This paper is concerned with the existence of a ground state solution for the following class of elliptic Kirchhoff-Boussinesq type problems given by Δ ^2 u ±Δ _p u + V(x)u= f(u) +γ |u|^2_**-2u in ℝ^N, where 2< p≤ 2^*= 2N/N-2 for N≥ 5 and 2_**= 2N/N-4 . The potential V can change sign and f is a continuous function with subcritical growth and behaves like |u|^q-2u with p< q< 2_** . We show the existence of a ground state solution using variational methods, Splitting Lemma and considering the subcritical case, i.e, γ =0 and the critical case, i.e, γ =1 .
The Riemann problem for the varying Chaplygin gas Euler equations with zeroth-order terms is investigated. By the varying Chaplygin gas, we mean that the fluid obeys the pressure-density-time relation where the pressure is negative and is the product of a function of time and the inverse of the density. It includes the Chaplygin gas as an exceptional case. Through the variable substitution method, the solution formed by two contact discontinuities and the solution with delta-shock are constructed in completely explicit forms. Under the generalized Rankine-Hugoniot relation and entropy condition, the position, propagation speed and strength of the delta-shock are uniquely determined. It is discovered that the shape of delta-shock in characteristic plane is either an exponential type curve or a straight line. An interesting phenomenon concerning the delta-shock is that even when the delta-shock is a straight line, its weight is no longer linear function of time. The phenomenon is attributed to the presence of the mass source term. It means that mass source term can play a role in adjusting the weight of the delta-shock. Additionally, it is proved that the constructed Riemann solutions tend to those of nonhomogeneous zero-pressure Euler equations with the same initial data as varying Chaplygin gas tends to zero, and the constructed Riemann solutions converge to the corresponding ones of Chaplygin gas Euler equations with same initial data as both source terms vanish simultaneously.
We consider the Eulerian droplet model with a varying generalized Chaplygin gas, which possesses negative pressure and is a product of a function of time and the inverse of a power of the density. The non-self-similar Riemann solutions are obtained, and then the limiting behavior of the Riemann solutions is investigated. As the parameter decreases to a critical value, the two-shock solution converges to a delta shock wave of the same system. Further, when the parameter goes to zero(vanishing pressure), the solution becomes exactly the delta shock wave solution of the zero-pressure Eulerian droplet model. Moreover, the two-rarefaction wave solution tends to a solution consisting of four contact discontinuities together with vacuum states, which is different from the vacuum solution of the zero-pressure Eulerian droplet model. The solution containing one rarefaction wave and one shock wave tends to the contact discontinuity solution of the zero-pressure Eulerian droplet model as pressure vanishes.
This work focuses on Nicholson’s blowflies model with mixed time-varying delays, a non-linear Immigration term, and (ς _1,ς _2) -pseudo-compact almost-automorphic parameters. The main contribution consists of improving and generalizing some existing results about (ς _1,ς _2) -pseudo compact almost-automorphic functions. Then, based on differential inequalities, the fixed point theorem, and the Halanay inequality, we establish some sufficient conditions for the existence, stability, and attractivity of positive solutions for the model considered. Further, we present a numerical example to support and illustrate the theoretical findings. Finally, a conclusion is drawn. Our findings are new and extend some existing results in the literature.
This paper studies the well–posedness and averaging for the dispersion–managed nonlinear Schrödinger equation (NLS) with general nonlinearity, where the dispersion map γ is assumed to be piecewise constant. We establish the local well–posedness of the equation in the Sobolev space H^m(ℝ^N) by applying a fixed–point theorem. Furthermore, we analyze the convergence of the solutions of the original equation to the solutions of the averaged equation. The analysis relies on linear propagators and Strichartz estimates to derive the main results. This study helps us improve our understanding of the dynamic behavior of dispersion–managed NLS and provides a mathematical framework that can be used in different physical systems.
In this paper we study a haptotaxis and diffusion model of cancer invasion. The model consists of a parabolic PDE coupled with an ODE. The PDE itself contains the gradients of both unknowns, as well as well as the potential to be degenerate. This degeneracy appearing in the PDE provides the main source of difficulty for this system. We derive new estimates showing solutions must be bounded which then enable us to prove the global existence of weak solutions.
