In this paper, we study a class of systems of semilinear wave equations modeling double-wall carbon nanotubes. Initially, we derive the local solution employing monotone operator theory. Additionally, we also establish the uniqueness of the weak solution and its continuous dependence on the initial data. Furthermore, through the construction of a family of potential wells (which introduced in [T. Q. Minh et. al. J. Differ. Equ. 418 (2025), 374-458.]), we establish the global existence, asymptotic behavior, and blow-up of solutions for subcritical initial energy and critical initial energy, respectively. One significant advantage of our approach is the stabilization estimate, which avoids generating lower-order terms and renders the proof of asymptotic dynamics more succinct and concise. Finally, we provide evidence for the finite-time blow-up of solutions in cases with arbitrarily positive initial energy. We also prove the existence of a global attractor and provide some its properties.
In this paper, a coefficient inverse problem for a non-local in time parabolic equation is considered. This equation arose when describing the chaotic motion of a polymer chain in a liquid. The role of time in it is played by the arc length parameter along the chain. The solution is the probability density function that a chain link is located at a particular point in space. Accordingly, the overdetermination condition for the inverse problem is that the integral over space of the solution is equal to one at every moment of time. The position of each chain link is affected by all other links, so the interaction potential included in the equation depends on the integral of the solution to the problem over the entire length of the chain. This means that the parabolic equation under consideration is essentially non-local in time and includes an integral of the solution over the entire time interval over which the problem is being solved. Therefore, to determine some of the coefficients in the equation, it is necessary to know the "future". This situation is unusual for parabolic equations. The solvability of the problem with homogeneous Neumann boundary conditions is proven without restrictions on the input data size, in particular, on the length of the time interval over which the problem is being solved. It is only assumed that the interaction potential is a bounded function.
The parabolic-parabolic-elliptic forager-exploiter model with kinetic terms is investigated in a smoothly bounded domain of the n dimensional space. It is demonstrated that if n= 2, or n >= 3 under weak chemotactic effect on exploiters, its classical solution exists globally and is also uniformly bounded. Furthermore, for different competitive scenarios: (i) weak exploiters versus weak foragers, the foragers and exploiters are shown to approach spatially homogeneous distributions; (ii) weak exploiters versus strong foragers, the exploiters tend to go extinct; (iii) strong exploiters versus weak foragers, the foragers will become extinct. This enriches the work of Wang and Xu (Differ. Integral Equ., 2023).
In this paper we consider a modified Kobayashi-Warren-Carter system describing one-dimensional grain boundary motion. We provide a new analytical framework by interpreting the state-dependent coefficient in the kinetic equation as a deformation of the underlying Hilbert space structure. More precisely, the system is formulated as a quasi-variational evolution inclusion on a family of twisted Hilbert spaces with state-dependent inner products. This formulation incorporates the nonlinear structure into the geometry of the state space and yields a unified and transparent description of the model. As a result, we establish the existence and uniqueness of global-in-time solutions to the corresponding initial-boundary value problem.
In this paper, we establish the existence and multiplicity of solutions for a class of quasilinear elliptic problems involving a double phase operator with variable exponents and perturbed concave and logarithmic convex nonlinearities. More precisely, we prove the existence of two distinct solutions: the first is obtained via the Mountain Pass Theorem, while the second is derived by applying Ekeland's Variational Principle.
We consider the nonlinear beam model introduced by Gao (1996), which describes large deflections under small strain assumptions. Although this model has been widely employed in applied mechanics and numerical simulations, a rigorous mathematical analysis has so far remained incomplete. In this work, we establish well-posedness of the associated initial value problem by means of semigroup theory. Moreover, in the presence of a single weak damping mechanism, we prove the existence of a finite-dimensional global attractor for the corresponding dynamical system, thus characterizing its long-time behavior. Finally, we investigate the associated stationary problem and obtain a multiplicity result, showing in particular that the global attractor is not reduced to a singleton.
In this paper, we study the stability of global attractors for semilinear parabolic equations in the Sobolev space W1,p. More precisely, we prove that, generically, the dynamical system induced by a reaction diffusion equation with the Robin boundary condition is Gromov-Hausdorff stable on its global attractor under Lipschitz perturbations of both the domain and the nonlinearity.
This paper investigates a nonlinear free boundary problem modeling the growth of triple-layered tumor with angiogenesis. The model incorporates a nonlinear boundary condition that captures the physiological process of angiogenesis, and involves a constant sigma(0) that represents the nutrient equilibrium between capillary supply and tumor consumption. Due to the lack of an explicit expression for the solution, we employ the maximum principle to rigorously analyze the properties of the solution and the two inner interfaces. We prove the existence of a unique radial stationary solution provided that the external nutrient level satisfies (sigma) over bar > (sigma) over tilde. Moreover, the global well-posedness of the problem and the asymptotic stability of stationary solutions are also studied. Our results further highlight the critical role of the nutrient equilibrium constant sigma(0) in the formation of triple-layered tumor.
