pth Moment Exponential Stability of Nonlinear Hybrid Stochastic Heat Equations

Abstract

We are concerned with the exponential stability problem of a class of nonlinear hybrid stochastic heat equations (known as stochastic heat equations with Markovian switching) in an infinite state space. The fixed point theory is utilized to discuss the existence, uniqueness, and 𝑝th moment exponential stability of the mild solution. Moreover, we also acquire the Lyapunov exponents by combining the fixed point theory and the Gronwall inequality. At last, two examples are provided to verify the effectiveness of our obtained results.

1. Introduction

There have been enormous theories for both definite and stochastic partial differential equations (SPDEs) since they can describe various phenomena in reality recently. Throughout the previous literature, besides the existence and uniqueness, the stability for SPDEs is a popular topic. Some results on exponential stability of SPDEs were investigated by Caraballo and Liu [1] and Luo [2].

At present, the hybrid SDEs and hybrid SPDEs (also known as SDEs and SPDEs with Markovian switching), for example, hybrid stochastic heat equations, have been paid much attention owing to their wide applications in natural science, engineering, biology, finance, and other areas. In addition, one important reason to consider hybrid SDEs or SPDEs is that real situations may exhibit sudden changes or go into different cases in different periods resulting in parameter transition and probably changes in branch structure so that we need Markov chains to characterize such systems. For example, [3, 4] discussed some properties of the solutions to SDEs with Markovian switching. So far, the 𝑝th moment exponential stability of stochastic heat equations without Markovian switching has been studied extensively, for example, [5, 6], while hybrid stochastic heat equations remain open due to the difficulty arising from Markov chains. As a consequence, it is more challenging and exciting to explore the stabilization of the systems with Markov chains.

Up to now, there have been many methods to study the stability of SPDEs as well as hybrid stochastic heat equations like Lyapunov’s function method, successive approximation approach, large deviation technique, and so on. For example, Deng et al. [7] obtained some sufficient conditions on the stability of hybrid stochastic differential equations by using the Lyapunov’s function method and stochastic feedback controls. In [8], Ubøe and Zhang constructed sequence solutions to approximate the exact solution successively. Gong and Qian studied the large deviation rate function of the Markov chains more completely in [9]. In [10], Bao et al. discussed the 𝑝th moment exponential stability of linear hybrid stochastic heat equations by employing the explicit formulae and large deviation technique. However, we cannot use the method of the linear case to obtain the stability property of a class of nonlinear hybrid stochastic heat equations. Generally speaking, it is very difficult to calculate the explicit formulae of solutions to nonlinear hybrid stochastic heat equations. Furthermore, the Markov chain has no unique stationary probability distribution when it is not irreducible in an infinite state space, which makes the analysis of Lyapunov exponents more difficult.

To overcome difficulties raised by what was mentioned above, we utilize a new method—the fixed point theory—to deal with the stability of a class of nonlinear hybrid stochastic heat equations in this paper. It is well known that the fixed point theory first put forward by Burton is usually applied to study the almost sure exponential stability and the 𝑝th moment exponential stability of both SDEs and SPDEs such as second-order stochastic evolution equations, for example, [11], nonlinear neutral SDEs, for example, [12], and stochastic Volterra-Levin equations, for example, [13–15]. Different from the method in [10], we use the fixed point theory to analyze the stability of nonlinear hybrid stochastic heat equations. Firstly, we prove the existence and uniqueness of the solution, as well as the 𝑝th moment exponential stability. Then, we obtain the 𝑝th moment Lyapunov exponents by using the Gronwall inequality instead of seeking the proper rate function which was complicated to compute in [10]. Moreover, we provide two examples and some comparisons to show that our results extend and improve those given in the previous literature.

The rest of this paper is organized as follows. In Section 2, we introduce the notations and the model of nonlinear hybrid stochastic heat equations along with some necessary assumptions. In Section 3, by applying the fixed point theorem, we prove the existence, uniqueness, and 𝑝th moment exponential stability of hybrid stochastic heat equations. Besides, based on the Gronwall inequality, we further obtain the Lyapunov exponents of the solutions and make some comparisons with the previous results. In Section 4, we provide two simple examples to verify the effectiveness of the obtained results. In the last section, we conclude the paper with some general remarks.

2. Preliminaries, Model, and Assumptions

In this section, we introduce some preliminaries and common notations for a more detailed description, and then give the model that we will deal with.

