In the present paper, we investigated the existence of normalized solutions for the following Kirchhoff equation with Choquard nonlinearity $-\left(a+b \int_{\mathbb{R}^{3}}|\nabla u|^{2} d x\right) \Delta u-\lambda u=\mu|u|^{q-2} u+\left(I_{\alpha} *|u|^{p}\right)|u|^{p-2} u, \quad x \in \mathbb{R}^{3}$with prescribed mass $\int_{\mathbb{R}^{3}}|u|^{2} d x=c^{2}$, where a, b, c > 0, µ ∈ $\mathbb{R}$, α ∈ (0, 3), 10/3 ≤ q < 6, (3+α)/3 ≤ p < 3+α and λ ∈ $\mathbb{R}$ is a Lagrange multiplier. We first considered the case of µ > 0 and obtained mountain pass type solutions. For the defocusing situation µ < 0, we proved the existence result by constructing a minimax characterization for the energy functional. Finally, we discussed the asymptotic behavior of normalized solutions obtained above as b → 0⁺ when µ > 0.

In this paper, we investigated the asymptotic symmetry and monotonicity of positive solutions to a reaction-diffusion equation in the unit ball, utilizing techniques from elliptic geometry. First, we discussed the properties of solutions in the elliptic space. Then, we established crucial principles, including the asymptotic narrow region principle. Finally, we employed the method of moving planes to demonstrate the asymptotic symmetry of the solutions.

One of the most classical and fundamental mathematical problems in kinetic theory is to study the diffusive limit of the neutron transport equation. As ε → 0, the phase space density u^ε(x,w)

converges to the interior solution U₀(x): in which $U_{0, \infty}^{B}$ is obtained by solving the Milne problem for the celebrated boundary layer correction U^B₀. The function g represents the inflow data, and n is the unit outward normal to the smooth bounded domain Ω. Surprisingly, we found [1,2] that the expected is invalid due to the grazing singularity of $U_{0}^{B}$. As a result, the corresponding well-known mathematical theory breaks down, and the diffusive limit has remained an outstanding question. A satisfactory theory was developed for convex domains [1–6] by constructing new boundary layers with favorable ε-geometric corrections. However, this approach is inapplicable in non-convex domains. In this paper, we settle this open question affirmatively in the L² sense. The convergence holds for general smooth domains, including non-convex ones. We achieve this by discovering a novel and optimal L² expansion theory that reveals a surprising $\varepsilon^{\frac{1}{2}}$ gain for the average of the remainder, and by choosing a test function with a new cancellation via conservation of mass. We also introduce a cutoff boundary layer $U_{0}^{B}$ and investigate its delicate regularity estimates to control the source terms of the remainder equation with the help of Hardy’s inequality. Notably, our new cutoff boundary layer $U_{0}^{B}$ determines U₀, despite its absence in the estimate.

In this paper, we study the following fully parabolic chemotaxis system with nonlocal growth and indirect signal production: $\left\{\begin{array}{ll} u_{t}=\nabla \cdot(D(u) \nabla u)-\nabla \cdot(u \nabla v)+f(u), & x \in \Omega, t>0, \\ v_{t}=\Delta v+w-v, & x \in \Omega, t>0, \\ w_{t}=\Delta w+u-w, & x \in \Omega, t>0, \\ \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=\frac{\partial w}{\partial n}=0, & x \in \partial \Omega, t>0, \\ u(x, 0)=u_{0}(x), v(x, 0)=v_{0}(x), w(x, 0)=w_{0}(x), & x \in \Omega, \end{array}\right.$ in a smooth bounded domain $\Omega \subset \mathbb{R}^{n}$(n ≥ 3), where D(u) ≥ D₁ u^γ,$f(u)=u\left(a_{0}-a_{1} u^{\sigma}+a_{2} \int_{\Omega} u^{\sigma} d x\right)$, D₁,a₁,a₂ are positive constants, a₀, $\gamma \in \mathbb{R}$, σ ≥ max{1, −γ} and a₁ − a₂|Ω| > 0. It is shown that the above system admits a globally bounded classical solution if $\frac{4}{n}+\frac{\gamma}{\sigma}>\frac{3-\sigma}{1+\sigma}$. Furthermore, by the method of Lyapunov functionals, the global stability of steady states with convergence rates is established.

In this paper, we consider the problem of optimal time control for a fourth-order parabolic-type equation describing the growth process of a thin film in a bounded three-dimensional space. The control function is defined on a certain part of a boundary. It is proved that the optimal time depends on the parameters of the growth process when the average value of the growth interface height of the thin film in the domain is close to the critical value.

