Abstract
This paper deals with the limiting behavior of random attractors of stochastic wave equations with supercritical drift driven by linear multiplicative white noise defined on unbounded domains. We first establish the uniform Strichartz estimates of the solutions with respect to noise intensity, and then prove the convergence of the solutions of the stochastic equations with respect to initial data as well as noise intensity. To overcome the non-compactness of Sobolev embeddings on unbounded domains, we first utilize the uniform tail-ends estimates to truncate the solutions in a bounded domain and then employ a spectral decomposition to establish the pre-compactness of the collection of all random attractors. We finally prove the upper semicontinuity of random attractor as noise intensity approaches zero.
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1 Introduction
In this paper, we investigate the stability of pullback random attractors of the stochastic supercritical wave equation driven by multiplicative noise defined on \({\mathbb {R}}^n\):
with initial data
where \(1 \le n \le 6\), \(\tau \in {\mathbb {R}}\), \(\alpha \) and \(\nu \) are positive constants, \(f: {\mathbb {R}}^n \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is a nonlinear term, \(g \in L^2_{loc}( {\mathbb {R}}, L^2({\mathbb {R}}^n) )\), \(\sigma \in (0,1) \) is a parameter representing the noise intensity, the symbol \(\circ \) stands for Stratonovich’s integration, and W(t) is a standard real-valued Wiener process on a probability space \(( \Omega , {\mathscr {F}}, {\mathbb {P}} )\).
The long term dynamics of the stochastic wave equation (1.1) depends on the growth rate p of the nonlinear term f(x, u) as \( |u| \rightarrow \infty \). In general, \(p={\frac{n}{n-2}}\) for \(n>2\) is called a critical exponent, and the equation is said to be subcritical, critical and supercritical when \(p<{\frac{n}{n-2}}\), \(p={\frac{n}{n-2}}\) and \(p>{\frac{n}{n-2}}\), respectively. In the subcritical or critical case, the nonlinear function f maps \(H^1({{{\mathcal {O}}}}) \) into \( L^2({{{\mathcal {O}}}})\) for a domain \({{{\mathcal {O}}}}\) in \({\mathbb {R}}^n\), which plays a key role for studying the random attractors of the stochastic wave equation, see, e.g., [11, 18, 32, 47, 48] for bounded domains and [40, 42, 43, 46, 49] for unbounded domains.
However, in the supercritical case, the nonlinear function f does not map \(H^1({{{\mathcal {O}}}}) \) into \( L^2({{{\mathcal {O}}}})\) any more, and the uniform Strichartz estimates must be used to study the random attractors in this case, see, e.g., [12, 13, 44, 45]. In particular, the existence of random attractors of system (1.1)–(1.2) with supercritical nonlinearity has been proved in [13] recently. In the present paper, we continue this line of research and further investigate the stability of these random attractors for supercritical stochastic wave equation driven by multiplicative noise as the intensity of noise \(\sigma \rightarrow 0\).
More precisely, we will prove the random attractors of (1.1)–(1.2) are upper semicontinuous at \(\sigma =0\). To that end, we first establish the Strichartz estimates of the solutions which are uniform with respect to noise intensity (see Lemma 3.1). We then prove the pathwise convergence of the solutions of the stochastic equation as \(\sigma \rightarrow 0\) (see Lemma 3.2). The main difficulty of the paper is to show the precompactness of the collection of all random attractors of (1.1)–(1.2) in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\). To overcome the non-compactness of Sobolev embeddings on unbounded domains, we first utilize the invariant property of random attractors as well as the uniform tail-estimates of the solutions to prove that all functions in random attractors are uniformly infinitesimal outside a large bounded domain. We then decompose the solution operator as two parts: one is linear and the other is nonlinear. We prove the linear part is convergent in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\) as \(\sigma \rightarrow 0\). For the nonlinear part in bounded domains, we use the spectral decomposition of the Laplace operator to further split the solutions as a sum of a finite-dimensional component and an infinite-dimensional component. By showing the infinite-dimensional component is uniformly small, we eventually obtain the precompactness of the collection of all random attractors of (1.1)–(1.2) in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\), see Lemma 4.1. Based on above analysis, we then prove by [41, Theorem 3.2] that random attractors of (1.1)–(1.2) are upper semicontinuous as the noise intensity \(\sigma \rightarrow 0\), see Theorem 5.1. For details on upper semicontinuity of attractors, we refer the reader to [1, 10, 23, 24, 30, 35] for deterministic equations and [6, 17, 26, 28, 29, 39, 50] for stochastic equations.
We mention that further information on existence of pathwise random attractors can be found in [5, 7,8,9, 15, 16, 22, 27, 36] and the references therein. For attractors of deterministic wave equations, the reader is referred to [2,3,4, 14, 21, 25, 31, 37, 38] for bounded domains and [19, 20, 33, 34] for unbounded domains. In particular, the existence of global attractors of the supercritical deterministic wave equation on \({\mathbb {R}}^3\) was examined in [20].
The paper is organized as follows. In Sect. 2, we review the existence and uniqueness of random attractors of (1.1)–(1.2). In Sect. 3, we prove the convergence of the solutions of (1.1)–(1.2) with respect to initial data and noise intensity. Section 4 is devoted to the precompactness of the collection of all random attractors of (1.1)–(1.2) in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\). In the last section, we prove the upper semicontinuity of random attractors as the noise intensity \(\sigma \) approaches 0.
