problem for the 2D Grushin operator $\triangle_X=\partial_{x_{1}}^2+x_{1}^2\partial_{x_{2}}^2$ on a bounded open set $\Omega$ in $\mathbb{R}^2$. The Grushin operator is an important class of Hörmander operators in the non-equiregular case, where the 2D Lebesgue measure of its singular degenerate set $H$ is zero ($|H|=0$), making Métivier's asymptotic formula no longer applicable. By utilizing the explicit expression of the global heat kernel and refined estimates for the error term of the Dirichlet heat kernel, we establish a Weyl asymptotic law with a logarithmic term: $\lambda_k \sim \frac{4\pi}{s_{\Omega}(0)} \frac{k}{\ln k}$. Furthermore, we show that $s_{\Omega}(0)$, the 1D Lebesgue measure of the projection of the singular degenerate set $H$ onto the $x_2$-axis, is a geometric spectral invariant characterizing the asymptotics of the Dirichlet eigenvalues for this operator.

This paper mainly introduces the multiplicity conjecture and Hofer-Wysocki-Zehnder conjecture for closed characteristics on compact star-shaped hypersurfaces in ${\mathbb{R}}^{2n}$, and some recent related progresses on them, furthermore we introduce the corresponding closed orbit problems for contact manifold and explain the research methods therein, which involve Variational Method and Morse Theory、Dynamical System、Symplectic Geometry, and so on.

This paper concerns a class of non-divergence non-linear elliptic equations driven by the cone degenerate Laplacian motivated by cone calculus, as follows $$t^{-2}{\rm div}_ \mathbb{B}(\nabla_ \mathbb{B}u) +t^{-2}(n-2)(t\partial_t u)+h(u)=f(t,x) \ \ \ \ (t,x) \in \mathbb{B}.$$ Using a special auxiliary function, we establish the comparison principle for the viscosity solutions under some assumptions on $f$ and $h$, and then obtain the uniqueness of the viscosity solutions for the corresponding Dirichlet problem.

This paper is devoted to the well-posedness and long-time behavior of initial-boundary value problems for a class of nonlinear parabolic equations associated with the Grushin operator $$u_t - \Delta_\alpha u = |u|^{p-2} u \log |u|,$$ where $\Delta_\alpha = \frac{\partial^2}{\partial x^2} + |x|^{2\alpha} \frac{\partial^2}{\partial y^2}$ is the Grushin operator, and $p > 2$ satisfies a subcritical growth condition. Via the semigroup theory in the framework of weighted Sobolev spaces, the existence and uniqueness of local solutions is proved. Subsequently, using the potential well method, the global dynamics of solutions is established. More precisely, when the initial energy satisfies $J(u_0) \leq d$ and the Nehari functional $I(u_0) > 0$, the equation admits a global solution whose energy decays exponentially;when the initial energy $J(u_0) \leq d$ and $I(u_0) < 0$, the solution blows up in finite time. For the case when the initial energy $J(u_0) > d$, by defining relevant invariant sets and functionals, the conditions are clarified under which the solution exists globally or blows up in finite time.

This paper is concerned with the global existence for a class of Keller-Segel model $$\begin{equation*} \begin{cases} u_t=\Delta(\gamma (v)u)+\rho u-\mu u^\alpha,&x\in\Omega,\,t>0,\\ v_t=\Delta v-v+u^\beta,&x\in\Omega,\,t>0, \end{cases} \end{equation*}$$ under homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega\subset\mathbb{R}^n\,(n\geqslant1)$. It is proved that for $\rho\in\mathbb{R},\,\mu>0$, $\alpha> 1$, $\beta>0$ satisfying certain additional relations, and under suitable assumptions on the motility function $\gamma$, the system admits a global classical solution for all sufficiently smooth initial data. This result improves recent ones established in [Lv W B, Wang Q Y. Proc Roy Soc Edinburgh, 2021, 151(2): 821-841], [Tao X Y, Fang Z B. Z Angew Math Phys, 2022, 73(3): Art 123].

This paper aims to sharp estimates of fully bubbling solutions of the Toda system with Cartan matrix $G_2$ in a compact Riemann surface, thereby providing a comprehensive understanding of the asymptotic behavior of such solutions. By using the non-degeneracy results of entire solutions, it proves that: 1) All fully bubbling solutions are approximated by a sequence of global solutions with precise error estimates; 2) the gradient of certain functions must approach zero with sufficient rate at the blow-up points, which uniquely determines their locations; 3) a corresponding $\partial_z^2$ condition exists.