Throughout this paper, the following notations will be used. Let {𝐵(𝑡),𝑡≥0} be a real-valued Brownian motion defined on the complete probability space {Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃} which has a filtration satisfying the usual conditions. We denote a bounded domain Θ⊂ℝ^𝑛, equipped with 𝒞^∞ boundary 𝜕Θ. In this paper, ℒ²(Θ) is defined as usual, that is, the family of all-real valued square integrable functions with their inner product ⟨𝑓,𝑔⟩∶=∫_Θ𝑓(𝑥)𝑔(𝑥)𝑑𝑥, 𝑓,𝑔∈ℒ²(Θ) and norm ‖𝑓‖∶=(∫_Θ𝑓²(𝑥)𝑑𝑥)^1/2, 𝑓∈ℒ²(Θ). Also, we can define 𝐻^𝑚(Θ), 𝑚=1,2, by 𝐻^𝑚(Θ)∶={𝑢∈ℒ²(Θ)∣𝐷^𝛼𝑢∈ℒ²(Θ),|𝛼|≤𝑚} and 𝐻^𝑚₀(Θ) by 𝐻^𝑚₀(Θ)∶={𝑢∈𝐻^𝑚(Θ)∣𝑢=0on𝜕Θ}. Then, we can denote the Laplace operator 𝐴∶=∑^𝑛_𝑖=1(𝜕²/𝜕𝑥²_𝑖), with domain 𝒟(𝐴)∶=𝐻¹₀(Θ)∩𝐻²₀(Θ), which generates a strongly continuous semigroup 𝑒^𝑡𝐴. Furthermore, let {𝑟(𝑡),𝑡≥0} be a right continuous Markov chain which takes values in a listed state space 𝑆={1,2,…,𝑁}, where 𝑁 is some positive integer or arrives at ∞. Moreover, we assume that the Markov chain {𝑟(𝑡),𝑡≥0} is independent of the Brownian motion {𝐵(𝑡),𝑡≥0}.

Model Definition. In this paper, we consider the following hybrid stochastic heat equation: 𝜕𝑢(𝑡,𝑥)()(𝑟(𝑡𝑓(()(𝑡(𝑟(𝑡𝑔(()(𝑡𝐵̇(𝑡),𝑢()𝑢()𝜕𝑡=𝐴𝑢𝑡,𝑥+𝛼))𝑡,𝑢𝑡,𝑥,𝑟))+𝛽))𝑡,𝑢𝑡,𝑥,𝑟))𝑥∈Θ,𝑡>0,𝑡,𝑥=0,𝑥∈𝜕Θ,𝑡>0,0,𝑥=𝑢₀(𝑥)𝑥∈Θ, (1) where 𝛼,𝛽 are mappings from 𝑆→ℝ and we take 𝛼_𝑖∶=𝛼(𝑖), 𝛽_𝑖∶=𝛽(𝑖) simply in this paper. Letting us fix an interval [0,𝑇],𝑇>0, then the operators 𝑓,𝑔∶[0,𝑇]×ℒ²(Θ)×𝑆∶→ℒ²(Θ) are ℱ_𝑡-measurable while the initial value 𝑢₀ is a 𝒟(𝐴)-valued, ℱ₀-measurable random variable, which is independent of 𝑟(⋅) and 𝐵(⋅), and, for any 𝑝>0, 𝐄‖𝑢₀‖^𝑝<∞.

Next, we will give the definitions of the mild solution to (1) and 𝑝th moment exponential stability as well as 𝐶_𝑝 inequality. For the convenience of writing, we will take 𝑢(𝑡)∶=𝑢(𝑡,⋅) and 𝑢∶={𝑢(𝑡,⋅)}_𝑡≥0 such that 𝑢 is a ℒ²(Θ)-valued stochastic process since 𝑢(𝑡) is a ℒ²(Θ)-valued random variable.

Definition 1. A ℒ²(Θ)-valued stochastic process 𝑢={𝑢(𝑡)}_{𝑡∈[0,𝑇]} is called a mild solution to (1) if the following conditions are satisfied: (i)𝑢∈𝒞([0,𝑇];ℒ²(Θ)) and for any 𝑡∈[0,𝑇], 𝑢(𝑡) is adapted to ℱ_𝑡 with 𝑃𝜔∶^𝑡₀‖𝑢(𝑠)‖^𝑝𝑑𝑠<∞=1; (2) (ii)stochastic integral equation 𝑢(𝑡,𝑥)=𝑒^𝐴𝑡𝑢₀+^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝛼(𝑟(𝑠𝑓(()(𝑠+))𝑠,𝑢𝑠,𝑥,𝑟))𝑑𝑠_𝑡0𝑒^{𝐴(𝑡−𝑠)}𝛽(𝑟(𝑠))𝑔(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))𝑑𝐵(𝑠) (3)holds a.s. for any 𝑡∈[0,𝑇] and 𝑥∈Θ.