Hereafter, we denote the inner product and the norm of \(L^2({\mathbb {R}}^n)\) by \((\cdot , \cdot )\) and \(\Vert \cdot \Vert \), respectively.
2 Preliminaries
In this section, we review existence of random attractors of the stochastic supercritical wave equation (1.1)–(1.2) on \({\mathbb {R}}^n\), which is needed for further proving the upper semicontinuity of these random attractors.
In the sequel, we assume \(f: {\mathbb {R}}^n \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is continuous and write \(F(x,r)= \int _0^r f(x,s) ds\) for all \(x \in {\mathbb {R}}^n\) and \(r \in {\mathbb {R}}\). Suppose f and F satisfy the conditions: for all \(x \in {\mathbb {R}}^n\) and \(u, u_1, u_2 \in {\mathbb {R}} \),
where \(p\ge 1\) for \(n=1,2\) and \(1\le p < \frac{n+2}{n-2}\) for \(3\le n \le 6\), \(\alpha _1, \alpha _2, \alpha _3 > 0\), \(\gamma \in (0,1]\), \(\varphi \in L^\infty ({\mathbb {R}}^n)\) for \(n=1,2\) and \(\varphi \in L^\infty ({\mathbb {R}}^n) \cap L^{\frac{2n}{(n-2)(p-1)}}({\mathbb {R}}^n) \) for \(3\le n \le 6\), \(\varphi _1, \varphi _2 \in L^1({\mathbb {R}}^n)\), \(\varphi _3, \varphi _4 \in L^1({\mathbb {R}}^n) \cap L^\infty ({\mathbb {R}}^n)\), and \(\varphi _5\in L^\infty ({\mathbb {R}}^n)\).
We mention that if \(f(x,u) = |u|^{p-1}u\) for \(x\in {\mathbb {R}}^n\) and \(u\in {\mathbb {R}}\), then f satisfies all conditions (2.1)–(2.6) for \(p\ge 1\).
For the probability space, we denote by \(( \Omega , {\mathscr {F}}, {\mathbb {P}} )\) the classical Wiener space, where \(\Omega =\{ \omega : {\mathbb {R}} \rightarrow {\mathbb {R}} \ is \text { continuous}, \ \omega (0) = 0 \}\). Given \(t\in {\mathbb {R}}\), denote by
Let \(y(\theta _t \omega )\) be the unique stationary solution of linear stochastic differential equation
Then there exists \(\theta _t\)-invariant set \({\tilde{\Omega }} \subseteq \Omega \) with \({\mathbb {P}}({\tilde{\Omega }}) = 1\) such that \(y(\theta _t \omega )\) is tempered and continuous in t for any \(\omega \in {\tilde{\Omega }}\), For convenience, \({\tilde{\Omega }}\) will be written as \(\Omega \).
Denote by \(V(t) = u(t) e^{-\sigma \int _0^t y(\theta _s \omega ) ds } \). Then equation (1.1) can be rewritten as follows:
with initial data
where \(V_0(x) = u_0(x) e^{ -\sigma \int _0^{\tau } y(\theta _s \omega ) ds }\) and \(V_{1,0}(x) = \left( u_{1,0}(x) - \sigma y(\theta _\tau \omega ) u_0(x) \right) e^{ -\sigma \int _0^{\tau } y(\theta _s \omega ) ds }.\)
As usual, a solution of (2.7)–(2.8) will be understood in the following sense.
Definition 2.1
Given \(\tau \in {\mathbb {R}}\), \(\omega \in \Omega \), \((V_0,V_{1,0}) \in H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\), a function \(V(\cdot ; \tau , \omega , (V_0, V_{1,0})): [\tau , \tau +T] \rightarrow H^1({\mathbb {R}}^n)\) is called a solution of (2.7) and (2.8) if
-
(i)
\( V \in L^\infty (\tau , \tau +T; H^1({\mathbb {R}}^n) ) \bigcap C([\tau , \tau +T], L^2({\mathbb {R}}^n) ), \) \( \partial _t V \in L^\infty (\tau , \tau +T; L^2({\mathbb {R}}^n) ) \bigcap C([\tau , \tau +T], H^{-1}({\mathbb {R}}^n) ), \) \(V(\tau )=V_0\) and \(\partial _t V(\tau ) = V_{1,0}\).
-
(ii)
\(V(t; \tau , \cdot , (V_0, V_{1,0})): \Omega \rightarrow H^1({\mathbb {R}}^n)\) is \(({\mathcal {F}}, {\mathcal {B}}(H^1({\mathbb {R}}^n))\)-measurable, and \(\partial _t V(t; \tau , \cdot , (V_0, V_{1,0})): \Omega \rightarrow L^2({\mathbb {R}}^n)\) is \(({\mathcal {F}}, {\mathcal {B}}(L^2({\mathbb {R}}^n))\)-measurable.
