This paper investigates the solvability, stability, and convergence of numerical solutions for nonlinear stochastic differential-algebraic equations (SDAEs) with time-varying coefficients. We establish sufficient conditions for solvability and stochastic asymptotic stability, and prove that the explicit Euler method converges as the mesh size tends to zero. Moreover, we show that the solvability conditions are also necessary in the linear case. Several examples are presented to illustrate the theoretical results. To the best of our knowledge, these results are new for time-varying SDAEs.
Let B be the backward shift on ω=CN0, endowed with the product topology. We prove that there is a dense Gδ set of hypercyclic vectors x for which every infinite suborbit has dense linear span. This gives a negative answer to the locally convex version of a question of Faghih-Ahmadi and Hedayatian about supercyclic vectors. The same vectors can be chosen so that every proper closed affine subspace contains only finitely many points of the orbit. For each such x, every nonzero continuous linear operator A on ω, and every y∈ω, the equation ABnx=y has only finitely many solutions n∈N0. Moreover, two continuous linear operators that agree on an infinite suborbit are equal.
Protests often spread through social networks and across cities, a phenomenon that can be modeled using reaction–diffusion systems. Building on the model in [10], which links protest activity and social tension, we study an extended model that also includes police protest management strategies. We focus on traveling wave solutions and examine how protest activity, social tension, and police density interact to shape the spread of protests [36]. Assuming the existence of traveling fronts in parameter regimes where the system is not monotone, we derive stability conditions for these traveling fronts, extending earlier results that assumed monotonicity. Through numerical simulations, we estimate how fast protests spread and identify threshold conditions, similar to epidemiological reproduction numbers, that determine whether protests grow or fade.
This paper considers a generalized Lotka-Volterra reaction-diffusion population system with mixed instantaneous and nonlocal delayed density dependence and subject to homogeneous Dirichlet boundary conditions. The existence and asymptotic expression of the positive steady-state bifurcating from the trivial one are established in virtue of the implicit function theorem. The associated eigenvalue problem is analyzed in detail and the local asymptotic stability and the existence of Hopf bifurcation of the positive steady-state are derived. According to the normal form method and the center manifold theorem for delayed reaction-diffusion equations, the direction of Hopf bifurcations and the stability of bifurcating periodic solutions are demonstrated. To verify the validity of the obtained theoretical predictions, some numerical simulations are also carried out.
We study the nonlocal parabolic-elliptic Keller-Segel system in the entire space, incorporating nonlinearities for both cell kinetics and chemical signal production. Our aim is to establish a global solvability result with small data and prove the Hölder regularity for the obtained mild solution. The proofs are based on Fourier analysis, resolvent theory and fixed-point arguments.