This study proposes a neural networked Gierer-Meinhardt model to characterize the spatiotemporal dynamics of glutamate and γ -aminobutyric acid (GABA) by incorporating both free-diffusion and cross-diffusion mechanisms. The primary objective is to investigate whether diffusive terms can destabilize an initially stable Hopf bifurcation periodic solution and subsequently trigger the emergence of irregular spatiotemporal patterns. Using Floquet multiplier analysis, we rigorously evaluate the potential of diffusion-driven perturbations to disrupt the periodic solution, thereby establishing a quantitative basis for identifying the onset of diffusion-induced pattern formation. Numerical simulations confirm our theoretical predictions and further explore the effect of networked cross-diffusion on the dynamics. By integrating reaction-diffusion dynamics with network topology, this framework provides a mathematical foundation for analyzing the interplay between excitatory and inhibitory neurotransmitter models in neural networks and their role in spatiotemporal pattern formations.
In this paper we consider the fourth-order elliptic equation describing the Kirchhoff-Love model for pure bending of a thin solid symmetric plate under a transverse load. We derive Hashin-Shtrikman bounds on the complementary energy, and calculate these bounds explicitly for mixtures of two isotropic materials in dimension d=2 . Moreover, we give bounds on the bulk and shear moduli of an isotropic two-phase composite material, and show that these bounds are optimal, i.e. that they are attainable by finite-rank sequential laminates. These results pave the way for numerous applications of the homogenization theory in optimal design problems for stationary elastic plates, for example the development of an optimality criteria method for two-dimensional compliance minimization problems.
This paper addresses the problems of global well-posedness and asymptotic stabilization for a viscoelastic Euler-Bernoulli beam model incorporating distributed damping and a nonlinear feedback with time delay acting at the right boundary as a shear force. The nonlinearity is merely assumed to be Lipschitz continuous and to vanish at the origin, without satisfying boundedness or monotonicity as in saturation or cone-bounded cases. By a change of variables to handle the delay, the system is reformulated as an evolutionary coupled beam-transport system in a suitable Hilbert space. We then show that the corresponding nonlinear operator generates a contraction semigroup, ensuring the global well-posedness of the system. Furthermore, by designing an appropriate Lyapunov functional and applying LaSalle’s invariance principle in infinite dimension, we establish the global asymptotic stability of the system. This work extends existing results by effectively addressing the complex interaction between a distributed damping mechanism, a broad class of nonlinearities, and a boundary time delay. A numerical simulation validating the proposed approach is provided.
In this paper, we study the degenerate beam equation on (0,1) with local damping. This damping is effective in a subset ω :=[x_1,x_2] of (0,1) and the damping coefficient may vanish in some subsets of (0,1) . As the first step, we prove the existence of a solution for the degenerate beam equation. Then, we derive the exponential stability result of the system.
This paper examines the inhomogeneous Neumann boundary value problem for a high-dimensional chemotaxis-consumption model with a logistic source, under the nonlinear boundary condition |u|^p , where 1 < p < 3/2 . The initial data u_0, v_0 are assumed to be nonnegative and satisfy k(k-1)/2(k+1) ( 4(k-1)(4k^2+n)v_0^2_L^∞(Ω )/k+1 )^1/k + 2(k + n -1)v_0_L^∞(Ω )^2/k+1 ( 8(k-1)(k+n-1)(4k^2 + n)v_0_L^∞(Ω )^4/k+1 )^k-1/2 < μ for some k > max{1,n/2} . We prove the existence of classical bounded global solutions for the system in bounded convex domains with a smooth boundary.
In this work, we propose a multiscale finite element method for solving heterogeneous nonlinear parabolic problems involving Duhem operators that model hysteresis in spatially varying media. A formulation of the method is introduced to facilitate the mathematical analysis, linking microscopic heterogeneities to macroscopic behavior. We establish the existence, uniqueness and boundedness of the numerical solution for both periodic and Dirichlet coupling scenarios, laying a strong foundation for the practical implementation of the multiscale method in computational settings.
This paper focuses on the convergence of invariant measures in the Wasserstein sense for stochastic FitzHugh-Nagumo lattice systems featuring one-sided dissipative nonlinearities satisfying f'(z)≤κ <0 in weighted spaces as the noise intensity tends to zero. By utilizing uniform estimates of solutions, we establish that the family of invariant measures of the stochastic systems converges to the invariant measure of the corresponding deterministic systems with respect to the Wasserstein metric. Additionally, we provide an estimation for this convergence rate.