In this paper, we establish a sharp Gagliardo-Nirenberg inequality involving combined nonlinearities and apply it to the study of normalized solutions for Kirchhoff-type equations. Unlike the classical Gagliardo-Nirenberg inequalities with a single nonlinear term, the inequality considered here contains two nonlinear terms with different homogeneities, which leads to a more delicate balance in the determination of the sharp constant and the associated minimizers. The main difficulty arises from the interaction between the two nonlinear terms. To address this issue, we introduce a scaling argument that allows the two nonlinear contributions to be estimated separately. This leads to a refinement of the minimization-sequence approach used in the classical Gagliardo-Nirenberg inequality.As an application, we investigate a class of Kirchhoff-type equations with combined nonlinearities under a prescribed L2-constraint. The sharp inequality allows us to derive a precise criterion for the existence and non-existence of normalized solutions. Our results show that the interaction between the two nonlinear terms plays a decisive role in the variational structure of the Kirchhoff functional and in the existence of normalized solutions.
We propose a new nonlinear system of balance laws arising in chromatography, where the adsorption capacity of the solid (stationary) phase is allowed to vary with time. The model is derived from a Langmuir isotherm framework with two components and includes a time-dependent saturation parameter n(t), leading to a nonautonomous system with both time-dependent flux and damping source terms. We analyze the hyperbolic structure of the resulting system, establishing strict hyperbolicity and genuine nonlinearity. The associated Riemann problem is solved explicitly, and the structure of shock waves, rarefaction waves, and contact discontinuities is characterized. Furthermore, we prove the existence and convergence of solutions to a regularized viscous system, and we show rigorously the existence of delta-shock wave solutions in the vanishing viscosity limit. The model recovers the classical chromatography system when the adsorption capacity n is constant, and thus provides a physically motivated extension that accounts for evolving adsorption dynamics. This work contributes to the mathematical theory of chromatography by incorporating realistic, time-dependent operating conditions, and includes a physical admissibility analysis and numerical experiments that illustrate the theoretical results.
In this paper, we investigate the global well-posedness of the 2D incompressible inhomogeneous magnetohydrodynamic (MHD) equations with variable electrical conductivity in the framework of Besov spaces. Specifically, for p, lambda & euro; [2,infinity) satisfying 1/(2) < 1/(p )+ 1/(lambda) <= 1 and p >= 2, we assume that -14 the initial velocity and magnetic fields (u(o). B-o) is an element of (L & sup2;(R-2) boolean AND B-p,1( -1+2/p)(R & sup2;))& sup2; are solenoidal vector fields. Additionally, the initial density p(o) is assumed to satisfy 1 rho B---1(0)2,1(1+alpha)(R-2) for a is an element of (0, 1) and K-1 <= P not less than K-2, where K-1, K-2 are two positive constants. Under these assumptions, we establish the global well-posedness of the system (2). Furthermore, if the electrical conductivity coefficient is a positive constant, we can prove the corresponding global existence result with 1 --rho(-1)(0)is an element of B-lambda,1(2/lambda) (R & sup2;), climinating the need for the coefficient a is an element of (0, 1). In other words, we achieve global solvability of the system (2) in the critical Besov spaces. However, the uniqueness result also require the condition 1 rho B---1(0)2,1(1+alpha)(R-2).
This paper is concerned with the well-posedness and asymptotic behavior of a plate equation with degenerate memory and localized frictional damping in the case that the exponent of nonlinear source term is at most critical. When the memory term is non-degenerate and the memory kernel decays exponentially, the corresponding dynamical system possesses a global attractor without requiring any additional dissipation. In this paper, we address the degenerate case by introducing a supplementary frictional damping. Despite the fact that the dissipation arises from two partial damping terms of different natures, neither of which necessarily satisfies a geometric control condition, we can still prove the well-posedness of such problem based on the nonlinear semigroup theory under some suitable assumptions. Moreover, we also prove the existence of a finite dimensional global attractor by verifying the existence of a strict Lyapunov functional and establishing the desired quasi-stability inequality.
The existence of global weak solution for a generalized two-component shallow water system with higher-order inertia operators is proved. A special kind of global weak solution but not strong solution—traveling wave solution, is then presented.