Definition 2. Equation (1) is called exponentially stable in the 𝑝th moment if there exists a pair of constants 𝛿>0 and 𝐾>0 such that 𝐄‖𝑢(𝑡)‖^𝑝≤𝐾𝐄‖𝑢₀‖^𝑝𝑒^−𝛿𝑡, 𝑡≥0.

Lemma 3 (𝐶_𝑝 inequality). Let {𝜉_𝑘,1≤𝑘≤𝑛} be an arbitrary family of random variables; then one has 𝐄^𝑛_𝑘=1𝜉_𝑘^𝑝≤𝐶_𝑝^𝑛_𝑘=1𝐄𝜉_𝑘^𝑝, (4) where 𝐶_𝑝=𝑛^𝑝−1, 𝑝≥1; 𝐶_𝑝=1, 0<𝑝<1.

Finally, we present some necessary assumptions to close this section.

Assumptions. Consider the following:(A₁)‖𝑒^𝑡𝐴‖≤𝑀𝑒^−𝛾𝑡 for some constants 𝑀>0 and 𝛾>0;(A₂)the operators 𝑓,𝑔 satisfy the following properties: for any 𝑢,𝑣∈𝐿²(Θ) and 𝑝≥2, there exist positive constants 𝐿_𝑓𝑖,𝐿_𝑔𝑖(𝑖∈𝑆) such that 𝑓()()𝑓(())(()𝑡,0,𝑖=0,𝑔𝑡,0,𝑖=0,𝑡,𝑢𝑡,𝑥,𝑖−𝑓𝑡,𝑣𝑡,𝑥,𝑖)≤𝐿_𝑓𝑖‖𝑢()(,𝑔(())(()𝑡,𝑥−𝑣𝑡,𝑥)‖𝑡,𝑢𝑡,𝑥,𝑖−𝑔𝑡,𝑣𝑡,𝑥,𝑖)≤𝐿_𝑔𝑖‖𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥)‖; (5)(A₃)sup_{1≤𝑖≤𝑁}|𝛼_𝑖|𝐿_𝑓𝑖<∞, sup_{1≤𝑖≤𝑁}|𝛽_𝑖|𝐿_𝑔𝑖<∞.

Remark 4. It is obvious to see that (1) has a trivial solution under Assumptions 𝐀_𝟏--𝐀_𝟑 when the initial value 𝑢₀ is equal to zero.

3. 𝑝th Moment Exponential Stability

In this section, we will use the fixed point principle to discuss the existence and uniqueness of the mild solution to (1) and prove the exponential stability result.

Theorem 5. Let 𝑝≥2 and suppose that Assumptions 𝐀_𝟏--𝐀_𝟑 hold. Then, (1) is exponentially stable in the 𝑝th moment if the following holds: 2^𝑝−1𝑀^𝑝𝛾^𝑝𝛼^∗𝑝𝐿^𝑝_𝑓∗+√2^𝑝−2𝑀^𝑝𝐽_𝑝𝛾^𝑝/2𝛽^∗𝑝𝐿^𝑝_𝑔∗∈(0,1), (6) where 𝐽_𝑝=(𝑝^𝑝+1/2(𝑝−1)^𝑝−1)^𝑝/2, |𝛼^∗|𝐿_𝑓^∗=sup_{1≤𝑖≤𝑁}|𝛼_𝑖|𝐿_𝑓𝑖, |𝛽^∗|𝐿_𝑔∗=sup_{1≤𝑖≤𝑁}|𝛽_𝑖|𝐿_𝑔𝑖.

