The neural operator theory offers a promising framework for efficiently solving complex systems governed by partial differential equations (PDEs for short). However, existing neural operators still face significant challenges when applied to spatiotemporal systems that evolve over large time scales, particularly those described by evolution PDEs with time-derivative terms. This paper introduces a novel neural operator designed explicitly for solving evolution equations based on the theory of operator semigroups. The proposed approach is an iterative algorithm where each computational unit, termed the single-step neural operator solver (SSNOS for short), approximates the solution operator for the initial-boundary value problem of semilinear evolution equations over a single time step. The SSNOS consists of both linear and nonlinear components: The linear part approximates the linear operator in the solution map; in contrast, the nonlinear part captures deviations in the solution function caused by the equations nonlinearities. To evaluate the performance of the algorithm, the authors conducted numerical experiments by solving the initial-boundary value problem for a two-dimensional semilinear hyperbolic equation. The experimental results demonstrate that their neural operator can efficiently and accurately approximate the true solution operator. Moreover, the model can achieve a relatively high approximation accuracy with simple pre-training.

The authors study the regularity of the p-Gauss curvature flow with flat sides. In their previous paper [Huang, G. G., Wang, X.-J. and Zhou, Y., Long time regularity of the p-Gauss curvature flow with flat side, https://arxiv.org/abs/2403.12292], they obtained the regularity of the interface, namely the boundary of the flat part. In this paper, they study the regularity of the convex hypersurface near the interface.

It is proved that there are many (positive Lebesgue measure) Kolmogorov-Arnold-Moser (KAM for short) tori at infinity and thus all solutions are bounded for the Duffing equations

with p_j(t)’s being time-quasi-periodic smooth functions.

The author considers the uniqueness of the following positive solutions of m-Laplacian equation:

where m > 1 is a constant. When p → m, the uniqueness of positive solutions of (*) is shown which is based on the essential uniqueness of first eigenfunction for m-Laplacian equation. Futhermore, it is proved that the uniqueness results when (*) is a perturbation of Laplacian equation.

In this paper, the authors give a characterization of finite Blaschke products with degree n. The main results are: (1) An n-dimensional complex vector can be the first n Taylor coefficients of a finite Blaschke product with degree no more than n − 1 if and only if the vector induces a lower triangular Toeplitz matrix with norm 1; (2) an n-dimensional complex vector can be the first n Taylor coefficients of an inner function if and only if the vector induces a lower triangular Toeplitz matrix with norm no more than 1. Möbius transformations acting on contraction matrices play an important role in the proofs.

The classical stochastic plate equation suffers from a lack of exact controllability, even with controls that are effective in both the drift and diffusion terms and on the boundary. To address this issue, a one-dimensional refined stochastic plate equation was previously proposed and established as exactly controllable in [Yu, Y. and Zhang, J. -F., Carleman estimates of refined stochastic beam equations and applications, SIAM J. Control Optim., 60, 2022, 2947–2970]. In this paper, the authors establish the exact controllability of the multidimensional refined stochastic plate equation with two interior controls and two boundary controls by a new global Carleman estimate. Furthermore, they show that at least one boundary control and the action of two interior controls are necessary for exact controllability.

The authors prove a rigidity result of Lagrangian translating solitons in ℝ²ⁿ, which extends the result of [Han, X. and Sun, J., Translating solitons to symplectic mean curvature flows, Ann. Global Anal. Geom., 38(2), 2010, 161–169] to higher dimension.

Assume that M is a closed, connected and smooth Riemannian manifold. The authors consider the evolutionary Hamilton-Jacobi equation

where φ ∈ C(M) and the stationary one where H(x,u,p) is continuous, convex and coercive in p, uniformly Lipschitz in u. By introducing a solution semigroup, the authors provide a representation formula of the viscosity solution of the evolutionary equation. As its applications, they obtain a necessary and sufficient condition for the existence of the viscosity solutions of the stationary equations. Moreover, they prove a new comparison theorem depending on the neighborhood of the projected Aubry set essentially, which is different from the one for the Hamilton-Jacobi equation independent of u.