-
(iii)
For each \(\zeta \in C_0^\infty ( (\tau ,\tau +T) \times {\mathbb {R}}^n )\) and a.s. \(\omega \in \Omega \),
$$\begin{aligned}&- \int _\tau ^{\tau +T} \left( \partial _t V, \partial _t \zeta \right) dt + \int _\tau ^{\tau +T} \left( \nabla V, \nabla \zeta \right) dt + \alpha \int _\tau ^{\tau +T} \left( \partial _t V, \zeta \right) dt + \nu \int _\tau ^{\tau +T} \left( V, \zeta \right) dt \nonumber \\&\qquad + \int _\tau ^{\tau +T} \int _{{\mathbb {R}}^n} f( x, e^{\sigma \int _0^t y(\theta _s \omega ) ds} V(t,x) ) \ e^{ - \sigma \int _0^t y(\theta _s \omega ) ds } \zeta (t,x) dxdt\ \nonumber \\&\quad = \int _\tau ^{\tau +T} e^{ - \sigma \int _0^t y(\theta _s \omega ) ds } \left( g(t,\cdot ), \zeta \right) dt - \sigma ^2 \int _\tau ^{\tau +T} y^2(\theta _t \omega ) \left( V(t), \zeta \right) dt \nonumber \\&\qquad - 2 \sigma \int _\tau ^{\tau +T} y(\theta _t \omega ) \left( \partial _t V(t), \zeta \right) dt. \end{aligned}$$
We recall from [13] the following existence and uniqueness of solutions to (2.7)–(2.8).
Theorem 2.1
Assume (2.1)–(2.4) hold, \(V_0 \in H^1({\mathbb {R}}^n)\), \(V_{1, 0} \in L^2({\mathbb {R}}^n)\), \(\tau \in {\mathbb {R}}\) and \(T>0\). Then problem (2.7)–(2.8) has a unique solution V on \([\tau , \tau +T]\) in the sense of Definition 2.1 which further satisfies Strichartz’s inequality:
for any \(p^*\) and \(q^*\) with
In addition, the solution V with (2.9) is continuous with respect to initial data \((V_0, V_{1,0})\) in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\), \( V \in C([\tau , \tau +T], H^1({\mathbb {R}}^n))\), \( V_t \in C([\tau , \tau +T], L^2({\mathbb {R}}^n))\), and for a.e. \(t\in ( \tau , \tau +T)\),
By Theorem 2.1, we find that for every \(\sigma \in (0, 1)\), \(\tau \in {\mathbb {R}}\) and \( (u_0, u_{1,0} ) \in H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\), problem (1.1)–(1.2) has a solution \(u_\sigma (t) = u_\sigma (t; \tau , \omega , (u_0, u_{1,0}) ) \) which is pathwise unique. Then a cocycle \(\Phi _\sigma : {\mathbb {R}}^+ \times {\mathbb {R}} \times \Omega \times H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n) \rightarrow H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\) associated with (1.1)–(1.2) is given by
for all \(s\in {\mathbb {R}}^+\), \(\tau \in {\mathbb {R}}\), \(\omega \in \Omega \) and \( (u_0, u_{1,0} ) \in H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\).
Denote by \( c_1 = 2 (1+\nu ^{-1}) \left( 16 + 2\alpha _1^2 \alpha _3^{-2} + \alpha _1^2 \Vert \varphi \Vert ^2_{L^\infty ({\mathbb {R}}^n) } \right) , \ c_2 = 8 (1+\nu ^{-1}). \) Since \(\alpha >0\) and \(\nu > 0\), it follows that there exists \(\varepsilon >0\) such that
Let \({\mathcal {D}}\) be the collection of all families \(D = \left\{ D(\tau ,\omega ):\tau \in {\mathbb {R}}, \omega \in \Omega \right\} \) of bounded nonempty subsets of \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\) such that
for every \(\tau \in {\mathbb {R}}\) and \(\omega \in \Omega \). For the function g, we assume
For convenience, we set
Then under (2.1)–(2.6) and (2.15), it follows from [13, Lemma 4.2] that for every \(\sigma \in (0, \sigma _0]\), the cocycle \(\Phi _\sigma \) has a closed measurable \({\mathcal {D}}\)-pullback absorbing set as given by
where
with \(M_1>0\) being a number independent of \(\tau , \omega \) and \(\sigma \), and
which satisfies
Furthermore, by [13, Theorem 6.1] we know that the cocycle \(\Phi _\sigma \) possesses a unique \({\mathcal {D}}\)-pullback random attractor \({\mathcal {A}}_\sigma = \left\{ {\mathcal {A}} _\sigma (\tau ,\omega ): \tau \in {\mathbb {R}}, \omega \in \Omega \right\} \in {\mathcal {D}}\) in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\). We will investigate the limit of the family \(\{{\mathcal {A}} _\sigma \}\) as \(\sigma \rightarrow 0\).
For \(\sigma =0\), the stochastic equation (1.1) becomes a deterministic equation on \({\mathbb {R}}^n\):
with initial data
Denote by \({\mathcal {D}}_0\) a collection of some families of bounded nonempty subsets of \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\) as given by
As for system (1.1)–(1.2), under (2.1)–(2.6) and (2.15), we know that for every \(\tau \in {\mathbb {R}}\) and \((u_0, u_{1,0}) \in H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\), problem (2.21)–(2.22) has a unique solution \((u, u_t) \in C([\tau , \infty ), H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n))\), which generates a continuous deterministic cocycle \(\Phi _0\) in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\). Moreover, the cocycle \(\Phi _0\) has a unique \({\mathcal {D}}_0\)-pullback attractor \({\mathcal {A}}_0 = \{ {\mathcal {A}}_0(\tau ): \tau \in {\mathbb {R}}\}\) in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\), and has a \({\mathcal {D}}_0\)-pullback absorbing set \(K_0 =\{ K_0(\tau ): \tau \in {\mathbb {R}}\}\) where \(K_0(\tau )\) is given by
and
with \(M_1>0\) being the same number as in (2.18).
