On the Initial Value Problem of Stochastic Evolution Equations in Hilbert Spaces

The present paper studies the initial value problem of stochastic evolution equations with compact semigroup in real separable Hilbert spaces. The existence of saturated mild solution and global mild solution is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions.The results obtained in this paper improve and extend some related conclusions on this topic. An example is also given to illustrate that our results are valuable.


Introduction
In recent years, the stochastic differential equations have attracted great interest because of their practical applications in many areas such as physics, chemistry, economics, social sciences, finance, and other areas of science and engineering.For more details about stochastic differential equations we refer to the books by Sobczyk [1], Da Prato and Zabczyk [2], Grecksch and Tudor [3], Mao [4], and Liu [5].One of the branches of stochastic differential equations is the theory of stochastic evolution equations.Since semilinear stochastic evolution equations are abstract formulations for many problems arising in the domain of engineering technology, biology, economic system, and so forth, stochastic evolution equations have attracted increasing attention in recent years and the existence, uniqueness, and asymptotic behavior of mild solutions to stochastic evolution equations have been considered by many authors; see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the references therein.Taniguchi et al. [6] discussed the existence, uniqueness, th moment, and almost sure Lyapunov exponents of mild solutions to a class of stochastic partial functional differential equations with finite delays by using semigroup methods.El-Borai et al. [7] studied exponentially asymptotic stability of stochastic differential equation in a real separable Hilbert space.More recently, Luo [8], Luo and Taniguchi [9], Bao et al. [10], and Sakthivel and Ren [11] discussed the exponential stability of mild solutions for stochastic partial differential equations by using the contraction mapping principle and stochastic integral technique, by the fixed point theorem, by introducing a suitable metric between the transition probability functions of mild solutions, and by using the stochastic analysis theory, respectively.Chang et al. [12][13][14] studied the existence and uniqueness of Stepanovlike almost automorphic mild solutions, the existence of square-mean almost automorphic mild solutions, and the existence and uniqueness of quadratic mean almost periodic mild solutions to nonlinear stochastic evolution equations in real separable Hilbert spaces, respectively.Moreover, the existence of mild solutions of stochastic evolution equations in Hilbert spaces has also been discussed in [15][16][17][18][19][20].
However, to the best of the authors' knowledge, most of the existing articles (see, e.g., [6,[12][13][14][15][16][17][18][19]) are only devoted to studying the local existence of mild solutions for stochastic evolution equations, and there are no results yet present on the existence of saturated mild solutions and global mild solutions for stochastic evolution equations in Hilbert spaces.Motivated by the abovementioned aspects, in this paper, by using Schauder's fixed point theorem, compact semigroup theory, and piecewise extension method, we investigate the existence of saturated mild solution and global mild solution for the initial value problem to a class of semilinear stochastic evolution equations in real separable Hilbert spaces.

Preliminaries
Let H and K be two real separable Hilbert spaces and let (K, H) be the space of all bounded linear operators from K into H.For convenience, we will use the same notation ‖ ⋅ ‖ to denote the norms in H, K, and (K, H) and use (⋅, ⋅) to denote the inner products of H and K without any confusion.Throughout this paper, we assume that (Ω, F, {F  } ≥0 , P) is a complete filtered probability space satisfying the usual condition, which means that the filtration is a right continuous increasing family and F 0 contains all P-null sets of F. Let {  ,  ∈ N} be a complete orthonormal basis of K. Suppose that {W() :  ≥ 0} is a cylindrical K-valued Wiener process defined on the probability space (Ω, F, {F  } ≥0 , P) with a finite trace nuclear covariance operator  ≥ 0, denote Tr() = ∑ ∞ =1   =  < ∞, which satisfies that   =     ,  ∈ N. So, actually, W() = ∑ ∞ =1 √  W  ()  , where {W  (),  ∈ N} are mutually independent onedimensional standard Wiener processes.We further assume that F  = {(), 0 ≤  ≤ } is the -algebra generated by W.

Main Results
In this section, we prove the existence of saturated mild solution and global mild solution to the IVP (2).Firstly, in order to obtain the existence of saturated mild solution to the IVP (2), we impose the following assumption to the nonlinear term .
Theorem 2. Assume that the condition (  ) is satisfied; then for every  0 ∈ H the IVP ( 2) has a saturated mild solution  on a maximal interval of existence [0, ).
Notice that () is compact for  > 0; we know that () is continuous by operator norm for  > 0. Combining this fact with the Lebesgue dominated convergence theorem we know that the third term of the right-hand side of (13) tends to zero as  2 −  1 → 0. Therefore, ‖()( 2 ) − ()( 1 )‖ 2 tends to zero independently of  ∈ Θ as  2 −  1 → 0, which means that  : Θ → Θ is equicontinuous.Hence by the Arzela-Ascoli theorem one has that  : Θ → Θ is a compact operator.Therefore, by Schauder fixed point theorem we obtain that  has at least one fixed point ( 0 ) ∈ Θ, which is in turn a mild solution of the IVP (6) on the interval [ 0 ,  0 + ℎ].
This completes the proof of Theorem 3.

An Application
In this section, we give an example to illustrate the applicability of our main results.Let  ⊂ R  be a bounded domain with a sufficiently smooth boundary .