This paper investigates the initial-boundary value problem for a class of coupled semilinear degenerate parabolic equations with singular potential term on manifolds with corner singularities. Under the condition of high initial energy, a sufficient condition is established to describe the global existence and finite-time blow-up of solutions to problem (1.1), respectively. Furthermore, by employing Levine's concavity method, we prove that solutions blow up in finite time for any initial energy, and derive an upper bound for the blow-up time.

We briefly review the $BV$ entropy theory for a class of flux-limited diffusion equations, including the notion and well-posedness of entropy solutions, the hyperbolic features and entropy conditions, and the finite speed of propagation. The purpose is to serve as a basic theoretical framework for subsequent investigations of nonlinear parabolic equations exhibiting large-gradient degeneracy.

In this paper, we investigate the 2D and 3D hyperbolic Prandtl equations. We prove that this system has a unique long-time solution with small initial data in Gevrey function space with index up to 2. The proof is based on a new cancellation mechanism with linear terms and the decay rate of the radius with respect to time.

In this paper, we prove the existence of two-dimensional hydroelastic solitary waves with constant vorticity by using spatial dynamics method. The flow beneath the wave is of finite depth, and critical layers are absent throughout the flow. The problem is converted to an equivalent dynamical system in which the horizontal spatial direction is the time like variable. Then application of a center-manifold reduction technique and normal-form theory yields the existence of homoclinic solutions to the reduced system, which correspond to solitary or generalized solitary waves of the hydroelastic problem.

Singularity formation for nonlinear PDEs has attracted much attention in recent years. In this survey article, some recent progress on the parabolic gluing method and its wide applications in investigating the singularity formation for parabolic flows, originating from geometry, mathematical physics and mathematical biology, will be introduced. Two model problems will be revisited to illustrate the key ideas of the gluing method in detail.

In this paper, we investigate, via variational methods, the existence of positive solution to a class of degenerate partial differential equations on the Heisenberg group involving critical Sobolev growth and logarithmic nonlinearity. In contrast to the classical perturbation term $\lambda u$ appearing in the [Brézis H, Nirenberg L. Comm Pure Appl Math, 1983, 36(4): 437-477], the logarithmic nonlinearity $\lambda u \log u^2$ leads to a weaker constraint on the parameter, yielding an improved existence result in the critical case.

When the Knudsen number approaches zero, the compressible Navier-Stokes-Poisson (NSP) system is not the limit of the dimensionless Vlasov-Poisson-Boltzmann (VPB) system; however, via the Chapman-Enskog expansion, it constitutes a second-order approximation to the VPB system. The purpose of this paper is to rigorously prove that if the difference between the local Maxwellian corresponding to the compressible NSP system and the initial value of the VPB system is a second-order small quantity in the Knudsen number, then the solutions of the two systems will maintain this order of magnitude difference for all times. The analysis in this paper is based on the energy estimates of the macroscopic equations, as well as the refined energy method within the macroscopic-microscopic decomposition framework.

In this paper, we prove that there exists at least one discretely self-similar solution to the steady Navier-Stokes equations in $\mathbb{R}^4\backslash\{0\}$ for any given locally Lipschitz discretely self-similar external force. If the external force is smooth away from the origin, the constructed discretely self-similar solution is also smooth away from the origin. Notably, the existence result does not require any smallness assumption on the external force.

In this paper, we study the following Hénon-type elliptic problem $$\begin{cases} -\Delta u = |x|^\alpha f(u) + \lambda |x|^\beta u^q, & \text{in } {B_1(0)}, \\ u > 0, & \text{in } {B_1(0)}, \\ u = 0, & \text{on } \partial {B_1(0)}, \end{cases} $$ where $\alpha > 0$, $\beta \geq 0$, $\lambda > 0$, $0<q<1$, ${B_1(0)}$ is the unit ball in $\mathbb{R}^N$ with $N \geq 3$, and $f$ satisfies $$0 \leq tf(t) \leq C_0 t^{2_\alpha^*}, t \in \mathbb{R}$$ with $2_\alpha^* = \frac{2(N+\alpha)}{N-2}$. Without any compactness conditions, we employ the Galerkin method to investigate the existence of positive solutions.The main result is that there exists a $\lambda_\ast > 0$such that for $\lambda \in (0, \lambda_\ast)$, the problem has a positive radial solution$u \in \ H_0^1( B_1(0)) \cap C^{1, \theta}(\overline{{B_1(0)}}), \theta \in (0, 1)$.