By (2.17)–(2.19) and (2.23)–(2.24), we see that
In order to investigate the upper semicontinuity of the pullback attractors \({\mathcal {A}}_\sigma \) as \(\sigma \rightarrow 0\), we first establish the convergence of solutions to (2.7)–(2.8) as \(\sigma \rightarrow 0\) in the next section.
3 Pathwise Convergence of Solutions
For convenience, for every \(\sigma \in (0,1)\), we write the solution of (2.7)–(2.8) as \(V_\sigma (t; \tau , \omega , (V_0, V_{1,0}) )\) = \(V_\sigma (t)\), and write the solution of (2.21)–(2.22) as \(u(t; \tau , (u_0, u_{1,0}) )=u(t)\). We first derive the uniform estimates of the solutions of (2.7)–(2.8).
Lemma 3.1
Assume (2.1)–(2.5) hold. Then for every \(\tau \in {\mathbb {R}}\), \(\omega \in \Omega \) and \(T>0\), there exists \(C_1=C_1( \tau , T, \omega )>0\) such that for all \((V_{0}, V_{1,0}) \in H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\), \(\sigma \in (0,1)\) and \(t \in [\tau , \tau +T]\), the solution \(V_\sigma (t)\) of (2.7)–(2.8) satisfies
and if \(\frac{n}{n-2}< p < \frac{n+2}{n-2}\) and \( q={\frac{2p}{(n-2)p -n}}\) with \(3\le n \le 6\), then
Proof
By Theorem 2.1, we have
For the third term on the right-hand side of (3.3), by (2.1), (2.3) and (2.4), we have
For the fourth term on the right-hand side of (3.3), by Hölder’s inequality, we obtain
From (3.3)–(3.5), it follows that
By (3.6), we find that there exists \(\lambda _1=\lambda _1 (\tau , T, \omega )>0\) such that for all \(t\in (\tau , \tau +T)\) and \(\sigma \in (0,1)\),
By (2.1), (2.3), (2.5), (3.7) and Gronwall’s inequality, we obtain that for all \(t \in [\tau , \tau +T]\) and \(\sigma \in (0,1)\),
where we have used the Sobolev embedding \(H^1({\mathbb {R}}^n) \hookrightarrow L^r({\mathbb {R}}^n)\) for \(2\le r < \frac{2n}{(n-2)} \).
Then it follows from (2.4) and (3.8) that there exists a constant \(C_1 =C_1(\tau , T, \omega )>0\) such that for all \(t \in [\tau , \tau +T]\) and \(\sigma \in (0,1)\),
which yields (3.1).
Next, we prove (3.2). By the Strichartz inequality given by Theorem 2.1, we find
and hence by [44, Lemma 3.4] we get for \(t \in [\tau , \tau +T]\) and \(\frac{n}{n-2}< p < \frac{n+2}{n-2}\),
where \(C_2\) is a positive constant depending on T and \(\omega \), but independent of \(\tau , \sigma \) and initial data.
For the second term on the right-hand side of (3.10), by (2.1) and (2.3) we have
By (3.10), (3.11), and the Hölder inequality, we obtain that for all \(t\in [\tau , \tau +T]\) and \(\sigma \in (0,1)\),
Denote
Then by (3.9) and (3.12), we have
Since \(p<{\frac{n+2}{n-2}}\), we see that \(q>p\). Then by (3.14) we infer that there exists \(\delta \in (0, T)\) independent of \(\sigma \) such that for all \(t\in [\tau , \tau +\delta ]\) and \(\sigma \in (0,1)\),
Repeating the above argument in \([\tau +\delta , \tau +2\delta ]\) if necessary, and after a finite number of steps, we can extend the inequality (3.15) to the entire interval \([\tau , \tau +T]\), which concludes the proof. \(\square \)
In what follows, we investigate the convergence of solutions of (2.7)–(2.8) as the noise intensity \(\sigma \) approaches zero.
Lemma 3.2
Assume (2.1)–(2.5) hold. Then for every \(\eta > 0\), \(\tau \in {\mathbb {R}}\), \(\omega \in \Omega \) and \(T>0\), there exist \(c=c(T,\omega )>0\), \(\sigma _1=\sigma _1(\eta , \tau ,\omega ,T) \in (0, \sigma _0)\) and \(M=M(\tau ,\omega ,T, u_0, u_{1,0})>0\) such that for all \(t\in [\tau ,\tau +T]\) and \(\sigma \in (0, \sigma _1)\),
where \((u_0, u_{1,0})\), \((V_0, V_{1,0}) \in H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\) with \(\Vert V_{0} - u_0\Vert _{H^1({\mathbb {R}}^n)} + \Vert V_{1,0} - u_{1,0}\Vert \le 1\).
