In this article, we investigate the existence and multiplicity of normalized solutions for the p -Laplacian Schrödinger–Poisson system: { − Δ p u + λ u p − 1 − κ ( | x | − 1 * | u | p ) u p − 1 = f ( u ) in R 3 , ∫ R 3 | u | p d x = a p , u > 0 in R 3 , where a > 0 represents the prescribed L p -norm, κ ∈ R ∖ { 0 } is a parameter, and λ ∈ R appears as an undetermined Lagrange multiplier. Our principal findings are summarized as follows. (i) For κ < 0 , under the conditions that f is odd and satisfies L p -supercritical yet Sobolev subcritical growth, we establish the existence of a normalized ground state solution for sufficiently small a > 0 by employing the Pohožaev manifold method combined with genus theory. In this setting, we prove that the problem admits infinitely many normalized solutions whose energies tend to infinity. The asymptotic behavior of the normalized ground state energy is also analyzed. (ii) For κ > 0 and the Sobolev critical nonlinearity f ( u ) = | u | p * − 2 u , we use a truncation technique together with the concentration-compactness principle to address the lack of lower boundedness of the energy functional. Under appropriate constraints on κ and a , we demonstrate the existence of infinitely many normalized solutions possessing negative energy. These results extend earlier work by some of the authors on p -Laplacian Schrödinger–Poisson systems.
In this article, we investigate the existence of two positive solutions to the following Hardy problem involving critical and variable exponent singular terms − Δ u − μ u | x | 2 = λ ( u − γ ( x ) + u q ) , u > 0 in Ω , u = 0 on ∂ Ω . Here Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with smooth boundary ∂ Ω such that 0 ∈ Ω , λ > 0 is a real parameter, 0 < μ < μ ¯ = ( N − 2 2 ) 2 , 0 < γ ( x ) ∈ C 1 ( Ω ¯ ) and 1 < q ≤ 2 * − 1 = N + 2 N − 2 . By employing the sub- and supersolution technique and Ekeland’s variational principle, we establish the existence of at least two positive weak solutions for a suitable range of parameter λ . To the best of our knowledge, these results remain new even in the constant exponent case γ ( x ) ≡ γ , particularly in the strongly singular range 1 ≤ γ < 3 .
We study the viscous Burgers’ equation with a family of initial data having infinite mass. After rescaling, the solution converges toward a bounded discontinuous profile in the long-time limit. We investigate the solution near this discontinuity point, and show that, by changing once again the scale depending on time near it, we find a new profile, that is also discontinuous. This process can be repeated any arbitrary number of times. In other words, if we think of the discontinuity in the limit profile as a boundary layer, we can change the scale in time near it to get a new profile, that will also have a boundary layer, and this repeats at infinity.
This article deals with a suspension bridge where the main cable is under Kelvin-Voigt damping and the deck is composed of a non-homogeneous flexible structure. The well-posedness is established using the Lumer–Phillips theorem. The uniform stabilization of the system is achieved by constructing a Lyapunov functional that provides an explicit exponential energy decay rate.
In this article, we first present a decomposition of the displacements of a straight beam P δ . The beam has length L and cross-section ω δ = δ ω where ω is a bounded Lipschitz domain in R 2 and δ a small parameter. The beam is clamped at atleast one end and subjected to simple loading. Using the tools of the dimension reduction method, we obtain the limit problems and error estimates (global error, of order δ 1 / 2 , L 2 global error and H 1 interior error with respect to δ ).
This paper focuses on the asymptotic behavior of the eigenvalues of the classical Steklov problem in a thin multidomain as it becomes increasingly narrow. The thin multidomain consists of two vertical cylinders, one positioned atop the other. We show that the eigenvalues tend to zero as the domain becomes thinner. In this case, the limiting eigenvalue problem in the upper cylinder corresponds to a one-dimensional eigenvalue problem, while the limiting problem in the lower cylinder is defined in an (n - 1) -dimensional ball. We conclude our investigation by considering the case in which the thinness of the upper and lower cylinders is equal.
In this article, we study the phenomenon of p-synchronization for a quasilinear system governed by p-Laplacian, p > 2. We obtain this result through the upper semicontinuity of the attractors associated with the systems and the synchronicity property of the limit system.
The goal of the paper is to provide a sequence of eigenfunctions that saturates the L p bounds obtained by Koch and Tataru for the multidimensional Hermite operator. More precisely, several such sequences of eigenfunctions have already been identified by Koch and Tataru, and we present an example in another range of p .