In this paper, we study the following fractional Choquard problem with shifting subcritical perturbation on bounded domains $$\begin{equation*} \left\{ \begin{aligned} &(-\Delta)^s u=\left(\int_{\Omega} \frac{u^{2^*_{\mu,s}}(y)}{|x-y|^\mu} \mathrm{d} y\right) u^{2^*_{\mu,s}-1}+g(x)\left[(u-k)^{+}\right]^{q-1}, &&x \in \Omega, \\ &u>0,\hspace{21em} &&x \in \Omega, \\ &u=0,\hspace{21em} &&x \in \mathbb{R}^N\backslash\Omega, \end{aligned} \right. \end{equation*}$$ where $N \geq 3$, $0<\mu<N$, $2^*_{\mu,s}=\frac{2N-\mu}{N-2s}$ is the fractional critical exponent in the sense of Hardy-Little-Wood-Sobolev inequality. Since the Choquard equation has non-local operator, we prove the existence of nontrivial solution $u_k$ for any $k\in(0,\infty)$ by energy estimation and variational method. What's more, the solutions $u_k$ are uniformly bounded when $k\to \infty$. At last, we get the concentration property of solutions.

In this paper, we study a Schrödinger-Bopp-Podolsky-Proca system with singular nonlinear terms in the context of closed 3-dimensional manifolds. Employing the $\varepsilon$-approximation techniques and variational methods, we establish the existence, uniqueness, and multiplicity of positive solutions, subject to appropriate conditions.

This paper establishes a spectral asymptotic formula for pseudodifferential operators on the quantum torus. Specifically, for a classical pseudodifferential operator $T \in \mathrm{C}\Psi^{-m}(C^\infty(\mathbb{T}_\theta^d))$ of order $-m < 0$ the spectral asymptotics $\lim_{t \to \infty} t^{\frac{m}{d}} \mu(t, T)$ is given by the $ L_{\frac d m}$-integral of its principal symbol $\sigma(T)_{-m}$ over the unit sphere. This result confirms a conjecture posed by McDonald and Ponge [Adv Math, 2023, 412: 108815]. As a corollary, we derive the Weyal law for pseudodifferential operators on the quantum torus, including the Weyl law for the Laplace-Beltrami operator.

This paper considers the rigidity of steady inhomogeneous incompressible Euler flows in two-dimensional annular domains. Under the assumption of no stagnation points and the slip boundary condition, with additional asymptotic conditions at infinity for unbounded domains and near the origin for punctured domains, the smooth fluids are proved to be circular shear flows, which extends the rigidity theorem for the homogeneous case to the inhomogeneous case. First, by establishing geometric properties of streamlines and the gradient of the stream function, the original system is transferred to a semilinear elliptic equation depending on the gradient terms. Then, by the moving plane method, the comparison principles are established in the corresponding domains, from which the radial symmetry properties of the stream function and streamlines are derived. Finally, for free boundary problems, a Serrin-type theorem for the inhomogeneous case is proved, based on which the rigidity theorem for contact discontinuity solutions is established.

As a typical model of kinetic equations, the Vlasov-Poisson-Boltzmann (VPB) equation describes the motion of charged particles in a plasma under the coupled effects of particle collisions and electric fields. For the scaled VPB equation, its hydrodynamic limit as the Knudsen number tends to zero is a problem of great importance in both physics and mathematics. In this paper, for the VPB equation on a half-space with the Maxwell reflection boundary condition, using the Hilbert expansion method, we formally analyze the fluid limit of the compressible Euler-Poisson system and its boundary layer when the Knudsen number is sufficiently small. By means of an inductive approach, we formally derive the macroscopic fluid equations for the interior and viscous layer at each order, as well as the equations governing the Knudsen layer, thereby revealing the coupling relations among the boundary layers and presenting a formal solution procedure.