1 Introduction

1 Introduction Stochastic partial differential equations with jump have been extensively studied in recent years. There exists a great number of works on the subject in literature,see the monograph^[1] and the references therein. The stability of stochastic partial differential equations has also been extensively studied; for its theory and various applications (such as applications in finance and automatic control),the reader is referred to Refs. ^{[2, 3, 4]} and the references therein. The stability of a linear equation with jump coefficient was studied in Ref. ^[4]. In Ref. ^[5],the stability of a semilinear stochastic differential equation with Wiener process was investigated. The exponen- tial stability of general nonlinear stochastic differential equations with Wiener processes was studied in Ref. ^[6]. The asymptotic and exponential stability of the nonlinear stochastic delay differential equations withWiener processes were considered in Ref. ^[7] and Ref. ^[8],respectively. In Ref. ^[9],the stability of infinite dimensional stochastic evolution equations with memory and Lévy jumps was studied.

In this article,the aim is to consider the following stochastic partial differential equation:

where L is a partial (or pseudo) differential operator of order (less than or equal to) two,e.g., L = Δ (or L = −(−Δ)_α,α ∈ (0,1]) on or an open subset thereof. The maps are assumed to fulfill certain monotonicity conditions. Q is a Hilbert-Schmidt operator valued Lipschitz map,and F denotes a Lipschitz map taking values in a Hilbert space. W stands for the cylindrical wiener process and is defined on a complete probability space (Ω ,F,P) with normal filtration F_t = σ{W(s) : s ≤ t},t ∈ [0,T]. N(dt,dz) represents the Poisson random measure associated with compensated (Ft)-martingale measure defined as where λ(dz),defined on the separable Banach space Z,is the intensity measure of N(dt,dz). In particular,if Φ = 0,Q = 0,F = 0,and Ψ(s) := |s|^r−1s for some r > 1,then (1) reduces back to the classical porous medium equation (e.g.,see Ref. ^[10]) as follows: Furthermore,if L is the Dirichlet Laplacian on an open bounded domain in ^d,it is well known that the solution decays algebraically fast in t. We refer to Refs. ^{[2, 10, 11]} and the references therein,also for historical remarks. In this article,under some monotonicity assumptions,we obtain the moment exponential stability and the almost certain exponential stability of the stochastic generalized porous media equations with jump.

In recent years,the stochastic version of the porous medium equation with Brownian motion has been studied intensively,see Refs. ^{[12, 13, 14, 15, 16, 17, 18, 19, 20, 21]} and the references therein. However,as far as we know,there have been few works that deal porous media equation with jump noise. As jump noise is different from wiener noise,this will bring various new difficulties from calculation and probability when stochastic equations with jump noise are considered. The method used in this article follows that Refs. ^{[22, 23]}. In Ref. ^[23],the existence and uniqueness of stochastic gener- alized porous media equations with L`evy jump were obtained,and in Ref. ^[24],the ergodicity of the same equation was obtained. In this article,the moment exponential stability is obtained, and then it is used to establish the almost certain exponential stability of the equations. 2 Some preliminaries

Let (,M,υ) be a separable probability space and (L,D(L)) a negative definite self-adjoint linear operator on L²(υ) having discrete spectrum with eigenvalues

and L²(υ)-normalized eigenfunctions {e_i} such that e_i ∈ L^r+1 (υ) for any i > 1,where r (> 1) is a fixed number throughout this article. A classical example of L is the Laplacian operator on a smooth bounded domain in a complete Riemannian manifold with Dirichlet boundary conditions.

Before stating our equation,we first introduce the state space of the solutions. Let

Define H to be its dual space with inner product <·,·>. Identify L²(υ) with its dual. We get the continuous and dense embedding We denote the duality between H and H¹ by <·,·>. Obviously,when restricted to L²(υ) × H¹, this coincides with the inner product in L²(υ),which is also denoted by <·,·>. It is clear that Let LHS be the space of all Hilbert-Schmidt operators from a separable Hilbert space U to L²(υ). Let W_t denote the cylindrical Brownian motion on U w.r.t. a complete filtered probability space (Ω ,F,P). Let Ψ and Φ be nonlinear deterministic continuous functions on satisfying where Denote A very simple example satisfying (3)-(4) is that Let be progressively measurable such that where μ > 0,and for arbitrary δ > 0,the nonnegative continuous function h(t) with t ∈ ⁺ satisfies i.e., and x,y ∈ H. From Kuratowski’s theorem,we know B(L²(υ)) ⊂ B(H). Let be B(⁺) × B(H) × B(Z)/B(L²(υ))-measurable and satisfy where η1(t)(t ∈ ⁺) is a nonnegative continuous function and for arbitrary δ > 0 satisfies i.e., and y,y₁,y₂ ∈ H.