Cervical intraepithelial neoplasia (CIN) is the development of abnormal cells on the surface of the cervix, caused by an infection with human papillomavirus. Although in most cases it is resolved by the immune system, a small percentage of people may develop to more severe CIN, which, if left untreated, can progress to cervical cancer. Cervical cancer is the fourth most common cancer among women worldwide. This work intends to develop a nonlinear mathematical model that describes the dynamics at the microscopic level of the epithelial cells (healthy, dysplastic, cancer and immune), and the viral particles, while accounting for a dual immunotherapy. Using the two-scale convergence method, we derive the effective macroscopic model, thus highlighting the influence of the microscopic properties of the tumor on the overall dynamics of CIN. The resulting model is new in the literature and is simpler to handle as far as the computational aspects are concerned.
In this paper, we consider the Cauchy problem for a highly nonlinear shallow water model arising from the full water waves with Coriolis effect. By the transport equation theory and the classical Friedrichs regularization method, the local well-posedness for this shallow water model in the critical Besov spaces B1+ 1p (p,1) is established, and the ill-posedness for this model in the critical Besov spaces B-s (p,infinity) with s > max {3/2 , 1 + 1/p} is also obtained. Moreover, the precise blow-up criteria for the strong solutions of this equation are determined in Besov spaces B-p,r(s). Finally, the Gevrey regularity and analyticity of the solution of this model are presented.
In this paper, we consider a singular Kirchhoff double phase problem with a right-hand side that consists of a singular and superlinear reaction term. Moreover, we allow critical growth on the nonlinear Neumann boundary condition. Using an equivalent norm in our function space and very general assumptions on the data, we prove the existence of two weak solutions whereby the first solution turns out to have negative energy sign while for the second one it is positive. Our proofs are based on variational methods and minimization of the associated energy functional over certain subsets of the Nehari manifold which are characterized by the corresponding fibering function.
In this paper, we characterize all polynomial Kolmogorov vector fields for which the standard #-sphere is invariant. We exhibit completely integrable Kolmogorov vector fields of degree mon S-n for any m > 2 Then, we show that there is no cubic Hamiltonian Kolmogorov vector field that makes an odd-dimensional sphere invariant. We examine the conditions under which a cubic Kolmogorov vector field has a Darboux first integral. In many cases, we determine whether they constitute necessary and sufficient conditions. Moreover, we study the complete integrability of cubic Kolmogorov vector fields having an invariant -sphere.
We study a system of reaction-diffusion equations posed on a bounded domain composed of subdomains separated by a connected network with a metric graph structure. The reaction-diffusion dynamics with anisotropic diffusion on the graph edges are coupled to well-mixed ODE dynamics occurring at the vertices by junction conditions, and to similar PDE dynamics occurring on adjacent subdomains through Robin-like boundary conditions. The resulting PDE-ODE system can be used in epidemiological and ecological settings to study population movement in between cluster centers along road-like structures and into the surrounding continuum. We employ a semi-Galerkin approximation to establish the well-posedness of weak solutions to the PDE-ODE system, and examine further properties such as regularity, boundedness and finite-time extinction.
We study a Cahn-Hilliard two-phase model describing the flow of two viscoelastoplastic fluids, which arises in geodynamics. A phase-field variable indicates the proportional distribution of the two fluids in the mixture. The motion of the incompressible mixture is described in terms of the volume-averaged velocity. Besides a volume-averaged Stokes-like viscous contribution, the Cauchy stress tensor in the momentum balance contains an additional volume-averaged internal stress tensor to model the elastoplastic behavior. This internal stress has its own evolution law featuring the nonlinear Zaremba-Jaumann time-derivative and the subdifferential of a non-smooth plastic potential. The well-posedness of this system is studied in two cases: Based on a regularization by stress-diffusion we obtain the existence of Leray-Hopf-type weak solutions. In order to deduce existence results also in the absence of the regularization, we introduce the concept of dissipative solutions, which is based on an estimate for the relative energy. We discuss general properties of dissipative solutions and show their existence for the viscoelastoplastic two-phase model in the setting of stress-diffusion. By a limit passage in the relative energy inequality for vanishing stress-diffusion, we conclude an existence result for the non-regularized model.
The FitzHugh-Nagumo system is a 4-parameter family of 3D vector fields used for modeling neural excitation and nerve impulse propagation. The origin represents a Hopf-zero equilibrium in the FitzHugh-Nagumo system for two classes of parameters. The results of this paper not only extend those of previous study by providing explicit generic conditions for the existence and stability of invariant tori, but also establishes new conditions for the existence of limit cycle. By employing the Lyapunov-Schmidt reduction, we demonstrate that under certain circumstances, the first-order averaging function admits a continuum of zeros, and we derive explicit conditions under which limit cycles bifurcate from such situations. Furthermore, we explicitly state the stability condition for the bifurcating limit cycles, which was not addressed in the previous work. Throughout these investigation, we detect analytically the complex dynamics that require a third-order analysis. These behaviors are typically difficult to identify using conventional numerical approaches.