Proof. Let 𝐻 be the Banach space of all ℱ_𝑡-adapted continuous processes consisting of functions 𝑢(𝑡,𝑥) such that 𝐄‖𝑢(𝑡,𝑥)‖^𝑝≤𝑀^∗𝐄‖𝑢₀‖^𝑝𝑒^−𝜂𝑡, 𝑡≥0, where 𝑀^∗>0, 0<𝜂<𝛾. Below we denote the norm in 𝐻 by ‖𝑢(𝑡,𝑥)‖_𝐻∶=sup_𝑡≥0𝐄‖𝑢(𝑡,𝑥)‖^𝑝.
Then, we derive an operator 𝜑∶𝐻↦𝐻 as follows: 𝜑(𝑢)(𝑡)=𝑒^𝐴𝑡𝑢₀+^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝛼(𝑟(𝑠𝑓(()(𝑠+))𝑠,𝑢𝑠,𝑥,𝑟))𝑑𝑠_𝑡0𝑒^{𝐴(𝑡−𝑠)}𝛽(𝑟(𝑠𝑔(()(𝑠(𝑠),𝜑(𝑢0)))𝑠,𝑢𝑠,𝑥,𝑟))𝑑𝐵𝑡>0,)(=𝑢₀. (7) It is easy to prove that the following holds by 𝐶_𝑝 inequality yields: 𝐄𝜑(𝑢)(𝑡)^𝑝≤3^𝑝−1𝐄𝑒^𝐴𝑡𝑢₀^𝑝+3^𝑝−1𝐄×^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝛼(𝑟(𝑠))𝑓(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))𝑑𝑠^𝑝+3^𝑝−1𝐄×^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝛽(𝑟(𝑠))𝑔(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))𝑑𝐵(𝑠)^𝑝∶=3^3𝑝−1_𝑖=1𝐄𝐼_𝑖(𝑡)^𝑝. (8) Next, we will divide the proof into three steps.
Claim  1. 𝜑 is continuous in the 𝑝th moment on [0,+∞).
Proof of Claim  1. Let 𝑢∈𝐻,𝑡₁≥0, and let |𝑟| be sufficiently small, 𝐄𝜑(𝑡₁+𝑟)−𝜑(𝑡₁)^𝑝≤3^3𝑝−1_𝑖=1𝐄𝐼_𝑖(𝑡₁+𝑟)−𝐼_𝑖(𝑡₁)^𝑝. (9)
Letting |𝑟|→0, it is easy to see that 𝐄𝐼₁𝑡₁+𝑟−𝐼₁𝑡₁^𝑝=𝐄𝑒^{𝐴(𝑡1+𝑟)}𝑢₀−𝑒^𝐴𝑡1𝑢₀^𝑝=𝐄𝑒^𝐴𝑡1𝑢₀𝑒^𝐴𝑟−1^𝑝⟶0(|𝑟|⟶0),𝐄𝐼₂𝑡₁+𝑟−𝐼₂𝑡₁^𝑝=𝐄^𝑡1₀^+𝑟𝑒^{𝐴(𝑡1+𝑟−𝑠)}𝛼(𝑟(𝑠𝑓(()(𝑠−))𝑠,𝑢𝑠,𝑥,𝑟))𝑑𝑠_𝑡10𝑒^{𝐴(𝑡1−𝑠)}𝛼(𝑟(𝑠))𝑓(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))𝑑𝑠^𝑝≤2^𝑝−1𝐄×^𝑡1₀𝑒^{𝐴(𝑡1−𝑠)}𝛼(𝑟(𝑠))𝑓(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))𝑒^𝐴𝑟−1𝑑𝑠^𝑝+2^𝑝−1𝐄×^𝑡1_𝑡^+𝑟₁𝑒^{𝐴(𝑡1+𝑟−𝑠)}𝛼(𝑟(𝑠))𝑓(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))𝑑𝑠^𝑝⟶0(|𝑟|⟶0). (10)(11)