Proof
Let \(z(t) = V_\sigma (t) - u(t)\). By (2.7) and (2.21) we have
Since both \(V_\sigma \) and u satisfy the Strichartz estimates, we see
Then by [44, Lemma 3.4] we find that for all \(t \in [\tau , \tau +T]\), \(\frac{n}{n-2}< p < \frac{n+2}{n-2}\) and \(q = {\frac{2p}{(n-2)p-n} }\),
where \(C_2\) is a positive constant depending on T and \(\omega \), but independent of \(\tau , \sigma \) and initial data.
For the second term on the right-hand side of (3.17), we have
Applying the assumption (2.1) to the first term on the right-hand side of (3.18), we have
For the first term on the right-hand side of (3.19), by the Hölder inequality, we obtain
Since \(\Vert V_{0} - u_0 \Vert _{H^1({\mathbb {R}}^n)} + \Vert V_{1,0} - u_{1,0} \Vert \le 1 \), we find by (3.2) that there exists a constant \(C_3>0\), depending on \(\tau , T, \omega \) and \((u_0, u_{1,0})\) but not on \(\sigma \) or \((V_{0}, V_{1,0})\), such that
Then by (3.20) and (3.21), we obtain
which implies that there exists \(\delta _1>0\), depending only on \(\tau , T, \omega \) and \((u_0, u_{1,0})\) but not on \(\sigma \) or \((V_{0}, V_{1,0})\), such that for all \(t \in [\tau , \tau +\delta _1]\) and \(\sigma \in (0,1)\),
Then by (3.19) and (3.22), we have
For the second term on the right-hand side of (3.18), by (2.1), we have
Given \(\tau \in {\mathbb {R}}\), \(\omega \in \Omega \), \(T > 0\) and \(\eta \in (0, 1)\), by the continuity of \(\int _0^s y(\theta _r \omega ) dr\) in \(s\in {\mathbb {R}}\), we find that there exists \(\sigma _1 \in (0, \sigma _0)\), depending only on \(\tau , \omega , T\) and \(\eta \), such that for all \(\sigma \in (0, \sigma _1)\) and \(s \in [\tau , \tau +T]\),
Then by (3.24) and (3.25), we obtain for all \(t \in [\tau , \tau +T]\) and \(\sigma \in (0, \sigma _1)\),
Similar to (3.1), we find that there exists \(C_4=C_4(\tau , T)>0\) such that for all \(t\in [\tau , \tau +T]\),
Then by (3.21), (3.26) and (3.27), we obtain for all \(t \in [\tau , \tau +T]\) and \(\sigma \in (0, \sigma _1)\),
For the third term on the right-hand side of (3.18), by (2.1) and (3.25), we get
which together with (3.21) and (3.27) yields that for all \(t \in [\tau , \tau +T]\) and \(\sigma \in (0, \sigma _1)\),
From (3.18), (3.23), (3.28) and (3.29), it follows that there exists \(L=L(\tau , T, \omega )>0\) such that for all \(t\in [\tau , \tau +\delta _1]\) and \(\sigma \in (0, \sigma _1)\),
By (3.17), (3.25) and (3.30), we obtain that for any \(\sigma \in (0, \sigma _1)\) and \(t \in [ \tau , \tau +\delta _1 ]\),
Then from the Gronwall inequality, it follows that for all \(\sigma \in (0, \sigma _1)\) and \(t \in [ \tau , \tau +\delta _1 ]\),
which along with (3.1) implies that there exists \(C_3=C_3(\tau , T, \omega , u_0, u_{1,0})>0\) such that for all \(\sigma \in (0, \sigma _1)\) and \(t \in [ \tau , \tau +\delta _1 ]\),
Repeating the above argument in \([\tau +\delta _1, \tau +2\delta _1]\) if necessary, and after a finite number of steps, we can extend the inequality (3.32) to the entire interval \([\tau , \tau +T]\), which concludes the lemma when \(p\in ( {\frac{n}{n-2}}, \ {\frac{n+2}{n-2}} )\) for \(3\le n \le 6\).
If \(1\le p\le {\frac{n}{n-2}}\) for \(3\le n \le 6\) or \(p\ge 1\) for \(1\le n \le 2\), then \(H^1({\mathbb {R}}^n) \hookrightarrow L^{2p} ({\mathbb {R}}^n)\) and hence the proof is much simpler in this case and thus is omitted. \(\square \)
In the next section, we prove the uniform precompactness of the family of random attractors \(\{{\mathcal {A}}_\sigma \}\) in \(H^1({\mathbb {R}}^n)\times L^2({\mathbb {R}}^n)\).
4 Uniform Precompactness of Random Attractors
To prove the precompactness of the set \(\{{\mathcal {A}}_\sigma \} _{\sigma \in (0, \sigma _0)} \), we introduce a family of subsets of \(H^1({\mathbb {R}}^n)\times L^2({\mathbb {R}}^n)\): for every \(\tau \in {\mathbb {R}}\) and \(\omega \in \Omega \), setting
where
and \(M_1\) is the same number as in (2.18).
Then by (2.20) and \(\lim _{t \rightarrow +\infty } \frac{1}{t} \int _{-t}^{0} y(\theta _{r} \omega ) dr = 0\), we know that
which implies \( K = \{ K(\tau ,\omega ): \tau \in {\mathbb {R}}, \omega \in \Omega \} \in {\mathcal {D}}. \)
Moreover, by comparing (2.17) with (4.1), we have for any \(\tau \in {\mathbb {R}}\) and \(\omega \in \Omega \),
We now prove the following compactness results on the family of attractors \(\left\{ {\mathcal {A}}_{\sigma } \right\} _{\sigma \in (0,\sigma _0)}\). To avoid confusion, if necessary, the derivatives of the solutions u and V with respect to time are also written as \({\dot{u}}\) and \({\dot{V}}\), respectively.
