We investigate non-traveling wave solutions of the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation with time-dependent coefficients, which describes the propagation of waves in incompressible fluids. We creatively combine the extended three-wave method with the generalized variable separation method and successfully obtain sixty exact non-traveling solutions including kink-like solutions, singular solitary wave-like solutions, periodic solitary wave-like solutions, periodic kink-like solutions, periodic cross-kink-like waves, homoclinic breather wave-like solutions and so on. The variable coefficients and arbitrary functions in the obtained solutions are easy to exhibit abundant soliton structures, which may be of great significance for explaining some practical physical phenomena. By contour plots, 2D plots, and 3D plots, we analyze the dynamic characteristics of periodic cross-kink-like solution, singular solitary wave-like solution, homoclinic breather wave-like solution. Additionally, we show changes of solutions under different tails to illustrate the influence of tails on solutions.

For a graph G of order n, its Q-walk matrix is defined by $W_{Q}(G)=\left[e, Q e, \cdots, Q^{n-1} e\right]$, where Q is the signless Laplacian matrix of G and e denotes the all-one column vector. Let $G \circ P_{k}$ represent the rooted product graph of G and a path P_k. In this note, we establish the relationship between determinants of $W_{Q}(G)$ and $ W_{Q}\left(G \circ P_{k}\right)$ for k=2,3.

In this paper, we characterize the rigidity of umbilical hypersurfaces by a Serrin-type partially overdetermined problem in space forms, which generalizes the similar results in Euclidean half-space and Euclidean half-ball. Guo-Xia first obtained these rigidity results when the Robin boundary condition on the support hypersurface is homogeneous, at this time the target umbilical hypersurface has orthogonal contact angle with the support. However, in this paper we can obtain any contact angle $\theta \in(0, \pi)$ by changing the Robin boundary condition to be inhomogeneous.

In this paper, under the asymptotically regular condition, we investigate the existence of a unique common fixed point for a pair of generalized nonlinear mappings in the framework of complete metric spaces. Additionally, we extend our analysis to the case involving two control functions. Our work generalizes some results in recent papers.

We give some generalized upper bounds for the numerical radius of off-diagonal 2×2 operator matrices. These inequalities are mainly based on the extension Buzano inequality and the generalized Young inequality. And our bounds refine and generalize the existing related upper bounds. Moreover, the conclusion is applied to the non-monic operator polynomials and gives a new bound for the eigenvalues of these operator polynomials.

We consider the incompressible inviscid elastodynamics equations with a free surface and establish the regularity of solutions for elastic system. Compared with the previous result on this free boundary problem [ Gu X and Wang F, Well-posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition, J. Math. Anal. Appl., 2020, 482(1): 123529] in space H³, we are able to establish the regularity in space $H^{2.5+\delta}$. It is achieved by reformulating the system into the Lagrangian formulation, presenting the uniform estimates for the pressure, the tangential estimates for the system, as well as the divergence and curl estimates.

In this paper we are interested in the existence of solutions for Dirichlet problem associated to the degenerate nonlinear elliptic equations

$ \begin{aligned}& -\sum_{j=1}^{n} D_{j}\left[\omega(x) \mathcal{B}_{j}(x, u, \nabla u)\right]-\sum_{i, j=1}^{n} D_{j}\left(a_{i j}(x) D_{i} u(x)\right)+b(x, u, \nabla u) \omega(x)+g(x) u(x) \\= & f_{0}(x)-\sum_{j=1}^{n} D_{j} f_{j}(x), \quad \text { in } \Omega\end{aligned}$

in the setting of the weighted Sobolev spaces $ W_{0}^{1, p}(\Omega, \omega).$

In this paper, we will study the sharp estimates of p-adic weighted Hardy operator on central and noncentral weighted p-adic Morrey spaces. Moreover, we can obtain the sharp bound of generalized p-adic Hardy operator. In addition, the commutator that is generated by the generalized p-adic Hardy operator and the central BMO function is also bounded on p-adic Morrey spaces.

This paper establishes the global existence of the pathwise solutions for the stochastic 3-D incompressible Euler equations with the axially symmetric initial data and axially symmetric additive noise.