In this paper, we investigate a semilinear wave equation that incorporates localized memory and delay effects. By imposing the delta -condition on the memory kernel and including a semilinear source term, we ensure that the energy of the system decays exponentially. This approach effectively localizes these effects, allowing them to vanish on a set of positive measure. While the viscoelastic dissipation is distributed around the boundary in accordance with the Geometric Control Condition, the delay term is localized such that it does not satisfy this condition. Our findings demonstrate a flexible geometric interaction between memory and delay effects, ensuring stability even when the instability occurs in uncontrolled regions.
We analyze a transmission problem for a string composed of three distinct types of materials: an elastic material with frictional damping, an elastic (non-dissipative) material, and a Kelvin-Voigt viscoelastic material. The dissipative mechanisms may act on separate parts of the string or overlap in certain regions. The decay rate of the solutions depends on the arrangement of these components. Specifically, if there exists an elastic segment that connects with the Kelvin-Voigt viscoelastic material without being in contact with the frictional damping region, exponential stability is not achieved. Instead, the solutions exhibit polynomial decay at an optimal rate of t(-2). In all other configurations, where both damping mechanisms are present, the solutions are exponentially stable.
We investigate a three-dimensional thermo-visco-elastic model with Kelvin-Voigt rheology under small strains confined to a thin domain. The model comprises a quasistatic linear momentum equation, with viscous stresses adhering to a Kelvin-Voigt viscosity law, coupled with a nonlinear heat equation governing temperature. The heat equation incorporates source terms arising from viscous dissipation and adiabatic heat sources due to thermal expansion. The model ensures thermodynamic consistency, maintaining energy conservation, positive temperature, and entropy production. We analyze the asymptotic behavior of solutions as the domain thickness approaches zero, deriving an effective two-dimensional model. This derivation involves rescaling the domain to a fixed thickness and establishing uniform a priori estimates relative to the plate's thickness. In the limit, the temperature becomes vertically constant, and displacement are of Kirchhoff-Love type, enabling meaningful interpretation of the limiting objects within the plate's two-dimensional cross-section. The mechanical equations consist of two parabolic equations, one for the membrane part and one for the bending part. Notably, the viscosity law in the limiting model departs from the Kelvin-Voigt form, reflecting nontrivial kinematic constraints on the rescaled out-of-plane strains. The bending of the plate does not depend on the temperature in the limit.
This paper aims to derive the time decay estimates for both weak and strong solutions to the incompressible magnetic B & eacute;nard system in a half-space. To address the technical challenges posed by the boundary effects in unbounded domains, we primarily employ L q - L r estimates, together with properties of the Stokes operator and a suitable decomposition of the nonlinear terms. We establish the L 2 -estimates of weak solutions and obtain the L q ( 1 <= q <= infinity ) -estimates of strong solutions and their first three derivatives. In addition, if the given initial data belongs to a suitable weighted space, more optimal decay effects are established.
In this article, we study the Poincar & eacute; compactification of the limiting planar semiflow of a coupled partial differential equation and ordinary differential equation system composed of a reaction-diffusion equation with large diffusion coupled with an ordinary differential equation by a boundary condition in a heating transition region. The nonlinear sources are dissipative polynomials. We guarantee conditions to apply the invariant manifold theorem in order to reduce the dimension of the partial differential equation and prove that the compactified vector fields are close in the C 1 -norm.
The Ladyzhenskaya fluid model is proposed to describe more complex physical phenomena of non-Newtonian fluid flow in 1960s. This paper is concerned with the determination of the nonautonomous three-dimensional Ladyzhenskaya model with shear-dependent viscosity equipped with periodic boundary conditions. Under suitable assumptions on the monotonicity of stress tensor, the boundedness of determining modes is estimated for our considered model via the generalized Grashof number by means of energy estimates and the properties of some operators.
