In this paper, nonlinear boundary controls associated with two control gain for a onedimensional 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.
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 sigma = Delta u, we reformulated the problem into a coupled system of second-order equations. Then we discretized the equations at tk- alpha 2 by a second-order backward differentiation formula (BDF2) with a fractional parameter alpha. 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(Delta t2 + hk+1 + H2k+2), where Delta 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.
In this paper, we establish C-1,C-alpha-regularity theory with an accurate estimate vertical bar vertical bar u vertical bar vertical bar(C1,alpha(B1/2)) <= C(b(-1) (vertical bar vertical bar f vertical bar vertical bar(L infinity(B1))) + vertical bar vertical bar u vertical bar vertical bar(W1,B(B1)) , where b(t) = ta(t), for weak solutions of the following general quasilinear elliptic equation with Orlicz growth in divergence form: -div(a(vertical bar del u vertical bar)del u) = f is an element of L-loc(infinity)(Omega) in Omega subset of R-n 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 focuses on two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with only fractional horizontal dissipation Lambda 2 alpha 1 u and Lambda 2 beta 1 b in the spatial domain Omega = T & times; 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 field A = (A1, A2) with A1, A2 is an element of R and established nonlinear stability in the Sobolev space H3(Omega). In this paper, we consider the stability problem in the lower regularity space H2(Omega). By applying the Ho & uml;lder's inequality with various anisotropic fractional exponents and some special anisotropic interpolation inequalities, we obtain the global stability of the system when alpha is an element of (0,1] and beta is an element of [0,1] in H2(Omega). In addition, we prove that the oscillation part (ue,be) of the solution in H1(Omega) decays to zero exponentially in time.
This paper studies the following Neumann initial-boundary problem of a quasilinear predator-prey model with nonlinear indirect signal production mechanism: u(1t) = del center dot(phi(1)(u(1))del u(1) + phi(1)(u(1))del v) + alpha(1)u(1)(1 - u(1)(k1-1) - beta(1)u(2)), x is an element of Omega, t > 0, u(2t) = del center dot(phi(2)(u(2))del u(2) -phi(2)(u(2))del v) + alpha(2)u(2) (1-u(2)(k2-1) + beta(2)u(1)), x is an element of Omega, t > 0, v(t) = Delta v - v+ w(rho), x is an element of Omega, t > 0, 0 = Delta w - w+ u(1)(theta) + u(2)(eta), x is an element of Omega, t > 0, in a smoothly bounded domain Omega subset of R-n(n >= 1). Assume that the parameters alpha(i), beta(i), rho, theta, eta > 0, phi(i) and phi(i) (i=1, 2) are nonlinear functions satisfying phi(i)(s) >= chi(1 + s)(delta) and vertical bar phi(i)(s)vertical bar <= xi s(1 + s)(sigma-1), respectively, for all s >= 0 with chi, xi > 0 and delta, sigma is an element of R. The main result shows that for all nonnegative initial data with appropriate regularity, if the power exponents satisfy k(1) > 1, k(2) > 2 and sigma + rho(theta + eta) < n+2/n + delta, the model possesses a bounded global classical solution.
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 E H1(R3) with V & centerdot; u0 = 0 that give rise to a strong solution of the generalized Navier-Stokes equations on the time interval [0, T] is open in H1(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 degrees degrees(0, T; H10,div(T3)) and L2(0, T; H kappa+1(T3) n H10,div(T3)).
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(1/c)(+) lies strictly between M-low:=root 1-root(1-beta)/(1+beta)+is an element of 0 and M-upp:=root 1+root(1-beta)/(1+beta)-is an element of 0, where beta:=c(A/c)(2)(2)and is an element of(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 Omega 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 Omega combined with a partition (Gamma 0, Gamma 1) 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 Omega is connected, this phenomenon does not show up and all assertions in the paper are correct.
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.
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 manuscript was dedicated to exploring the existence and uniqueness of global solutions pertaining to a specific class of stochastic pseudo-parabolic equations. These equations have logarithmic nonlinearity and are driven by Brownian motion. By utilizing the Galerkin method, Prokhorov's theorem, and Skorohod's embedding theorem, we proved the existence of a global solution in the weak probabilistic sense. Subsequently, by leveraging the uniqueness of solutions and applying the Yamada-Watanabe theorem, the global existence and uniqueness of a probability strong solution were established. A notable finding, in contrast to the stochastic heat equation, was that as the pseudoparabolic coefficient & micro; increased, the growth condition imposed on the noise coefficient within the stochastic pseudo-parabolic equation could be significantly relaxed.
