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

Abstract

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.

1. 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 [6–20] 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. [12–14] studied the existence and uniqueness of Stepanov-like 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 [15–20].

However, to the best of the authors’ knowledge, most of the existing articles (see, e.g., [6, 12–19]) 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.

2. Preliminaries

Let and be two real separable Hilbert spaces and let be the space of all bounded linear operators from into . For convenience, we will use the same notation to denote the norms in , , and and use to denote the inner products of and without any confusion. Throughout this paper, we assume that is a complete filtered probability space satisfying the usual condition, which means that the filtration is a right continuous increasing family and contains all -null sets of . Let be a complete orthonormal basis of . Suppose that is a cylindrical -valued Wiener process defined on the probability space with a finite trace nuclear covariance operator , denote , which satisfies that , . So, actually, , where are mutually independent one-dimensional standard Wiener processes. We further assume that is the -algebra generated by .

For , , we define , where is the adjoint of the operator . Clearly, for any bounded operator , If , then is called a -Hilbert-Schmidt operator.

In this paper, we consider the following initial value problem (IVP) of stochastic evolution equations:where the state takes values in the real separable Hilbert space , is a closed linear operator, and generates a compact -semigroup in ; is a continuous nonlinear mapping, .

The collection of all strongly measurable, square-integrable -valued random variables, denoted by , is a Banach space equipped with the norm , where the expectation is defined by with . An important subspace of is given by Let denote the closed subset of the interval . We denote by the space of all continuous -adapted measurable processes from to satisfying . Then it is easy to see that is a Banach space endowed with the supnorm

Definition 1. By a mild solution of the IVP (2), we mean that a continuous -adapted stochastic process defined from to satisfies the following:(i),(ii) has cádlág paths on almost surely and for each , satisfies the integral equation

3. 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 .The function is continuous and the set is bounded for any bounded .

Theorem 2. Assume that the condition () is satisfied; then for every the IVP (2) has a saturated mild solution on a maximal interval of existence . If then .

Proof. We first prove the local existence of mild solution for the initial value problem (IVP) of stochastic evolution equationson interval , where , , and will be given later. Consider the operator defined by From the continuity of nonlinear term one can easily see that the operator is continuous. By Definition 1, the mild solution of the IVP (6) on is equivalent to the fixed point of operator defined by (7). LetDenote ; then is a closed ball in with center and radius . SetFor any and , by (7)–(9), we know thatTherefore, . Thus, we proved that is a continuous operator.
Now, we demonstrate that is a compact operator. To prove this, we first show that is relatively compact in for every . This is clear for since . For , , and , we define the operator by Since is compact for every , the set is relatively compact in for every . Moreover, for every , by (7), (8), and (11) we get that Therefore, we have proved that there are relatively compact sets arbitrarily close to the set in for . Hence, the set is also relatively compact in for . And therefore, we have the compactness of in for all . We continue to show that is an equicontinuous family of functions in . For any and , we haveSince is continuous on , it is uniformly continuous, and therefore the first term of the right-hand side of (13) tends to zero as . It is obvious that the second term of the right-hand side of (13) tends to zero as . Notice that is compact for ; we know that is continuous by operator norm for . 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 . Therefore, tends to zero independently of as , which means that is equicontinuous. Hence by the Arzela-Ascoli theorem one has that is a compact operator. Therefore, by Schauder fixed point theorem we obtain that has at least one fixed point , which is in turn a mild solution of the IVP (6) on the interval .
From the local existence of mild solutions we have just proved it follows that there exists interval such that the IVP (2) has a mild solution , and it can be extended to a large interval with by defining on , where is the mild solution of the initial value problemwhere depends only on , , and . Therefore, repeating the above procedure and using the methods of steps, we can prove that there exists a maximal interval such that is a saturated mild solution of the IVP (2).
Next, we show that if then . To do so we first prove that implies . Indeed, if and , then there exists a constant such that . Denote and  . For , we have The right-hand side of (15) tends to zero as , as a consequence of the continuity of in the uniform operator topology for which in turn follows the compactness of for . Therefore, exists. Let ; by the local existence of mild solutions we proved above we know that the stochastic evolution equation presents a mild solution on , where is a constant. This means that the mild solution of the IVP (2) can be extended beyond , which contradicts with being a saturated mild solution of the IVP (2). Therefore, the assumption implies that . To conclude the proof we will show that implies . If this is not true, then there exist a constant and a sequence such that for all . Let and . By the fact that is continuous and one can find a sequence such that for and . By direct calculation, we have From (17) we conclude that , which contradicts with the definition of . Therefore, we have proved that if then . This completes the proof of Theorem 2.

Next, we discuss the existence of global mild solution to the IVP (2). To this end, we need to replace the assumption () by the following assumption:The function is continuous and there exist two functions , such that

Theorem 3. Assume that the condition is satisfied; then for every the IVP (2) has a global mild solution .

Proof. It is obvious that . Therefore, by Theorem 2 we know that the IVP (2) has a saturated mild solution on a maximal interval of existence . If , then the result has been proved. If , by the proof process of Theorem 2 one can see that the IVP (2) has a global mild solution if for every in the interval of existence of . Therefore, in order to prove Theorem 3 we only need to prove for every . Let By Definition 1, the assumption , and (19) we know that for any , where Therefore, by (20) and Gronwall’s inequality, we know that This completes the proof of Theorem 3.

4. An Application

In this section, we give an example to illustrate the applicability of our main results. Let be a bounded domain with a sufficiently smooth boundary . We consider the initial-boundary value problem of stochastic parabolic partial differential equation where is the Laplace operator, denotes a one-dimensional standard cylindrical Wiener process defined on a stochastic space , and is -measurable.

Let with the norm . We define an operator in Hilbert space byIt is well known from [21, 22] that generates a compact -semigroup in . Let , , and ; then the initial-boundary value problem of stochastic parabolic partial differential equation (23) can be transformed into the abstract form of IVP (2).

Theorem 4. The initial-boundary value problem of stochastic parabolic partial differential equation (23) has a global mild solution .

Proof. From the definition of nonlinear term , we can easily verify that the assumption is satisfied with . Therefore, our conclusion follows from Theorem 3. This completes the proof.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

Research supported by The Science Research Project for Colleges and Universities of Gansu Province (2015A-213, 2015A-003), NNSF of China (11501455, 11261053), and Project of NWNU-LKQN-14-3.