| In this paper,we study the existence and uniqueness of the random attractor of the first-order random lattice differential equation with nonlinear color noise at each node.We prove that the equation can generate a random dynamic system by applying the equation to the abstract theory of the general random evolution equation,we prove the existence of the random attractor by constructing the random compact absorbing set and applying the compact embedding theorem.Then,we prove that the existence and uniqueness of the random attractor of the first-order random lattice differential equation with local Lipschitz conditions.This paper is divided into four parts:In chapter 1,we mainly introduce the research background and development of this paper,and briefly illustrate the research direction of this paper.In chapter 2,we review some related concepts and some preparatory knowledge about random attractors of random dynamic systems,and introduce the judgment theorem of the existence of random attractors.In chapter 3,we first give an abstract theory of general random evolution equation.In order to study the long-term behavior of first-order random lattice differential equation with global Lipschitz condition,we apply it to the abstract theory of random evolution equation as a special example.By rewriting the equation as a random evolution equation to prove that there is a weak solution,it is proved that it generates a random dynamic system.Finally,we prove the existence of unique random attractors by constructing a random compact absorbing set and using the compact embedding theorem.In this way,we avoid the calculation of a uniform estimate of the tail of the solution.In chapter 4,in order to study the existence and uniqueness of random attractors with local Lipschitz conditional first-order random lattice differential equations,we relax the conditions of the abstract theory of the general random evolution equation given in chapter 3.We study and prove the existence and uniqueness of the weak solution of this general random differential equation defined in the Gelfand triple in Hilbert space.By applying the idea of stopping time and using truncation function to deal with the local Lipschitz term in the abstract equation,it guarantees the existence and uniqueness of the solution,and it is proved that it generates a random dynamic system.Then applies the studied equation to this abstract theory,and proves the existence of its unique random attractor in the same way as in Chapter 3. |