This paper considers the application of the orthogonality sampling method (OSM) with single and multiple sources for a fast identification of small objects in the limited-aperture inverse scattering problem. First, we apply the OSM with a single source and demonstrate that the indicator function of the OSM with a single source can be expressed by the Bessel function of order zero of the first kind, an infinite series of Bessel functions of nonzero integer order of the first kind, the range of the signal receiver, and the emitter location. We then explain that the objects can be identified using the OSM with a single source; however, the identification is strongly influenced by the location of the source and the applied frequency. To realize effective improvement, we consider the OSM with multiple sources. Based on the identified structure of the OSM with a single source, we propose an indicator function for the OSM with multiple sources and demonstrate that it can be expressed by the square of the Bessel function of order zero of the first kind and an infinite series of the square of the Bessel function of nonzero integer order of the first kind. This result shows that the locations of objects can be uniquely identified using the designed OSM. Simulation results with experimental data provided by the Institute Fresnel demonstrate the advantages and disadvantages of the OSM with a single source and how the proposed OSM with multiple sources behaves.
This paper investigates the existence, energy decay, and finite-time blow-up of solutions for a class of fourth-order viscoelastic evolution equations involving variable exponent nonlinearities, logarithmic terms, and strong damping effects. Such models arise naturally in the mathematical description of heterogeneous viscoelastic media and nonlinear plate equations with memory. By applying the Nehari manifold method and suitable energy estimates, we establish the global existence of weak solutions under appropriate assumptions on the initial data and the variable exponents. Moreover, using a perturbed energy method combined with a carefully constructed Lyapunov functional, we prove that the solutions exhibit general energy decay rates depending on the properties of the relaxation kernel and the spatially variable nonlinearities. Finally, we derive sufficient conditions ensuring finite-time blow-up for solutions with negative initial energy, thereby highlighting the competing effects of strong damping, viscoelastic memory, and logarithmic nonlinearities.
In recent years, fractional-order dynamical systems have attracted significant attention because of their ability to describe memory effects and hereditary properties found in many physical and engineering processes. Fractional calculus extends the concept of differentiation and integration to non-integer orders, providing a more flexible mathematical framework for modeling and analyzing complex system behaviors. Within this framework, the study of chaotic dynamics in fractional-order systems has become an active research area since many well-known chaotic models show new and qualitatively different behaviors when generalized to fractional derivatives. This study revisits a classical chaotic system originally defined in the integer-order setting and analyzes its behavior using fractional-order calculus. The main objective is to examine how fractional differentiation influences the system's chaotic dynamics and sensitivity to initial conditions. To achieve this, numerical simulations are performed to demonstrate the effects of different fractional orders on the development and evolution of chaotic behavior.
This study focuses on a class of fractional sublinear operators, denoted as S-ohm,S-alpha,S- and their commutators S-ohm,S-alpha,S-b with rough kernels. We establish the boundedness of these operators on newly emerging vanishing generalized Morrey spaces that are characterized by (V-infinity) or (V-& lowast;). The primary innovation of this paper lies in the novel approach of controlling S-ohm,S-alpha via the Riesz potential I alpha and the management of S ohm,alpha,b through fractional maximal commutators with rough kernels M-ohm,M-alpha-epsilon,M-b and M-ohm,M-alpha+epsilon,M-b for some epsilon > 0.
In this article, a heat equation with an interior logarithmic source and a nonlinear dynamic boundary condition is considered. After establishing the local well-posedness, the global existence and finite time blow-up results of the solutions with different initial energy levels are given. Moreover, the lower bound of the maximal existence time of the weak solution is deduced for the special case m = 2.
In this paper, we investigate the stability and instability of standing waves for the Schrodinger-Choquard equation with mixed fractional Laplacians. We first establish the existence and stability of normalized standing waves in the L2-subcritical case. Subsequently, we prove the existence and strong instability of normalized ground state standing waves in the L2-supercritical case.
