In this section, we will first deduce the exponential stability of the basic solution of the homogeneous equation of (2.1), and then we will use the fixed point principle to discuss the existence and uniqueness of the mild solution to (2.1) and prove the exponential stability results.
Theorem 1
Let
\(p \ge2 \)
and suppose that Assumption
A
holds. Then (2.1) is exponentially stable in the
pth moment if
$$ \frac{{{2^{p - 1}}{{\tilde{M}}^{p}}}}{{{\gamma^{p}}}} {\bigl\vert {{\alpha^{*}}} \bigr\vert ^{p}} + \frac{{{2^{p - 1}}{{\tilde{M}}^{p}}{J_{p}}}}{{{{(2\gamma)}^{\frac{p}{2}}}}} {\bigl\vert {{\beta^{*}}} \bigr\vert ^{p}} \in(0,1), $$
(3.1)
where
\({J_{p}} = (p^{p + 1}/2(p - 1)^{p - 1})^{\frac{p}{2}}\), \(\vert {{\alpha^{*}}} \vert = \sup _{1 \le i \le N} \vert {{\alpha_{i}}} \vert \), \(\vert {{\beta^{*}}} \vert = \sup _{1 \le i \le N} \vert {{\beta_{i}}} \vert \), and
M̃
is some constant.
Proof
Let H be the Banach space of all \({\mathcal{F}_{t}} \)-adapted continuous processes consisting of functions \(u(t,x) \) such that \(\mathbf{E}{\Vert {u(t,x)} \Vert ^{p}} \le {M^{*}}\mathbf{E}{\Vert {{u_{0}}} \Vert ^{p}}{e^{ - \eta t}}\), \(t \ge0 \), where \({M^{*}} > 0\), \(0 < \eta< \gamma\). We denote the norm in H by \({\Vert {u(t,x)} \Vert _{H}}: = \sup_{t \ge0} \mathbf{E}{\Vert {u(t,x)} \Vert ^{p}} \). Next, we divide the proof into two parts.
In part 1, we deduce the exponential stability of the basic solution of the homogeneous equation of (2.1), and in part 2, we employ the fixed point principle to discuss the case of the nonhomogeneous equation.
Part 1: Analysis of the basic solution.
The mild solution of (2.1) can be expressed as follows:
$$\begin{aligned} u(t,x) =& {e^{(A - {A^{2}})t}} {u_{0}} + \int_{0}^{t} {{e^{(A - {A^{2}})(t - s)}}\alpha\bigl(r(s) \bigr)u(s,x)\,ds} \\ &{}+ \int_{0}^{t} {{e^{(A - {A^{2}})(t - s)}}\beta\bigl(r(s) \bigr)u(s,x)\,dB(s)}, \end{aligned}$$
(3.2)
where \({e^{(A - {A^{2}})t}}{u_{0}} = \int_{\Theta}{G(x - y){u_{0}}(y)\,dy} \) is the solution of
$$ {u_{t}} = Au - {A^{2}}u $$
(3.3)
with the initial value \({u_{0}} \), and \(G(t,x) \) is the basic solution of (3.3).