We investigate the long-time dynamics of a thermoelastic Lam & eacute; system with memory, where heat conduction is governed by a Gurtin-Pipkin type law featuring a fading memory kernel. The model incorporates a viscoelastic relaxation term characterized by a small parameter omega > 0, representing the thermal relaxation time. Our main goal is to analyze the asymptotic behavior of the global attractors as omega -> 0. We first establish the well-posedness of the system and prove the existence of a finite-dimensional global attractor using the quasi-stability method. We then show that, as the memory kernel is rescaled and the relaxation parameter vanishes, the global attractors converge in the sense of upper semicontinuity to the global attractor of the limiting thermoelastic Lam & eacute; system governed by Fourier's law. Our analysis overcomes the challenges posed by the lack of strong compactness induced by the memory term and provides uniform in omega bounds for trajectories on the attractor. These results contribute to the understanding of singular perturbation problems in thermoelasticity with memory and establish robust convergence properties of the associated attractors.
In this article, we consider the following double-phase problems with Choquard-type nonlinearity: { - & varepsilon;(p)Delta(p)v - & varepsilon;(2) Delta v + V(x)( v+|v|(p - 2) v) = f(v) + (I-alpha (*) |v|(2)) v in R-2 , v > 0 , v is an element of W-1,W-p(R-2) boolean AND W-1,W-2 (R-2) , where 1 < p < 2 , & varepsilon; > 0 is a parameter, V is an element of C (R-2 , R ) , f : R -> R is a continuous function with critical exponential growth condition, I alpha is the Riesz potential with 0 < alpha < 2 . Under some hypotheses, the existence and concentration behavior of positive solutions are established by variational methods.
We investigate the long-time dynamics of a non-autonomous stochastic Benjamin-Bona-Mahony equation on a high-dimensional unbounded channel domain O subset of R d , driven by an operator-type Laplace-multiplier noise. We construct a non-autonomous random dynamical system on H 0 1 ( O ) and prove the existence of a compact pullback random attractor, which is cocycle-invariant and pullback attracts every tempered random set in H 0 1 ( O ) . The lack of compactness of Sobolev embeddings on unbounded domains is overcome by combining spectral decomposition on bounded truncations with uniform tail estimates outside large bounded sets.
In this work, we investigate the well-posedness and asymptotic behavior of a system of laminated beams subjected to thermal effects governed by the Maxwell-Cattaneo model, acting through the shear force. All results are established using the semigroup theory of linear operators. In particular, we apply the Gearhart-Huang-Pr & uuml;ss Theorem to prove that exponential decay occurs if and only if the stability number Upsilon tau vanishes. Otherwise, we invoke the Borichev-Tomilov Theorem to demonstrate polynomial decay. The results significantly enhance the understanding of the interplay between thermal relaxation governed by the Maxwell-Cattaneo law and structural mechanisms in the stabilization process, while also addressing a gap in the literature on the dynamic behavior of laminated beams under thermoelastic shear effects.
In the current work, we propose a discrete version of the Ricci-Bourguignon flow, based on the notion of Ollivier-Ricci curvature, to investigate the evolution of weighted graphs. This flow generalizes the Ollivier-Ricci flow in community detection problems. By deriving essential Lipschitz estimates for curvature operators and using standard ODE theory, we prove local results, including existence, uniqueness, continuous dependence, and a blow-up criterion for solutions to the proposed equation. In addition, we derive global-in-time results for some specific cases with idleness parameter larger than 1 / 2 . Precisely, considering a path graph of length 3 and the Wasserstein distance defined with the resistance mobility function, we obtain not only the global well-posedness of the solution but also its asymptotic behaviors. For a star graph with one center and the Wasserstein distance defined with the linear mobility function, we show that the proposed flow coincides with a normalized Ollivier-Ricci flow which preserves the total volume of the graph. Furthermore, we prove that under the action of the flow, the graph converges to a uniform star graph.
The quantitative analysis of stochastic homogenization problems has been a very active field in the last 15 years. Whereas the first results were motivated by applied questions (namely, the numerical approximation of homogenized coefficients), the more recent achievements in the field are much more analytically driven and focus on the subtle interplay between partial differential equation analysis (in particular, elliptic regularity theory) and probability (concentration, stochastic cancellations, and scaling limits). The aim of this article is threefold. First, we provide a complete and self-contained analysis for the popular example of log-normal coefficients with possibly fat tails in dimension d = 1 , establishing new results on the accuracy of the two-scale expansion and characterizing fluctuations (in the perspective of uncertainty quantification). Second, we work in a context where explicit formulas allow us to bypass analytical difficulties and therefore mostly focus on the probabilistic side of the theory. Last, the one-dimensional setting gives intuition on the available results in higher dimensions (provided the results are correctly reformulated), to which we give precise entries to the recent literature.