Remark 1 Since h(t) = o(e^μt) and η₁(t) = o(e^μt),there exists a positive constant γ such that h(t)e^-μt ≤ γ,η₁(t)e^-μt ≤ γ for all t ∈ ⁺.

We say mapping : [0,∞[→ H is right continuous and left hand limits (RCLL) if it is right continuous and its left limits exist. Then,we introduce the definition of the solution to (1).

Defintion 1 An H-valued RCLL (F_t)-adapted process X(t) is called a solution to (1),if X ∈ L^r+1([0,T] ×Ω × ; ds × P × υ) such that for any i ∈,

We define V :=L^r+1(υ). As H is a Hilbert space,we identify H^* with H via the Riesz isomorphism. Then,we have

continuously and densely. In order to rewrite (9) in vector form,we need two lemmas from Ref. ^[15]. For the reader’s convenience,we cite them in the following.

Lemma 1^[15] The linear operator

defines an isometry from L^(r+1)/r(υ) to V^* with a dense domain. Its unique continuous ex- tension L to all of L^(r+1)/r(υ) is an isometric isomorphism from L^(r+1)/r(υ) onto V^* such that Lemma 2^[15] Let (L⁻¹)′ : L^(r+1)/r(υ) → L^(r+1)/r(υ) be the dual operator of L⁻¹ : L^r+1(υ) → L^r+1(υ). Then,the operator extends the natural inclusion L²(υ) ⊂ H ⊂ V^*,and for all f ∈ L^(r+1)/r(υ),g ∈ L^r+1(υ) According to the above lemmas,we have In view of (3)-(4),Lemma 1,and Lemma 2,the integral is well defined in V^*. Thus,(9) can be rewritten in vector form as Before giving our main result,we need a theorem from Ref. ^[23],which for convenience is cited here.

Theorem 1^[23] Assume that conditions from (3) to (8) hold,and let X₀ ∈ L²(Ω ,F₀,P;H). Then,there exists a unique solution to (1) in the sense of Definition 1. Moreover,

3 Main results

With all preparations above,we are now in the position of stating our main results in this paper.

Theorem 2 In addition to conditions (3)-(8),we assume that Ψ(0) = 0,and Φ(0) = 0. Then,if X_t is a global strong solution to (1),there exists C > 0 such that

Proof Let ε be a very small positive constant such that μ − ε > 0. Then,Itô’s formula implies

By (4) and Ψ(0) = 0,as well as Φ(0) = 0,we get Thus, where the last inequality follows the Young inequality. Since the last two terms on the right hand side of (13) are martingales,we have Since from (14) and the Gronwall inequality,we have Finally,we get In the following,we intend to investigate the almost certain stability,which is in most situations the kind of stability one usually wants to have in practical applications,of the trivial solution of (1).

Theorem 3 Under the conditions in Theorem 2,there exist positive constants M,ε,and a subset Ω₀ ⊂ Ω with P(Ω ₀) = 1 such that,for each w ∈ Ω ₀,there exists a positive random number T (w) such that

Proof For 0 ≤ s ≤ t,by (5) and (11),we have where Similarly,by (7) and (11),we have where In the following,we prove that there exists a positive constant M > 0 such that By Itô’s formula,we have By Burkholder-Davis-Gundy’s inequality and (15),we get for any T ∈ ⁺, where M₁ and M₂ are two positive constants and are independent of T . By the Burkholder-Davis-Gundy inequality and (16),we have where M₃ and M₄ are independent of T . By Itô’s formula and (4),after some elementary calculations,we get By (11),(18),and (19),we have where Thus,we have Since for arbitrary δ > 0,h(t)e^δt and η₁(t)e^δt are continuous and subexponential growth,by the Fatou lemma,we get Finally,we are going to finish the proof. By Itô’s formula,after some elementary calculations, we get In the following,the positive constant C will change from line to line. By Burkholder-Davis- Gundy’s inequality,(15),and (16),we get Similar to the derivation of (19),we get where the second inequality follows (16). From (20)-(22),we get Thus,it follows that where α = min{2θ,μ − ε}. From (20),we have Let From (21)-(23),we get Thus,by Borel-Cantelli’s lemma,we complete the proof.

Acknowledgements The authors would like to thank the editor and the anonymous referees for their careful reading of the manuscript and valuable suggestions.