Taking the Fourier transform of (3.3), we obtain
$$ {\hat{u}_{t}} + \bigl({\vert \xi \vert ^{2}} + {\vert \xi \vert ^{4}}\bigr)\hat{u} = 0. $$
(3.4)
Hence, we have
$$ \hat{G}(t) (\xi) = {e^{ - ({{\vert \xi \vert }^{2}} + {{\vert \xi \vert }^{4}})t}}, $$
(3.5)
and then
$$ G(t,x) = {F^{ - 1}}\bigl({e^{ - ({{\vert \xi \vert }^{2}} + {{\vert \xi \vert }^{4}})t}}\bigr) = {G_{1}}(t,x) * {G_{2}}(t,x). $$
(3.6)
Here \({F^{ - 1}} \) denotes the inverse Fourier transform, ‘∗’ denotes the convolution of \({G_{1}}(t,x) \) and \({G_{2}}(t,x)\), \({G_{1}}(t,x) \) is the basic solution of
$$ {u_{t}} - Au = 0 $$
(3.7)
with \(\hat{G}_{1} (t)(\xi) = {e^{ - {{\vert \xi \vert }^{2}}t}} \), whereas \({G_{2}}(t,x) \) is the basic solution of
$$ {u_{t}} + {A^{2}}u = 0 $$
(3.8)
with \(\hat{G}_{2} (t)(\xi) = {e^{ - {{\vert \xi \vert }^{4}}t}} \). By [11] we see that the decay behavior of solution for (3.8) is as follows:
$$ {{\bigl\Vert {{G}_{2}}(t,x) \bigr\Vert }_{p}}\le C(p){{t}^{-\frac{1}{4}(1-\frac{1}{p})}} \quad(p\ge1). $$
(3.9)
We denote \({{e}^{At}}:={{G}_{1}}(x,t) \), \({{e}^{-{{A}^{2}}t}}:={{G}_{2}}(x,t) \), and \({{e}^{(A-{{A}^{2}})t}}:=G(x,t) \). Then it follows from (3.6), (3.9), Assumption (A1), and the Young inequality with convolution that
$$\begin{aligned} \bigl\Vert {{e^{(A - {A^{2}})t}}} \bigr\Vert &= \bigl\Vert {G(t, \cdot)} \bigr\Vert = \bigl\Vert {{G_{1}}(t, \cdot)*{G_{2}}(t, \cdot)} \bigr\Vert \\ &\leq\bigl\Vert {{G_{1}}(t, \cdot)} \bigr\Vert {\bigl\Vert {{G_{2}}(t, \cdot)} \bigr\Vert _{1}} \leq C\bigl\Vert {{G_{1}}(t, \cdot)} \bigr\Vert \\ &=C\bigl\Vert {{e^{tA}}} \bigr\Vert \leq\tilde{M}{e^{ - \gamma t}}. \end{aligned}$$
(3.10)
Hence, we get the exponential stability of the basic solution \({e^{(A - {A^{2}})t}} \) of (3.3).
Part 2: We will discuss the existence, uniqueness, and exponential stability of the mild solution to (2.1).
We derive the operator \(\phi: H\to H \) as follows:
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} \phi(u)(t) = {e^{(A - {A^{2}})t}}{u_{0}} + \int_{0}^{t} {{e^{(A - {A^{2}})(t - s)}}\alpha(r(s))u(s,x)\,ds}\\ \hphantom{\phi(u)(t) =}{}+ \int_{0}^{t} {{e^{(A - {A^{2}})(t - s)}}\beta(r(s))u(s,x)\,dB(s)},\\ \phi(u)(0) = {u_{0}}. \end{array}\displaystyle \right . \end{aligned}$$
It is easy to prove that the following holds by the \({C_{p}} \) inequality:
$$\begin{aligned} \mathbf{E} {\bigl\Vert {\phi(u) (t)} \bigr\Vert ^{p}} \le{}&{3^{p - 1}}\mathbf {E} {\bigl\Vert {{e^{(A - {A^{2}})t}} {u_{0}}} \bigr\Vert ^{p}} + {3^{p - 1}}\mathbf {E} {\biggl\Vert { \int_{0}^{t} {{e^{(A - {A^{2}})(t - s)}}\alpha\bigl(r(s) \bigr)u(s,x)\,ds} } \biggr\Vert ^{p}} \\ &{} + {3^{p - 1}}\mathbf{E} {\biggl\Vert { \int_{0}^{t} {{e^{(A - {A^{2}})(t - s)}}} \beta\bigl(r(s) \bigr)u(s,x)\,dB(s)} \biggr\Vert ^{p}} \\ :={}&{{3}^{p-1}}\sum_{i=1}^{3}{ \mathbf{E} {{\bigl\Vert {{I}_{i}}(t) \bigr\Vert }^{p}}}. \end{aligned}$$
(3.11)
Next, we will divide the proof into three steps.
