The initial-boundary value problem of the planar radiative magnetohydrodynamic equations in the half-line was studied. We considered the Neumann boundary condition on the transverse magnetic field which was initially introduced by Kazhikhov in 1987. For the density-dependent viscosity and the degenerate heat-conductivity, we obtained the uniform upper and lower bounds of the density and the temperature, thus the global existence of a strong solution with large initial data to the magnetohydrodynamic system in the half-line was established.

This work is devoted to the analysis of well-posedness and stabilization properties of both the classical and truncated Timoshenko beam models resting on a two-parameter elastic foundation. We rigorously investigate the effects of the elastic subgrade on the dynamic behavior of the systems and consider the influence of viscous damping mechanisms acting on the angular rotation. For each model, we establish the existence, uniqueness, and continuous dependence of solutions. Moreover, we analyze the long-time behavior of the solutions, proving exponential or polynomial energy decay depending on the parameters of the wave speeds. The results highlight the differences in stability between the classical and truncated models and contribute to the mathematical theory of damped elastic structures interacting with elastic media.

This paper investigates the approximate controllability of Hilfer integro-differential neutral dynamical systems with infinite delay governed by almost sectorial operators. The mild solution of the proposed system is derived using the Laplace transform technique. Subsequently, the approximate controllability of the system is established by the Bohnenblust–Karlin - fixed point theorem. Furthermore, the controllability of the associated neutral system is analyzed in detail. Finally, a numerical example is presented to demonstrate the validity and applicability of the theoretical results and our system explained via a filter system.

In this paper, we present an analysis of the KH instability in 2D magnetohydrodynamic (MHD) flows, providing rigorous confirmation that a parallel magnetic field can have a destabilizing effect on this instability. When the Mach number M := v₊ / c lies strictly between M_low := sqrt((1 - q)/(1 + beta)) * (1 + epsilon_0) and M_upp := sqrt((1 + q)/(1 - beta)) * (1 - epsilon_0), where beta := c_A^2 / c^2 and epsilon_0 > 0 is a small but fixed constant, we prove the linear and nonlinear ill-posedness of the KH problem for compressible MHD flows.

This note corrects an inaccuracy in the paper ”Nontrivial solutions for the Laplace equation with a nonlinear Goldstein–Wentzell boundary condition” by the author, which appeared in Commun. Anal. Mech. 15 (2023), no. 4, 811–830. In particular, we point out that the bounded open set Ω must be connected for some of the main results in the paper to hold. To motivate this claim, we give an example of a disconnected open set Ω combined with a partition (Γ₀, Γ₁) of its boundary, for which the potential–well depth associated to the problem vanishes. When this occurs, the framework developed in the paper breaks down. We also show that, conversely, when Ω is connected, this phenomenon does not show up and all assertions in the paper are correct.

In this paper, we first consider the robustness of strong solutions for 3D generalized Navier- Stokes equations, i.e., we show the set of all initial conditions u₀ ∈ H¹(R³) with ∇·u₀ = 0 that give rise to a strong solution of the generalized Navier-Stokes equations on the time interval [0,T] is open inH¹(R³). Moreover, we prove that the Galerkin approximations of a strong solution of the 3D generalized Navier- Stokes equations converge strongly to u in L^∞(0,T;H¹_0,div(T³)) and L²(0,T;H^κ+1(T³) ∩ H¹_0,div(T³)).

This paper focuses on two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with only fractional horizontal dissipation Λ^2α₁ u and Λ^2β₁ b in the spatial domain Ω= T × R (where T = [0, 1] is a periodic box, and R is the whole line). For this system, Feng, Wang, and Wu assessed the global stability of perturbations near the steady solution given by a background magnetic fieldA = (A₁, A₂) with A₁, A₂ ∈ R and established nonlinear stability in the Sobolev space H³(Ω). In this paper, we consider the stability problem in the lower regularity space H²(Ω). By applying the Hölder’s inequality with various anisotropic fractional exponents and some special anisotropic interpolation inequalities, we obtain the global stability of the system when α ∈ (0, 1] and β ∈ [0, 1] inH²(Ω). In addition, we prove that the oscillation part (e u, e b) of the solution in H¹(Ω) decays to zero exponentially in time.

In this paper, we establish C^1,α-regularity theory with an accurate estimate

where b(t) = ta(t), for weak solutions of the following general quasilinear elliptic equation with Orlicz growth in divergence form: for n ≥ 2. Its prototypes are the nonhomogeneous elliptic p-Laplacian equations with and without a logarithmic term, respectively. Meanwhile, we also present the local optimal (1 + s^′ a)-cap continuity for the above problem.

This paper studies the following Neumann initial-boundary problem of a quasilinear predator-prey model with nonlinear indirect signal production mechanism:

in a smoothly bounded domain $\Omega \subset \mathbb{R}^{n}$(n ≥ 1). Assume that the parameters $\alpha_{i}, \beta_{i}, \rho, \vartheta,$ η > 0, $\phi_{i}$ and $\varphi_{i}$ (i=1, 2) are nonlinear functions satisfying $\phi_{i}(s) \geq \chi(1+s)^{\delta}$ and $\left|\varphi_{i}(s)\right| \leq \xi s(1+s)^{\sigma-1}$, respectively, for alls ≥ 0 with χ, ξ > 0 and $\delta, \sigma \in \mathbb{R}$. The main result shows that for all nonnegative initial data with appropriate regularity, if the power exponents satisfy k₁ > 1, k₂ > 2 and $\sigma+\rho(\vartheta+\eta)＜\frac{n+2}{n}+\delta$, the model possesses a bounded global classical solution.

In this article, we developed a two-grid mixed virtual element method (MVEM) for solving a time-dependent fourth-order reaction-diffusion equation with a Riemann-Liouville fractional derivative and a nonlinear reaction term. By introducing an auxiliary variable σ = ∆u, we reformulated the problem into a coupled system of second-order equations. Then we discretized the equations at t_k−α by a second-order backward differentiation formula (BDF2) with a fractional parameter α. The method employed a two-grid strategy: First, we solved a nonlinear problem via Newton iterations on a coarse ploygonal grid, and subsequently, leveraging the coarse-grid solutions, we solved a linearized problem on a fine virtual element space. We established the stability of this method and derived a priori L²-norm error estimates with the optimal convergence result O(∆t² + h^k+1 + H^2k+2), where ∆t denotes the time step size, h and H are the fine and coarse spatial mesh sizes, respectively, and k is the polynomial degree of the virtual element space. Numerical experiments validated the theoretical convergence rates, and it was demonstrated that applying the two-grid MVEM is computationally more efficient than solving the nonlinear system directly on the fine grid.