Then, by Burkhölder-Davis-Gundy inequality, the following holds when |𝑟|→0: 𝐄𝐼₃𝑡₁+𝑟−𝐼₃𝑡₁^𝑝=𝐄^𝑡1₀^+𝑟𝑒^{𝐴(𝑡1+𝑟−𝑠)}𝛽(𝑟(𝑠𝑔(()(𝑠(𝑠)−))𝑠,𝑢𝑠,𝑥,𝑟))𝑑𝐵_𝑡10𝑒^{𝐴(𝑡1−𝑠)}𝛽(𝑟(𝑠))𝑔(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))𝑑𝐵(𝑠)^𝑝≤2^𝑝−1𝐄^𝑡1₀𝑒^{𝐴(𝑡1−𝑠)}𝛽(𝑟(𝑠(()(𝑠𝑒))×𝑔𝑠,𝑢𝑠,𝑥,𝑟))^𝐴𝑟−1𝑑𝐵(𝑠)^𝑝+2^𝑝−1𝐄^𝑡1_𝑡^+𝑟₁𝑒^{𝐴(𝑡1+𝑟−𝑠)}×𝛽(𝑟(𝑠))𝑔(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))𝑑𝐵(𝑠)^𝑝≤2^𝑝−1𝐽_𝑝𝐄×^𝑡1₀𝑀²𝑒^{−2𝛾(𝑡1−𝑠)}×𝛽(𝑟(𝑠))𝑔(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))𝑒^𝐴𝑟−1²𝑑𝑠^𝑝/2+2^𝑝−1𝐽_𝑝𝐄^𝑡1_𝑡^+𝑟₁𝑀²𝑒^{−2𝛾(𝑡1+𝑟−𝑠)}×𝛽(𝑟(𝑠))𝑔(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))²𝑑𝑠^𝑝/2⟶0(|𝑟|⟶0). (12) Hence, from (10)–(12), we see that 𝜑 is 𝑝th continuous on [0,+∞).
Claim  2. 𝜑(𝐻) is contained in 𝐻.
Proof of Claim  2. It follows from (8) that 𝐼₁(𝑡)=𝐄𝑒^𝐴𝑡𝑢₀^𝑝≤𝑀^𝑝𝑒^{−𝑝𝛾𝑡}𝐄𝑢₀^𝑝≤𝑀^𝑝𝑒^−𝜂𝑡𝐄𝑢₀^𝑝. (13) By Assumptions 𝐀_𝟏--𝐀_𝟑 and the Hölder inequality, we have 𝐼₂(𝑡)=𝐄^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝛼(𝑟(𝑠))𝑓(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))𝑑𝑠^𝑝≤𝑀^𝑝𝐄^𝑡₀𝑒^{−𝛾(𝑡−𝑠)}|𝛼(𝑟(𝑠))|𝑓(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))𝑑𝑠^𝑝≤𝑀^𝑝𝐄^𝑡₀𝑒^{−𝛾(𝑡−𝑠)}𝑑𝑠^𝑝−1×^𝑡₀𝑒^{−𝛾(𝑡−𝑠)}|𝛼(𝑟(𝑠))|^𝑝×𝑓(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))^𝑝≤𝑀𝑑𝑠_𝑝𝛾^𝑝−1𝐄^𝑡₀𝑒^{−𝛾(𝑡−𝑠)}|𝛼(𝑟(𝑠))|^𝑝𝐿^𝑝_𝑓(𝑟(𝑠))‖𝑢(𝑠,𝑥)‖^𝑝≤𝑀𝑑𝑠_𝑝𝛾^𝑝−1𝐄𝛼^∗𝑝𝐿^𝑝_𝑓∗^𝑡₀𝑒^{−𝛾(𝑡−𝑠)}‖𝑢(𝑠,𝑥)‖^𝑝≤𝑀𝑑𝑠_𝑝𝛼^∗𝑝𝐿^𝑝_𝑓∗𝛾^𝑝−1𝑡₀𝑒^{−𝛾(𝑡−𝑠)}𝐄‖𝑢(𝑠,𝑥)‖^𝑝≤𝑀𝑑𝑠_𝑝𝑀^∗𝛼^∗𝑝𝐿^𝑝_𝑓∗𝛾^𝑝−1𝐄𝑢₀^𝑝𝑡₀𝑒^{−𝛾(𝑡−𝑠)}𝑒^−𝜂𝑠≤𝑀𝑑𝑠_𝑝𝑀^∗𝛼^∗𝑝𝐿^𝑝_𝑓∗𝛾^𝑝−1𝐄𝑢𝛾−𝜂^0𝑝𝑒^−𝜂𝑡,𝐼₃(𝑡)=𝐄^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝛽(𝑟(𝑠))𝑔(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))𝑑𝐵(𝑠)^𝑝≤𝑀^𝑝𝐽_𝑝𝐄×^𝑡₀𝑒^{−2𝛾(𝑡−𝑠)}𝛽(𝑟(𝑠))²𝑔(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))²𝑑𝑠^𝑝/2≤𝑀^𝑝𝐽_𝑝𝐄^𝑡₀𝑒^{−2𝛾(𝑡−𝑠)}𝑑𝑠^(𝑝/2)−1×^𝑡₀𝑒^{−2𝛾(𝑡−𝑠)}𝛽(𝑟(𝑠))^𝑝×𝑔(𝑠,𝑢(𝑠,𝑥),𝑟(𝑠))^𝑝𝑑𝑠≤𝑀^𝑝𝐽_𝑝2𝛾^(2−𝑝)/2𝛽^∗𝑝𝐿^𝑝_𝑔∗^𝑡₀𝑒^{−2𝛾(𝑡−𝑠)}𝐄‖𝑢(𝑠,𝑥)‖^𝑝𝑑𝑠≤𝑀^𝑝𝑀^∗𝐽_𝑝2𝛾^(2−𝑝)/2𝛽^∗𝑝𝐿^𝑝_𝑔∗𝐄𝑢₀^𝑝𝑡₀𝑒^{−2𝛾(𝑡−𝑠)}𝑒^−𝜂𝑠≤𝑀𝑑𝑠_𝑝𝑀^∗𝐽2𝛾−𝜂^𝑝2𝛾^(2−𝑝)/2𝛽^∗𝑝𝐿^𝑝_𝑔∗𝐄𝑢₀^𝑝𝑒^−𝜂𝑡. (14)