Lemma 4.1
Assume (2.1)–(2.6) and (2.15) hold, \(\tau \in {\mathbb {R}}\), \(\omega \in \Omega \), and \(\sigma _k \rightarrow 0\). If \((u_k,v_k) \in {\mathcal {A}}_{\sigma _k}(\tau , \omega )\), then the sequence \(\{ (u_k,v_k) \}_{k=1}^\infty \) is precompact in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\).
Proof
Without loss of generality, we may assume \(\sigma _k\le \sigma _0\) for all \(k\in {\mathbb {N}}\). It follows from the invariance of \({\mathcal {A}}_{\sigma _k}\) that for every \(k\in {\mathbb {N}}\), there exists \((\varphi _k, \psi _k ) \in {\mathcal {A}}_{\sigma _k} (\tau -k,\theta _{-k} \omega )\) such that
Claim 1. \(\{(u_k, v_k)\}_{k=1}^\infty \) is uniformly infinitesimal outside a large ball in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\).
By the transform \(V(t) = u(t) e^{-\sigma \int _0^t y(\theta _s \omega ) ds }\), we have
and
where \({\widetilde{\varphi }}_k = \varphi _k e^{-\sigma _k \int _0^{\tau -k} y(\theta _{s-\tau } \omega ) ds}\) and \({\widetilde{\psi }}_k = e^{-\sigma _k \int _0^{\tau -k} y(\theta _{s-\tau } \omega ) ds} \left( \psi _k - \sigma _k y(\theta _{-k} \omega ) \varphi _k \right) \).
For any \(\tau \in {\mathbb {R}}\) and \(\omega \in \Omega \), let
where
Then \( {\bar{K}} = \{ {\bar{K}}(\tau ,\omega ): \tau \in {\mathbb {R}},\omega \in \Omega \} \in {\mathcal {D}} \) due to (4.2). By \((\varphi _k, \psi _k ) \in {\mathcal {A}}_{\sigma _k} (\tau -k,\theta _{-k} \omega ) \subseteq K(\tau -k, \theta _{-k}\omega )\), we find \(( {\widetilde{\varphi }}_k, {\widetilde{\psi }}_k ) \in {\bar{K}}(\tau -k,\theta _{-k} \omega )\). Then together with (4.5) and (4.6), it follows from [13, Lemma 5.1] on the uniform tail-estimates of the solutions of (2.7) that for every \(\epsilon '>0, \tau \in {\mathbb {R}}\) and \(\omega \in \Omega \), there exist \(T_0 = T_0(\epsilon ', \tau , \omega , K)>0\) and \(m_0 = m_0(\epsilon ', \tau , \omega ) \ge 1\) such that for all \(k \ge T_0\), \(m \ge m_0\) and \(k \in {\mathbb {N}}\),
So by (4.4) and (4.8), we obtain that for all \(k \ge T_0\) and \(k \in {\mathbb {N}}\),
where \({\mathcal {O}}_{m_0}^c\) is the complement of \({\mathcal {O}}_{m_0} = \{ x \in {\mathbb {R}}^n: \ |x| \le m_0\}\). By (4.4) and (4.9), we obtain for all \(k \in {\mathbb {N}}\),
Claim 2. \(\{(u_k, v_k) |_{{\mathcal {O}}_{m_0}}\}_{k=1}^\infty \) is precompact in \(H^1({\mathcal {O}}_{m_0}) \times L^2({\mathcal {O}}_{m_0})\).
Let \(V_{\sigma _k}(t) =: V_{\sigma _k} ( t; \tau -k, \theta _{-\tau }\omega , ({\widetilde{\varphi }}_k, {\widetilde{\psi }}_k ) ) \). Then by the uniqueness of solutions of (2.7)–(2.8), \(V_{\sigma _k}(t)\) can be decomposed into \(V_{\sigma _k}(t) = {\widetilde{V}}_{\sigma _k}(t) + {\widehat{V}}_{\sigma _k}(t)\), where \({\widetilde{V}}_{\sigma _k}(t)\) and \({\widehat{V}}_{\sigma _k}(t)\) are the solutions of the following random wave equations:
and
In what follows, we will prove Claim 2 in three steps.
Step 1: prove \(\{ ({\widetilde{V}}_{\sigma _k}(\tau ), \partial _t {\widetilde{V}}_{\sigma _k}(\tau )) \}_{k=1}^\infty \) is a Cauchy sequence in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\).
