In this paper, we study Liouville-type theorems for the threedimensional stationary Boussinesq system. Using the Littlewood-Paley theory and frequency-space decomposition, we derive a new sufficient condition ensuring that the only finite-energy solution is the trivial one. Specifically, we prove that if the low-frequency components of the velocity and temperature fields satisfy a certain localized integrability condition, then both fields must vanish identically. Our result generalizes recent progress on the stationary Navier-Stokes equations and highlights the critical role played by the behavior of solutions near the origin in frequency space.
In this paper, we deal with a chemotaxis-haptotaxis model that incorporates porous medium diffusion triangle u(m). { ut = triangle u(m)-chi V & centerdot; (u(k)Vv)-xi V & centerdot; (u(k)Vw), x is an element of ohm, t > 0, vt= triangle v + u-v, x is an element of ohm,t > 0, wt=-vw +eta w(1-u-w), x is an element of ohm,t > 0, partial derivative u(m)/partial derivative nu-chi u(k) partial derivative u/ partial derivative nu -xi u(k) partial derivative w/partial derivative nu| (partial derivative ohm )= 0, partial derivative u/ partial derivative nu|(partial derivative ohm) = 0, t > 0, u(x, 0) = u(0)(x), v(x, 0) = v(0)(x), w(x, 0) = w(0)(x), x is an element of ohm, where ohm subset of R-N is a bounded domain with smooth boundary partial derivative ohm. It is proved that (i) If m-k >= N-1+k/ N+1 , N >= 2, k is an element of (0, 1), the weak solution will always exist globally. (ii) If m-k < N-2/ N , N >= 2, k > 0, there exist blow-up solutions. (iii) If N-2/N< m-k< N-1+k/ N+1 , N >= 2/ 1-k, N is an element of N+, k is an element of (0, 1), the weak solution will exist globally.
. We investigate a dynamic thermo-visco elastic contact problem involving a deformable body in frictional interaction with a thermally conductive foundation. The model incorporates infinite memory effects through a hereditary integral term, a thermal normal damped response law, Tresca's friction condition, and a regularized heat exchange mechanism at the contact interface. We first derive a weak variational formulation of the problem and establish the existence and uniqueness of weak solutions. Global solvability is then obtained using nonlinear semigroup theory. We prove the exponential stability of the associated energy, showing that solutions converge exponentially to equilibrium. Finally, these analytical results are illustrated through simplified numerical simulations inspired by the thermo-viscoelastic dynamic contact problem with infinite memory under consideration.
Quasi-Steady State Approximation approach is used to derive reduced models of facilitated diffusion for the cases where adjusted concentration of mediators located on the boundary separating two compartments is much smaller compared with the concentration of particles in these compartments transported across the boundary via carrier-mediated transport mechanism. The cases of a generic mediator kinetics scheme as well as of more realistic mediator reactions are studied for symmetric boundary permeable in both directions. It is shown that for the generic case, under condition of sinks and sources absence, the newly derived reduced facilitated diffusion formula can be transformed to a model of an ordinary diffusion between compartments following Fick's Law. However, for the more realistic kinetics case, such transformation is not possible due to the presence of non-linearity in the corresponding reduced model. For semi-permeable boundary, i.e., when the particles can be transported across the boundary only in one direction, from one compartment to another, but not in the opposite direction, the derived formulas can be further reduced to produce, as a special case, the classical Michaelis-Menten-Henri kinetics approximation. The derived approximate formulas are intended for use in multiple medical, neuroscience, biological, and other applications.
In this paper, we propose to study a new partially discontinuous dynamical system governed by a discontinuous non-convex sweeping process and a differential inclusion with subdifferentials. A new concept of the couple solution occurs. Through mixing a catching-up algorithm and a perturbed differential inclusion of subdifferential type, we construct two sequences that converge to the required couple solution. Then, we derive an existence result to a discontinuous second-order sweeping process coupled by an evolution inclusion with subdifferentials via a fixed point approach.
We consider the wave equation on an N-edge star of strings with fixed outer endpoints and a Neumann-type control at the central node, imposed by Sigma(N)j=1 uj,xspace(t, 0) = h(t). We derive a Green-type identity adapted to this central actuation and obtain the moment formulationspace integral0(T) betanspacee(i lambda ntspace)h(t) dt = u1,nspace- i lambdanspaceu0,n,space space space betan = thetan(0),spacespace space where (lambda(2)n, thetan) are eigenpairs of the graph Laplacian. The spectrum is encoded by an entire generating function q, whose real zeros are precisely the eigenfrequencies. From the properties of these eigenfrequencies, we place the analysis in the broader framework of nonharmonic Fourier series. This connects our moment method approach to the Ingham-Beurling type observability results with weakened gap conditions developed for networks with a common node. Using the Paley-Wiener biorthogonal family associated with q, we construct controls in L-2(0, T) for every T > 2 Sigma(N)j = 1 & ell;j.space space Assuming Q-linear independence of the lengths, which ensures that the eigenmo des have nonzero values at the central node, we obtain spectral controllability.space space Finally, we relate the control cost to the quantities betan and q '(lambdan), showing that modes close to Dirichlet frequencies on individual edges can require large controls, and we confirm this behavior numerically in MATLAB.
