In this paper, nonlinear boundary controls associated with two control gain for a one-dimensional continuous porous elastic system are designed, with the primary goal of uniformly stabilizing the model. The nonlinear boundary controllers were designed based on the spillover effect generated by high-frequency problems that generate so-called spurious oscillations. The existence and uniqueness of the solutions are proven using the Galerkin approximation method with a special basis constructed using the Gram-Schmidt process. Uniform stabilization is achieved using the method introduced by Lasiecka-Tataru.

We study a thermoelastic wave–heat system on a smooth bounded domain in which the mechanical velocity is damped both by a spatially variable-exponent power and by a time-varying delay control. The exponent field s(·) is log-Hölder with 2 ≤ s₋ ≤ s(x) ≤ s₊, where s₊ lies strictly below a scaling-critical bound; the delay is uniformly positive with derivative < 1; the delayed coefficient may change sign but is small relative to the primary damping profile. Encoding the delayed velocity through a first-order transport equation, we construct a Lyapunov functional that controls the elastic, thermal, and delay contributions and yields coercive dissipation. Using a Faedo–Galerkin approximation, compactness arguments in variable-exponent spaces, and the Minty–Browder method, we obtain global existence and uniqueness of strong solutions for admissible data. The same functional, together with a Komornik-type integral inequality, yields an explicit exponential decay rate determined by the time integral of the primary damping profile. The approach also adapts to Cattaneo heat conduction and to distributed delays.

This paper investigates the well-posedness of local smooth solutions for the vacuum free boundary problem of the spherically symmetric non-isentropic compressible Euler equations. At the free boundary, the system degenerates into a hyperbolic system while exhibiting a singularity at the spherical center. To overcome these difficulties, a Lagrangian coordinate transformation is introduced, converting the problem into an initial-boundary value problem with fixed boundaries. Within this framework, the degeneracy of the system can be explicitly expressed by the function ρ^γ−1₀ (γ > 1) of the initial density, whose value equals the distance function near the boundaries. Consequently, by employing the distance-dependent Hardy’s inequality and weighted Sobolev spaces, the degeneracy and singularity of the system are successfully resolved, and a priori estimates for the local smooth solutions are established. Specifically, by building upon high-order time derivative estimates, the high-order spatial derivative estimates are rigorously proved via mathematical induction. Crucially, the influence of entropy introduces additional higher-order terms, necessitating boundedness of initial entropy derivatives up to a certain order. Finally, we establish the existence and uniqueness of local smooth solutions.

Due to the importance in physical applications and mathematics, the stability problems of partially dissipative hydrodynamic system have attracted considerable attention in recent years and achieved significant progress. However, research on the stability analysis for the partial fractional dissipative system is relatively limited. The present study demonstrates the stability of a two-dimensional (2D) incompressible magnetohydrodynamic (MHD) system with fractional vertical dissipation and magnetic damping within the Sobolev space H²(R²). In view of the presence of fractional order operators, classical tools and techniques designed for the fully dissipated MHD system no longer work; therefore, we develop new anisotropic inequalities that involve fractional derivatives to address this issue. Our result extends and improves the work in [Nonlinearity, 2021, 34(4): 2527].

We investigated the boundedness of pullback attractors for non-Newtonian fluids with delay in two-dimensional bounded domains. Under suitable assumptions on the delay term, we established the boundedness of the pullback attractor, defined in the phase space L², which was bounded in H⁴ by means of refined energy estimates.

The hypergraph Laplacian, which was introduced to study the structure of complicated networks in the information science, is defined as a set-valued operator on a finite-dimensional Euclidean space. In previous works, we have shown that a nonlinear ordinary differential equation governed by the hypergraph Laplacian describes the diffusion process on a hypergraph, and its behavior closely resembles the heat equation. In this paper, we consider some generalized doubly nonlinear equation governed by the hypergraph Laplacian in order to deal with various diffusion phenomena on discrete domains.

In this paper, we consider the following f-Laplacian equation with a gradient term and a nonlinear term

on smooth metric measure spaces with nonnegative m-Bakry-èmery Ricci curvature, where W(t) is a continuous function for t > 0 and F(t) is a differentiable function in (0, ∞). Under certain assumptions on W and F, by using the Saloff-Coste’s Sobolev inequality and the Nash-Moser iteration method, we derive some Liouville-type theorems for positive solutions to the above equation. Our results extend those of Lu (2025) and McCoy (2007).

In this work, we investigate two distinct control-theoretic problems associated with a linear Rao–Nakra-type sandwich beam. The first problem concerns stabilization under time-varying distributed damping and delayed boundary feedback, while the second addresses boundary controllability under dynamical boundary interactions. Both systems consist of three coupled equations that represent the longitudinal displacements of the outer layers and the transverse displacement of the composite beam, all of which are coupled with boundary conditions. In the first problem, time-varying interior damping and static boundary conditions with time-dependent delays are considered. Then, we establish the existence and uniqueness of solutions for the Cauchy problem associated with the damped system using semigroup theory and a classical result by Kato. Furthermore, employing a Lyapunov-based approach, we prove that the system’s energy decays exponentially, despite the presence of time-varying weights and delays. In the second problem, we consider a boundary linear control system with dynamical boundary conditions, and prove its well-posedness. By deriving an observability inequality for the adjoint system and applying the Hilbert Uniqueness Method (HUM), we show that the system is null controllable. A key contribution of this work lies in handling the full three-equation coupled system, which involves a challenging difficulty due to the delayed or dynamical boundary conditions, resolved via appropriately constructed Lyapunov functionals and intermediate observability inequalities.

In this paper, a fully implicit Crank–Nicolson discontinuous Galerkin method is proposed for solving the Ginzburg–Landau equation. By leveraging a novel analytical technique, we rigorously establish the unique solvability of the constructed numerical scheme as well as its unconditionally optimal error estimates under both the L²-norm and the energy norm. The core of the proof hinges on the L²-norm boundedness of the numerical solution and the refined estimation of the cubic nonlinear term. Finally, four numerical examples are presented to validate the theoretical findings.

This paper studied a time-optimal control problem for a pseudo-parabolic equation in a bounded multidimensional domain. The equation involved a uniformly elliptic operator and a small parameter characterizing memory effects. The control acted through a distributed source term, where the spatial part was fixed and the time-dependent part was a bounded measurable function. The goal was to steer the system from a zero initial state to a prescribed spatially averaged temperature in minimal time and maintain it thereafter. Using spectral decomposition, we derived an explicit solution representation and reduced the control problem to a Volterra integral equation. Under natural positivity assumptions on the Fourier coefficients, we established that the kernel was continuous, positive, and monotone. These properties yielded an explicit upper bound for the minimal time in terms of the first eigenvalue, the relaxation parameter, and the system constraints. A comparison with the parabolic case showed that the pseudo-parabolic term led to longer minimal times, reflecting the memory effects inherent in such models.