For any \(k,l \in {\mathbb {N}}\) with \(k < l\), let \( V_{k,l}(t) = {\widetilde{V}}_{\sigma _k}(t) - {\widetilde{V}}_{\sigma _l}(t). \) Then \(V_{k,l}\) satisfies the following equation:
with initial data
Since \(g \in L^2(\tau , \tau +T; L^2({\mathbb {R}}^n) )\) and \({\widetilde{V}}_{\sigma _l} \in L^\infty (\tau , \tau +T; H^1({\mathbb {R}}^n) ), \partial _t {\widetilde{V}}_{\sigma _l} \in L^\infty ( \tau , \tau +T; L^2({\mathbb {R}}^n) ) \), it follows from the energy equation associated with (4.13) that
For the first term on the right-hand side of (4.14), we have
For the last two terms on the right-hand side of (4.14), we have
For the second and third terms on the right-hand side of (4.14), we have
Then it follows from (4.14)–(4.17) that
Since \(\varepsilon (1+\nu ^{-1}) \le 1\) due to (2.13), we can obtain
Then it follows from (2.13), (4.18) and (4.19) that
Solving the above inequality (4.20) on \([\tau -k, \tau ]\), we get
Since \({\varepsilon }\le 1\) and \({\varepsilon }\le \nu \) we have
We now deal with the first term on the right-hand side of (4.23), for which we have
As in [13, (6.14)], one can verify
where \(C_1=C_1(\nu , \alpha )>0\).
By (2.16) we see that if \(\sigma \le \sigma _0\), then \(\sigma \le {\frac{p\alpha {\varepsilon }\gamma \nu }{(p+1)(1+\nu ) (24+50\alpha )}} \), and hence
Since \( \lim \limits _{t\rightarrow \infty } {\frac{1}{t}} \int ^0_{-t} y^2(\theta _r \omega ) dr = {\frac{1}{2\alpha }} \), we know that there exists \(T_1=T_1(\omega )>0\) such that
Since \(\sigma _k\le \sigma _0\), by (4.27)–(4.28) we get for \(k\ge T_1\),
For the first term on the right-hand side of (4.26), by (4.29) we have for all \(k\ge T_1\),
For the second term on the right-hand side of (4.26), by (4.28) and (4.29) we have for all \(l > k \ge T_1\),
Since \(\sigma _k \rightarrow 0\), there exists \(k_0=k_0(\omega )\ge T_1\) such that for all \(l>k\ge k_0\),
and thus for all \(l>k\ge k_0\),
By (4.31) and (4.33) we obtain as \(l > k \rightarrow \infty \),
By (4.28) and (4.32) for \(l > k \ge k_0 \), we have
Similarly, by (4.28) and (4.32) for \( k \ge k_0 \), we have
By (2.15), (4.35) and (4.36) for \( l>k\ge k_0 \), we obtain
It follows from (4.26), (4.30) (4.34) and (4.37) that for \(l>k\ge k_0\),
as \(k\rightarrow \infty \).
For the second term on the right-hand side of (4.23), by (2.15), (4.32) and the Lebesgue dominated convergence theorem we obtain
For the last term on the right-hand side of (4.23), we have
As in [13, (4.16)], we also have
where \(C_2=C_2(\nu , \alpha )>0\). By (4.40)–(4.41) we get
For the first term on the right-hand side of (4.42), we have for \(l > k \ge k_0\),
which can be verified by (4.32) as before.
For the second term on the right-hand side of (4.42), we have for \(l > k \ge k_0\),
which can be verified by (2.15) and (4.32). By (4.42)–(4.44) we see that for \(l>k\) with \(k\rightarrow \infty \),
It follows from (4.23), (4.38), (4.39) and (4.45) that
and hence \( \left\{ \big ( {\widetilde{V}}_{\sigma _k}(\tau ), \dot{{\widetilde{V}}}_{\sigma _k}(\tau ) \big ) \right\} _{k=1}^\infty \) is a Cauchy sequence in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\).
Step 2: prove for every \(\tau \in {\mathbb {R}}\), \(k\in {\mathbb {N}}\) and \(\omega \in \Omega \), \( \Big \{ \Big ( {\widehat{V}}_{\sigma _k} (\tau ;\tau -k,\omega ,0), \ \dot{{\widehat{V}}}_{\sigma _k} (\tau ;\tau -k,\omega ,0) \Big ) \Big \}_{k=1}^\infty \) is precompact in \({H^1( {\mathcal {O}}_{m} )} \times {L^2( {\mathcal {O}}_{m} )}\) for each \(m \in {\mathbb {N}}\).
The idea of the proof is to first truncate the solutions in a bounded domain and then use the spectral decomposition of the Laplace operator with homogeneous Dirichlet boundary condition.
Let \(\xi \) be a smooth function on \({\mathbb {R}}^n\) such that \(0 \le \xi (x) \le 1\) for all \(x \in {\mathbb {R}}^n\), and \(\xi (x)=1\) for \(|x| \le 1\) and \(\xi (x)=0\) for \(|x| \ge {\frac{3}{2}}\). For each \(m \in {\mathbb {N}}\), let \(\xi _m (x) = \xi (\frac{x}{m})\), and \({\mathcal {O}}_m=\{ x \in {\mathbb {R}}^n: |x| < m \}\).
Let \({\widehat{V}}_{k,m} (t) = \xi _m {\widehat{V}}_{\sigma _k}(t)\), then by (4.12) we find \({\widehat{V}}_{k,m}\) satisfies the following initial and boundary value problem:
Let \(A = - \triangle \) with domain \(H^2({\mathcal {O}}_{2\,m})\bigcap H^1_0({\mathcal {O}}_{2\,m})\). Then \(L^2({\mathcal {O}}_{2\,m})\) has an orthonormal basis \(\{e_j\}_{j=1}^\infty \) such that \(A e_j=\lambda _j e_j\) and \( 0<\lambda _1\le \lambda _2 \le \dots \le \lambda _n \rightarrow \infty . \) Given \(n\in {\mathbb {N}}\), let \(P_n: L^2({\mathcal {O}}_{2\,m}) \rightarrow \text {span}\{e_1,\cdots , e_n\}\) be the projection operator.