From (13)-(14), it is easy to see that 𝐄‖𝜑(𝑢)(𝑡)‖^𝑝≤𝑘𝐄‖𝜑(𝑢)(0)‖^𝑝𝑒^−𝜂𝑡, where 𝑘=3^𝑝−1𝑀^𝑝(1+(𝑀^∗/𝛾^𝑝−1(𝛾−𝜂))|𝛼^∗|^𝑝𝐿^𝑝_𝑓∗+(𝑀^∗𝐽_𝑝(2𝛾)^(2−𝑝)/2/(2𝛾−𝜂))|𝛽^∗|^𝑝𝐿^𝑝_𝑔∗), which implies 𝜑(𝐻)⊆𝐻.
Claim  3. 𝜑 is contractive for arbitrary 𝑢,𝑣∈𝐻 with 𝑢(0,𝑥)=𝑢₀(𝑥)=𝑣(0,𝑥).
Proof of Claim  3. Consider the following: 𝐄sup_0≤𝑡<∞𝜑(𝑢)(𝑡)−𝜑(𝑣)(𝑡)^𝑝≤2^𝑝−1𝐄sup_0≤𝑡<∞^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝛼(𝑟(𝑠𝑓(()(𝑠−))𝑠,𝑢𝑠,𝑥,𝑟))𝑑𝑠_𝑡0𝑒^{𝐴(𝑡−𝑠)}𝛼(𝑟(𝑠))𝑓(𝑠,𝑣(𝑠,𝑥),𝑟(𝑠))𝑑𝑠^𝑝+2^𝑝−1𝐄×sup_0≤𝑡<∞^𝑡₀𝑒^{𝐴(𝑡−𝑠)}𝛽(𝑟(𝑠𝑔(()(𝑠−))𝑠,𝑢𝑠,𝑥,𝑟))𝑑𝑠_𝑡0𝑒^{𝐴(𝑡−𝑠)}𝛽(𝑟(𝑠))𝑔(𝑠,𝑣(𝑠,𝑥),𝑟(𝑠))𝑑𝑠^𝑝≤2^𝑝−1𝑀^𝑝𝛾^1−𝑝𝛼^∗𝑝𝐿^𝑝_𝑓∗𝐄×sup_0≤𝑡<∞^𝑡₀𝑒^{−𝛾(𝑡−𝑠)}‖𝑢(𝑠,𝑥)−𝑣(𝑠,𝑥)‖^𝑝𝑑𝑠+2^𝑝−1𝑀^𝑝𝐽_𝑝2𝛾^(2−𝑝)/2𝛽^∗𝑝𝐿^𝑝_𝑔∗𝐄×sup_0≤𝑡<∞^𝑡₀𝑒^{−2𝛾(𝑡−𝑠)}‖𝑢(𝑠,𝑥)−𝑣(𝑠,𝑥)‖^𝑝𝑑𝑠≤2^𝑝−1𝑀^𝑝𝛾^1−𝑝𝛼^∗𝑝𝐿^𝑝_𝑓∗𝐄sup_0≤𝑡<∞‖𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥)‖^𝑝×^𝑡₀𝑒^{−𝛾(𝑡−𝑠)}𝑑𝑠+2^𝑝−1𝑀^𝑝𝐽_𝑝2𝛾^(2−𝑝)/2𝛽^∗𝑝𝐿^𝑝_𝑔∗×𝐄sup_0≤𝑡<∞‖𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥)‖^𝑝×^𝑡₀𝑒^{−2𝛾(𝑡−𝑠)}𝑑𝑠≤2^𝑝−1𝑀^𝑝𝛾^−𝑝𝛼^∗𝑝𝐿^𝑝_𝑓∗𝐄sup_0≤𝑡<∞‖𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥)‖^𝑝+2^𝑝−1𝑀^𝑝𝐽_𝑝2𝛾^−𝑝/2𝛽^∗𝑝𝐿^𝑝_𝑔∗×𝐄sup_0≤𝑡<∞‖𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥)‖^𝑝=𝑘𝐄sup_0≤𝑡<∞‖𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥)‖^𝑝, (15) where 𝑘=(2^𝑝−1𝑀^𝑝/𝛾^𝑝)|𝛼^∗|^𝑝𝐿^𝑝_𝑓∗+(√2^𝑝−2𝑀^𝑝𝐽_𝑝/𝛾^𝑝/2)|𝛽^∗|^𝑝𝐿^𝑝_𝑔∗.
Recalling condition (6) and noting that 𝑘∈(0,1), we see that 𝜑 is a contraction mapping. By the fixed point theory, we derive that 𝜑 has a unique fixed point 𝑢(𝑡,𝑥) in 𝐻, which is also exponentially stable in the 𝑝th moment from the proof of three claims. Therefore, the desired assertion in Theorem 5 is completed.