The multiplier method is a vital implement for investigating the decay and long-time behavior of solutions to hyperbolic equations with nonlinear damping. This paper examines the role of the strong damping term on the decay estimate of solutions, which extends our previous work (Li and Li in Evol. Equ. Control Theory 13(1):116–127, 2024). For the subcritical cases, energy estimates combined with Komornik inequality yield exponential decay under strong damping – a sharp contrast to the algebraic decay induced by single nonlinear damping. For the supercritical case, to overcome the failure of the embedding H_0^1(Ω )↪ L^m(· )(Ω ) , we establish a priori estimate and utilize a weighted multiplier method to demonstrate energy decay estimates, taking into account the effects of both single nonlinear damping and mixed damping. The main results are compiled in Table 1, and a comparative summary of the decay laws is provided in the final section.
This article presents an asymptotic analysis of a periodic micropolar fluid flow in an infinite thin channel with an impervious wall and an elastic stratified stiff wall in the context of fluid-structure interaction problems. The considered model was introduced in a previous article (Panasenko et al. in Math. Model. Anal. 29(4):641–668, 2024) and it depends on a small parameter, ε , defined as the ratio between the thickness of the elastic structure and that of the fluid layer. We perform an asymptotic analysis with respect to this small parameter for a suitable scaling of the physical data depending on ε . Corresponding to this choice, the densities are of order ε^0 , the elasticity coefficients (the Young’s moduli of the stratified elastic structure) are of order ε^-3 and the external forces acting on the elastic material are of order ε^-1 . We define an asymptotic solution of order J expressed by means of matrix-valued, vectorial and scalar functions and then we construct and solve the problems for these unknown functions. We provide next a rigorous justification of the asymptotic construction. More precisely, we show that the error between the solution of the physical problem and the asymptotic solution of order J with respect to suitable norms is of order 𝒪(ε^J+υ) for any J ≥ 0 and υ a positive fixed number. This means that the asymptotic solution of order J represents a good approximation of the exact solution even for J=0 , that fully justifies our asymptotic construction.
Fourier spectral methods demonstrate an exponential convergence rate when approximating an analytic function. However, for non-smooth and discontinuous functions, this exponential convergence rate diminishes to first order in the smooth regions and produces oscillatory behaviour known as the Gibbs phenomenon. Due to the global nature of spectral methods, these oscillations extend to the whole domain of the function. In our work, we construct a mollifier based on a summability kernel, known as the Rogosinski kernel. Further, we proved that convolving the oscillatory Fourier approximation with the proposed mollifier produces spectral-like convergence. The proof is established by splitting the total error into truncation and regularization errors. Additionally, the parameters are optimised, and the results are numerically compared with those of the existing adaptive Dirichlet mollifier.
This study presents two main contributions concerning the oscillatory behavior of Duffing-type nonlinear RLC circuits with a nonlinear voltage-charge capacitor. First, using the well-known Riccati-type technique, we establish a comparison theorem linking the oscillation of nonlinear differential equations to that of corresponding linear equations, showing that classical linear oscillation results can be extended to a wide class of nonlinear systems under suitable structural conditions. Second, we apply this theorem to Duffing-type nonlinear RLC electrical circuit models, deriving explicit conditions under which all solutions oscillate. This approach demonstrates how linear oscillation theory can be rigorously applied to nonlinear circuits. Moreover, the proposed approach is not restricted to circuit models and is applicable to differential equations with periodic coefficients, including special cases of nonlinear Mathieu-type differential equations. Such equations are known as models describing coefficient excitation and parametric resonance phenomena. Overall, these results indicate that linear oscillation theory provides a unified and effective analytical framework for the study of nonlinear differential equations arising in applied sciences and engineering.
This study investigates an epidemic framework governed by differential equations featuring delays of generalized piecewise constant structure (DEGPCD). The primary objective is to construct an invariant region and to demonstrate the existence and uniqueness of solutions by employing integral equation techniques under appropriately defined conditions. Additionally, an auxiliary lemma is derived, establishing a precise connection between the functional values of the state variable within the delay argument and the temporal domain. To investigate the model’s dynamic behavior, the Lyapunov–Razumikhin framework is utilized, tailored to accommodate the structural features introduced by the DEGPCD. The stability of the infection-free state is rigorously examined, while the positive steady state is transformed into an equivalent zero equilibrium to facilitate the analysis. Sufficient criteria are then formulated to guarantee uniform asymptotic stability for both equilibria, offering theoretical insights that enhance the applicability of DEGPCD-based epidemic modeling.
We consider the Cauchy problem for chemotaxis-Navier-Stokes equations with nonlinear diffusion Δ n^m in ℝ^2 . By exploring the new a priori estimates, we establish the global existence of weak solutions to the chemotaxis-Navier-Stokes equations with m>1 . This result extends the bounded domain in reference (Tao and Winkler in Discrete Contin. Dyn. Syst. 32:1901–1914, 2012) to the entire space.
