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.