Remark 6. If 𝑝=2, then it is obvious that (1) is mean square exponentially stable.

Remark 7. In Theorem 5, we apply the fixed point theory to obtain the existence and uniqueness of the solution for a class of nonlinear hybrid stochastic heat equations. Obviously, our results extend and improve those given in [3, 5, 10].

Remark 8. According to the proof of Theorem 5 and based on the Gronwall inequality, we can easily obtain the Lyapunov exponents of the solution as follows: limsup_𝑡→∞1𝑡log𝐄‖𝑢(𝑡)‖^𝑝≤𝐶−𝛾, (16) where 𝐶=3^𝑝−1𝑀^𝑝𝛾^1−𝑝|𝛼^∗|^𝑝𝐿^𝑝_𝑓∗+3^𝑝−1𝑀^𝑝𝐽_𝑝(2𝛾)^(2−𝑝)/2|𝛽^∗|^𝑝𝐿^𝑝_𝑔∗.
In particular, we derive that 𝐶<𝛾 when 𝑘<1 in Theorem 5 such that the solution of (1) is exponentially stable in the 𝑝th moment.

Remark 9. In [10], Bao et al. discussed the Lyapunov exponents of linear hybrid stochastic heat equations by employing the explicit formula of solution. However, we consider a class of nonlinear hybrid stochastic heat equations in this paper. Obviously, the method in [10] fails in our result since the explicit formula of solution cannot be obtained in the nonlinear case. So we apply succesfully the fixed point principle to study the 𝑝th moment exponential stability of a class of nonlinear hybrid stochastic heat equations, which obviously generalizes the linear case.

4. Two Examples

In this section, we provide two examples of heat equations with Markovian switching as the applications of our main results and give some remarks compared to the previous articles.

