Narrow Results

We study the nonlinear elliptic equation Δu(x) + a(x)u(x) = g(x)f(u(x)) on the Sierpinski gasket and with zero Dirichlet boundary condition. By extending a method introduced by Faraci and Kristály in the framework of Sobolev spaces to the case of function spaces on fractal domains, we establish the existence of infinitely many weak solutions.

In this paper, the bifurcation and attractor of the stochastic Rabinovich system with jump are discussed, and some new results for the system are presented. First, the sufficient condition and necessary condition for stochastic stability of the system are given. Second, the estimation of the global attractive set of system is obtained. The existence of random attractors of the stochastic Rabinovich system with jump is also discussed. Finally, stochastic bifurcation behavior for the system is analyzed. It is hoped that the investigation of this paper can help understanding the rich dynamic of the stochastic Rabinovich system and the true geometrical structure of the original amazing Rabinovich attractor.

Non-autonomous dynamical systems generate cocycles w.r.t. a flow. We give conditions for the existence of attractors for cocycles based on the so-called pull back convergence. These conditions will be applied to the non-autonomous Navier Stokes equation. In particular. we do not need compactness assumption for the time dependent coefficients of this equation.