We are concerned with the time growth of the highest-order energy of three-dimensional inhomogeneous incompressible isotropic elastodynamics. Utilizing Klainerman’s generalized energy method, refined weighted estimates, and the Keel-Smith-Sogge estimate [J. Anal. Math., 87: 265–279, 2002], it is justified that the highest-order generalized energy is uniformly bounded for all time.

We study the subcritical approximations to Li–Lin’s open problem, proposed by Li and Lin (Arch Ration Mech Anal 203(3): 943-968, 2012). By applying the variational method, we obtain two positive solutions. We establish a nonexistence theorem for positive solutions. Finally, through the combination of the variational method and the sub-supersolution method, we find a global bifurcation phenomenon for positive solutions.

In this paper, we extended the port-Hamiltonian framework by introducing the concept of Stokes-Lagrange structure, which enables the implicit definition of a Hamiltonian over an N-dimensional domain and incorporates energy ports into the system. This new framework parallels the existing Dirac and Stokes-Dirac structures. We proposed the Stokes-Lagrange structure as a specific case where the subspace is explicitly defined via differential operators that satisfy an integration-by-parts formula. By examining various examples through the lens of the Stokes-Lagrange structure, we demonstrated the existence of multiple equivalent system representations. These representations provide significant advantages for both numerical simulation and control design, offering additional tools for the modeling and control of port-Hamiltonian systems.

This work reports on initial value problems to hyperbolized compressible Navier-Stokes equations. Local existence and global existence for small data are addressed as well as singular limits for vanishing relaxation parameters. Then blow-up results for one- and multi-dimensional models are presented using ideas from T.C. Sideris from the 1980s.

The global well-posedness theory and viscosity vanishing limit of the initial-boundary value problem on two/three-dimensional (2D/3D) incompressible Navier-Stokes (NS) equations and/or Boussinesq equations with nonlinear boundary conditions are studied. The global existence of weak solution to the initial boundary value problem for 2D/3D incompressible NS equation with one kind of boundary of pressure-velocity’s relation and the global existence and uniqueness of the smooth solution to the corresponding problem in 2D case for large smooth initial data are proven. The viscosity vanishing limit of the corresponding initial-boundary value problem for 2D/3D incompressible NS equations in the bounded domain is also established. And the corresponding results are extended to the 2D/3D incompressible Boussinesq equations.

We review the formulation of a Lorentz-covariant bispinorial wave function and wave equation for a single photon on a flat background. We show the existence of a 10-dimensional set of conservation laws for this equation, and prove that 8 of these can be used to obtain global, gauge-invariant, ADM-like quantities that together define a covariantly constant self-dual bispinor.

In light of the advantages of the Caputo–Hadamard fractional derivative in characterizing ultra-slow diffusion phenomena, this paper proposes a second-order approximation scheme to approxi- mate it. Then, for the Allen–Cahn equation with the Caputo–Hadamard fractional derivative in time, a numerical algorithm is designed. This algorithm employs the proposed second-order formula for time discretization. Considering the potential anisotropic behavior of the solution in space, the anisotropic nonconforming quasi-Wilson finite element method is utilized for spatial approximation. The error in the L²-norm and the superclose error in the H¹-norm of this algorithm are analyzed. The global superconvergence in the H¹-norm is demonstrated through interpolation postprocessing techniques. Numerical examples are given to verify the theoretical results and further investigate the influence of different time derivatives on the dynamic behavior of the solution.

Within the self-consistent Maxwell-Pauli theory, a nonlinear Schrödinger equation with a short-range compensating field was derived. Stationary and nonstationary solutions of the obtained nonlinear Schrödinger equation for the hydrogen atom were investigated. It is shown that spontaneous emission and the associated rearrangement of the internal structure of the atom, which is traditionally called a spontaneous transition, have a simple and natural description within the classical field theory without any quantization and additional hypotheses. The solution of the nonlinear Schrödinger equation shows that, depending on the frequency of spontaneous emission, the compensating field behaves differently. At relatively low frequencies of spontaneous emission, there is no radiation (waves) of the short-range compensating field, and this field does not carry away energy. In this case, the damping rate of the spontaneous emission coincides with that obtained in quantum electrodynamics (QED). At relatively high frequencies of spontaneous emission, radiation (waves) of the compensating field arises, which, along with electromagnetic radiation, carry away some of the energy. In this case, the damping rate of the spontaneous emission is greater (up 1.5 times) than that predicted by QED.