Since \({\widehat{V}}_{\sigma _k}\) is the solution of (4.12), as in [44, Lemma 3.4] we have
where \(C_4 > 0\) depends on \(\tau , k, T\) and \(\omega \), but not on \(\sigma _k\), n or m. As in [13], after detailed calculations, we find that for every \(\eta >0\), there exists \(n_1=n_1(\eta , \tau , k, T, \omega )\ge 1\) and \(m_1=m_1(\eta , \tau , k, T, \omega ) \ge 1\) such that for all \(n\ge n_1\) and \(m\ge m_1\),
By (4.47) we have
for all \(t \in [\tau -k, \tau -k+T]\), where \(C_5\) is a positive constant depending on \(\tau , k, T, \omega \), but not on \(\sigma _k\), n or m. Then for every \(\tau \), k, \(\omega \), n and m, by (4.49) we find that the set
is precompact in a finite-dimensional space, which along with (4.48) shows that the set
has a finite cover of radius \(\eta \) in \({H^1( {\mathcal {O}}_{2\,m} )} \times {L^2( {\mathcal {O}}_{2\,m} )}\) for \(m\ge m_1\).
Since \({\widehat{V}}_{k,m} (\tau ;\tau -k,\omega ,0) = {\widehat{V}}_{\sigma _k} (\tau ;\tau -k,\omega ,0)\) for \(|x| \le m\), we see that the set
has a finite cover of radius \(\eta \) in \({H^1( {\mathcal {O}}_{m} )} \times {L^2( {\mathcal {O}}_{m} )}\) for \(m\ge m_1\). Therefore
is precompact in \({H^1( {\mathcal {O}}_{m} )} \times {L^2( {\mathcal {O}}_{m} )}\) for each \(m \in {\mathbb {N}}\).
Step 3: prove \(\{(u_k, v_k) |_{{\mathcal {O}}_{m}}\}_{k=1}^\infty \) is precompact in \(H^1({\mathcal {O}}_{m}) \times L^2({\mathcal {O}}_{m})\) for \(m\in {\mathbb {N}}\).
Since \(V_{\sigma _k}(t) = {\widetilde{V}}_{\sigma _k}(t) + {\widehat{V}}_{\sigma _k}(t)\), by Steps 1 and 2 we see that for each \(\tau \) and \(\omega \), the sequence
is precompact in \({H^1( {\mathcal {O}}_{m} )} \times {L^2( {\mathcal {O}}_{m} )}\) for every \(m \in {\mathbb {N}}\). Since \(\sigma _k \rightarrow 0\), it follows from (4.5) and (4.6) that for every \(\tau \) and \(\omega \), the sequence
is precompact in \({H^1( {\mathcal {O}}_{m} )} \times {L^2( {\mathcal {O}}_{m} )}\) for each \(m \in {\mathbb {N}}\), which along with (4.4) yields that \(\{ (u_k, v_k) |_{{\mathcal {O}}_{m_0}} \}_{k=1}^\infty \) is precompact in \(H^1({\mathcal {O}}_{m_0}) \times L^2({\mathcal {O}}_{m_0})\). This concludes Claim 2.
Therefore, by Claims 1 and 2, we find \(\{(u_k, v_k)\}_{k=1}^\infty \) has a finite cover of radius \(3 \epsilon '\) in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\) for every \(\epsilon '>0\), and hence is precompact in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\), as desired. \(\square \)
5 Upper Semicontinuity of Random Attractors
In this section, we prove the upper semicontinuity of random pullback attractors of (1.1) when the noise intensity \(\sigma \rightarrow 0\).
Theorem 5.1
Assume (2.1)–(2.6) and (2.15) hold. Then for every \(\tau \in {\mathbb {R}}\) and \(\omega \in \Omega \),
Proof
We first note that if a sequence \(\sigma _k \rightarrow 0\) and \((u_{0}^{\sigma _k}, u_{1,0}^{\sigma _k}) \rightarrow (u_0, u_{1,0})\) in \(H^1({\mathbb {R}}^n) \times L^2({\mathbb {R}}^n)\), then for any \(t \ge 0\), \(\tau \in {\mathbb {R}}\) and \(\omega \in \Omega \),
which follows immediately from (2.12), Lemma 3.2 and the transformation
By (2.25), (5.1) and Lemma 4.1, we find that all conditions of [41, Theorem 3.2] are fulfilled, from which we obtain that for every \(\tau \in {\mathbb {R}}\) and \(\omega \in \Omega \),
This completes the proof. \(\square \)
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The work is partially supported by the NNSF of China (11471190, 11971260).
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Chen, Z., Wang, B. Limiting Behavior of Random Attractors of Stochastic Supercritical Wave Equations Driven by Multiplicative Noise. Appl Math Optim 88, 59 (2023). https://doi.org/10.1007/s00245-023-10030-4
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DOI: https://doi.org/10.1007/s00245-023-10030-4
Keywords
- Stochastic wave equation
- Unbounded domain
- Supercritical exponent
- Strichartz’s inequality
- Random attractor
- Upper semicontinuity