We study discrete-time fractional evolution equations of Hilfer type on Banach spaces. For the associated linear problems, we establish existence, uniqueness, and explicit representation formulas under a simple spectral invertibility condition without assuming that the governing operator generates a strongly continuous semigroup. Sharp bounds for the solution families are obtained in terms of discrete Mittag-Leffler sequences and are then used to prove p-well-posedness pound for semilinear perturbations via a fixed point argument. In addition, we develop maximal p-regularity pound and a priori estimates for the in-homogeneous linear problem under standard multiplier assumptions in UMD spaces. Building on these estimates, we introduce controlled Hilfer-type dynamics with an additive input term Bg, derive admissibility and input-to-state bounds, and prove the existence of optimal controls for a natural p pound tracking functional. Several examples, including discrete Laplacians, elliptic-type operators, and diagonal models, illustrate the scope of the abstract framework.
. A recent contribution extended the Fourier transform and the fractional Laplacian to the context of fractional calculus with respect to a function. Building upon this work, we further explore this line of research by introducing a weight and showing new properties. We study the weighted Fourier transform with respect to a function, and proved that it satisfies the key property that the transform of the convolution is the product of the transforms. We introduce and develop the theory on the weighted fractional Laplacian: principal-value and proper-integral definitions, function space domain, transmutation relation, Fourier transform, and integration by parts. For the first time, we introduce and study the Riesz and Bessel potentials in this context. We illustrate the operators in the context of fractional partial differential equations.
This paper deals with global existence and blow-up for an attraction-repulsion chemotaxis system with diffusion rate m >= 1 in the fair-competition regime. For either the case of linear diffusion m = 1 in d = 4, or nonlinear diffusion m = m(& lowast;) = 2-4/d >= 1 in d >= 5, there exists a critical exponent M-c> 0. If the total mass M > 0 of cells satisfies M < Mc, solutions exist globally. However, if M > M-c, solutions will blow up in finite time. Moreover, the global well-posedness is established when M is equal to Mc, specifically for the case when m = 1 in d = 4.
. We establish new Halanay-type inequalities for systems involving the Caputo-Hadamard fractional derivative with unbounded time-varying delays, extending the results of [J. Math. Anal. Appl., 525(1):127145, 2023]. For the linear case, we derive a sharp and explicit decay rate under delay condition (D), thereby resolving the open problem posed in [Fract. Calc. Appl. Anal. 25(6):2420-2445, 2022]. For homogeneous power-type nonlinearities, we obtain refined upper bounds and, in the superlinear regime, also establish matching lower bounds, demonstrating that the decay explicitly depends on both the fractional order and the nonlinearity exponent. As an application, we prove Mittag-Leffler stability for a class of Hopfield-type neural networks with proportional delays. These results extend and sharpen existing inequalities in the literature and provide effective tools for the stability analysis of fractional-order systems with delays.
. This paper investigates a thermo elastic suspension bridge system governed by Fourier heat conduction law, incorporating visco elastic memory effect in the cross-sectional rotation angle and internal damping in the main cable. By introducing a relative history function to transform the memory term, the system is reformulated as an abstract Cauchy problem in a properly constructed Hilbert phase space. Utilizing the theory of operator semigroups, two key results are established: first, the system is well-posed, admitting a unique mild solution continuously dependent on initial data (and a classical solution for smooth initial data); second, the system exhibits exponential stability without requiring the conventional equal wave speeds condition. The analysis extends existing results on suspension bridge models by integrating visco elastic memory, providing new insights into the dynamic behavior of thermo elastic structural systems with memory effects.
. In this work, we investigate a dynamic contact problem between a piezo-elastic body and a rigid foundation. The interaction is governed by a unilateral contact condition combined with Coulomb friction. We derive a variational formulation of the model and establish the existence and uniqueness of a weak solution by employing tools from second-order nonlinear evolution variational inequalities, together with the Lax-Milgram theorem and a fixedpoint argument. A fully discrete finite element scheme is then proposed to approximate the variational problem, and corresponding error estimates for the discrete solution are obtained. Finally, we present numerical simulations for a two-dimensional test case that illustrate the performance of the proposed methods.
. In this paper, we deal with a feedback optimal control problem for the stochastic Cahn-Hilliard equation with a polynomial of degree 3 free energy and Wiener multiplicative noise. More precisely, we investigate the existence of optimal feedback controls for the stochastic Cahn-Hilliard equation, controlled by different external forces, which are feedback controls. In addition, using the Galerkin's approximation method, we show that the optimal cost can be approximated by a sequence of finite dimensional optimal costs, showing the existence of the & varepsilon;-optimal feedback controls for the stochastic Cahn-Hilliard equation.