Example 1. Consider the following stochastic heat equation with Markovian switching: 𝜕𝑢(𝑥,𝑡)()(𝑟(𝑡𝑢()𝜕𝑡=𝐴𝑢𝑥,𝑡+𝛼))sin𝑡,𝑥3^𝑟(𝑡)(𝑟(𝑡𝑢()𝐵̇(𝑡)()𝑢()()𝑢()=+𝛽))𝑡,𝑥𝑥∈0,𝜋,𝑡>0,𝑡,0=𝑢𝑡,𝜋=0,𝑡>0,0,𝑥2𝜋cos𝑥𝑥∈(0,𝜋), (17) where {𝑟(𝑡),𝑡≥0} is a right continuous Markov chain which takes values in an infinite state space 𝑆={1,2,…}. Take 𝛼_𝑖=𝑖, 𝛽_𝑖=𝐽_𝑝^−1/𝑝/𝑖, 𝑖∈𝑆. Below we recall that 𝐴 is an infinitesimal generator with a strongly continuous semigroup 𝑒^𝑡𝐴,𝑡≥0 so that the eigenfunctions of −𝐴 are 𝑒_𝑛(𝑥)=2/𝜋sin𝑛𝑥∈𝒟(𝐴), 𝑛=1,2,…, and the relevant eigenvalues of 𝐴 are 𝜆_𝑛=𝑛². Observing that 𝑒^𝑡𝐴𝑢=∑^∞_𝑛=1𝑒^−𝑛2𝑡⟨𝑢,𝑒_𝑛⟩_𝐻𝑒_𝑛, 𝑢∈𝐻, and ‖𝑒^𝑡𝐴‖≤𝑒^−𝜋2𝑡, 𝑡≥0, we have 𝛾=𝜋², 𝑀=1.
Since 𝑓(()=𝑢()𝑡,𝑢𝑡,𝑥,𝑖)sin𝑡,𝑥3^𝑖≤13^𝑖‖𝑢(𝑡,𝑥)‖,𝑖∈𝑆, (18) we can choose 𝐿_𝑓𝑖=1/3^𝑖 and similarly 𝐿_𝑔𝑖=1. It is easy to calculate that 𝑘=(1/3𝜋^2𝑝)+(√2^𝑝−2/𝜋^𝑝)∈(0,1) with 𝛼^∗𝐿_𝑓^∗=1/3, 𝛽^∗𝐿_𝑔∗=𝐽_𝑝^−1/𝑝. Therefore, by Theorem 5 we see that (17) is 𝑝th moment exponentially stable.

Remark 10. The state space 𝑆 in (17) is infinite in which Markov chain {𝑟(𝑡),𝑡≥0} takes values while generally we assume the state space to be finite to gain the 𝑝th moment exponential stability of the solution, so our results generalize those in the previous literature (e.g., see [4, 10]).

Remark 11. If 𝑝=2, then (17) is mean square exponentially stable.

Example 2. We compare the following linear stochastic heat equation with that in [10]: 𝜕𝑢(𝑥,𝑡)()(𝑟(𝑡𝑢()(𝑟(𝑡𝑢()𝐵̇(𝑡)()𝑢()()𝑢()=𝜕𝑡=𝐴𝑢𝑥,𝑡+𝛼))𝑡,𝑥+𝛽))𝑡,𝑥𝑥∈0,𝜋,𝑡>0,𝑡,0=𝑢𝑡,𝜋=0,𝑡>0,0,𝑥2𝜋sin𝑥𝑥∈(0,𝜋), (19) where {𝑟(𝑡),𝑡≥0} takes values in 𝑆={1,2} with the generator Γ=(𝛾_𝑖𝑗)_2×2: −𝛾₁₁=𝛾₁₂=1,−𝛾₂₂=𝛾₂₁=𝑞>0, (20) which implies that the Markov chain {𝑟(𝑡),𝑡≥0} has a unique invariant measure 𝜋=(𝜋₁,𝜋₂)=(𝑞/(𝑞+1),1/(𝑞+1)). Take 𝛼(1)=𝑎, 𝛼(2)=𝑏, 𝛽(1)=𝑐, 𝛽(2)=𝑑,𝑎,𝑏,𝑐,𝑑∈ℝ. Then, it is easy to compute 𝑘=(2^𝑝−1/𝜋^2𝑝)|𝑎∨𝑏|^𝑝+(√2^𝑝−2𝐽_𝑝/𝜋^𝑝)|𝑐∨𝑑|^𝑝. So we can conclude that the system of (19) is 𝑝th moment exponentially stable if 𝑘<1.

Remark 12. Example 2 is also considered by Bao et al. [10] based on the large deviation technique, in which a proper 𝑞 should be found to calculate the rate function 𝐼(𝜇) to testify whether the solution of (19) is 𝑝th moment exponentially stable or not. However, it is very difficult and complicated in comparison with our conditions, especially in the nonlinear hybrid stochastic heat equations. So our results generalize and imporove those given in [10].

5. Conclusion

In this paper, we have studied the stability problem of a class of nonlinear hybrid stochastic heat equations. Based on the fixed point theory and Burkhölder-Davis-Gundy inequality, we not only establish the existence and uniqueness of the equation in an infinite state space, but also prove the 𝑝th moment exponential stability of the system. Moreover, we give two simple examples to verify all our conditions at the end of this paper. In the future work, we will focus on the stability of more complicity models such as neutral stochastic differential equations with Markov chains and variable time delay.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China (61374080), the Natural Science Foundation of Zhejiang Province (LY12F03010), the Natural Science Foundation of Ningbo (2012A610032), and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.