Claim 1: ϕ is continuous in the pth moment on \([ 0, +\infty ) \).
Proof of Claim 1: Let \(u \in H\), \({t_{1}} \ge0 \), and \(\vert r \vert \) be sufficiently small. Then we get
$$\begin{aligned}& \begin{aligned}[b] \mathbf{E} {\bigl\Vert {{I_{1}}({t_{1}} + r) - {I_{1}}({t_{1}})} \bigr\Vert ^{p}} &= \mathbf{E} {\bigl\Vert {{e^{(A - {A^{2}})({t_{1}} + r)}} {u_{0}} - {e^{(A - {A^{2}}){t_{1}}}} {u_{0}}} \bigr\Vert ^{p}}\\ &\le\mathbf{E} {\bigl\Vert {{e^{(A - {A^{2}}){t_{1}}}} {u_{0}} \bigl({e^{(A - {A^{2}})r}} - 1\bigr)} \bigr\Vert ^{p}} \to0 \quad\bigl( \vert r \vert \to 0\bigr); \end{aligned} \end{aligned}$$
(3.12)
$$\begin{aligned}& \begin{aligned}[b] \mathbf{E}\bigl\Vert I_{2}(t_{1}+r)-I_{2}(t_{1}) \bigr\Vert ^{p}={}&\mathbf{E}\biggl\Vert \int_{0}^{t_{1}+r}e^{(A-A^{2})(t_{1}+r-s)}\alpha\bigl(r(s) \bigr)u(s,x)\,ds\\ &{}- \int_{0}^{t_{1}}e^{(A-A^{2})(t_{1}-s)}\alpha\bigl(r(s) \bigr)u(s,x)\,ds\biggr\Vert ^{p}\\ ={}&\mathbf{E}\biggl\Vert \int_{0}^{t_{1}}e^{(A-A^{2})(t_{1}+r-s)}\alpha \bigl(r(s) \bigr)u(s,x)\,ds\\ &{}+ \int_{t_{1}}^{t_{1}+r}e^{(A-A^{2})(t_{1}+r-s)}\alpha \bigl(r(s) \bigr)u(s,x)\,ds\\ &{}- \int_{0}^{t_{1}}e^{(A-A^{2})(t_{1}-s)}\alpha\bigl(r(s) \bigr)u(s,x)\,ds\biggr\Vert ^{p}\\ \le{}&2^{p-1}\mathbf{E}\biggl\Vert \int_{0}^{t_{1}}e^{(A-A^{2})(t_{1}-s)}\alpha \bigl(r(s) \bigr)u(s,x) \bigl(e^{(A-A^{2})r}-1\bigr)\,ds\biggr\Vert ^{p}\\ &{} +2^{p-1}\mathbf{E}\biggl\Vert \int _{t_{1}}^{t_{1}+r}e^{(A-A^{2})(t_{1}+r-s)}\alpha\bigl(r(s) \bigr)u(s,x)\,ds\biggr\Vert ^{p}\\ \rightarrow{}&0 \quad\bigl(\vert r\vert \rightarrow0\bigr). \end{aligned} \end{aligned}$$
(3.13)
Then, by the BDG inequality (Lemma 1) and (3.10), the following holds as \(\vert r \vert \to0 \):
$$\begin{aligned} &\mathbf{E}\bigl\Vert I_{3}(t_{1}+r)-I_{3}(t_{1}) \bigr\Vert ^{p} \\ &\quad=\mathbf{E}\biggl\Vert \int_{0}^{t_{1}+r}e^{(A-A^{2})(t_{1}+r-s)}\beta\bigl(r(s) \bigr)u(s,x)\,dB(s)- \int_{0}^{t_{1}}e^{(A-A^{2})(t_{1}-s)}\beta\bigl(r(s) \bigr)u(s,x)\,dB(s)\biggr\Vert ^{p} \\ &\quad\le 2^{p-1}\mathbf{E}\biggl\Vert \int_{0}^{t_{1}}e^{(A-A^{2})(t_{1}-s)}\beta \bigl(r(s) \bigr)u(s,x) \bigl(e^{(A-A^{2})r}-1\bigr)\,dB(s)\biggr\Vert ^{p} \\ &\qquad{} +2^{p-1}\mathbf{E}\biggl\Vert \int _{t_{1}}^{t_{1}+r}e^{(A-A^{2})(t_{1}+r-s)}\beta\bigl(r(s) \bigr)u(s,x)\,dB(s)\biggr\Vert ^{p} \\ &\quad\le 2^{p-1} J_{p} \mathbf{E} \biggl( \int_{0}^{t_{1}}\tilde {M}^{2}e^{-2(t_{1}-s)} \bigl\Vert \beta\bigl(r(s)\bigr)u(s,x) \bigl(e^{(A-A^{2})r}-1\bigr)\bigr\Vert ^{2}\,ds \biggr)^{\dfrac{p}{2}} \\ &\qquad{} +2^{p-1} J_{p} \mathbf{E} \biggl( \int_{t_{1}}^{t_{1}+r}\tilde {M}^{2}e^{-2(t_{1}+r-s)} \bigl\Vert \beta\bigl(r(s)\bigr)u(s,x)\bigr\Vert ^{2}\,ds \biggr)^{\dfrac{p}{2}} \\ &\quad\rightarrow 0 \quad\bigl(\vert r\vert \rightarrow0\bigr). \end{aligned}$$
(3.14)
Hence, we see that ϕ is pth continuous on \([0, +\infty ) \).
Claim 2: \(\phi(H) \) is contained in H.
Proof of Claim 2: It follows from (3.10) and (3.11) that
$$ \mathbf{E} {\bigl\Vert {{I_{1}}(t)} \bigr\Vert ^{p}} = \mathbf {E} {\bigl\Vert {{e^{(A - A{}^{2})t}} {u_{0}}} \bigr\Vert ^{p}} \le{\tilde{M}^{p}} {e^{ - p\gamma t}}\mathbf{E} {\Vert {{u_{0}}} \Vert ^{p}} \le{\tilde{M}^{p}} {e^{ - \eta t}}\mathbf{E} {\Vert {{u_{0}}} \Vert ^{p}}. $$
(3.15)
By Assumptions (A1) and (A2), (3.10), and by the Hölder inequality we have
$$\begin{aligned} \mathbf{E} {\bigl\Vert {{I_{2}}(t)} \bigr\Vert ^{p}} &= \mathbf{E} {\biggl\Vert { \int _{0}^{t} {{e^{(A - {A^{2}})(t - s)}}\alpha\bigl(r(s) \bigr)u(s,x)\,ds} } \biggr\Vert ^{p}} \\ &\le{\tilde{M}^{p}} \mathbf{E} {\biggl( \int_{0}^{t} {{e^{ - \gamma(t - s)}}\bigl\vert {\alpha \bigl(r(s)\bigr)} \bigr\vert \bigl\Vert {u(s,x)} \bigr\Vert }\,ds \biggr)^{p}} \\ &= {\tilde{M}^{p}} \mathbf{E} {\biggl( \int_{0}^{t} {{e^{ - \gamma(t - s)(1 - \frac {1}{p})}} {e^{ - \gamma(t - s)\frac{1}{p}}} \bigl\vert {\alpha\bigl(r(s)\bigr)} \bigr\vert \bigl\Vert {u(s,x)} \bigr\Vert }\,ds\biggr)^{p}} \\ &\le{\tilde{M}^{p}} \mathbf{E}\biggl( \int_{0}^{t} {{e^{ - \gamma(t - s)}}\,ds \biggr){^{p - 1}} \int_{0}^{t} {{e^{ - \gamma(t - s)}}} {{\bigl\vert { \alpha\bigl(r(s)\bigr)} \bigr\vert }^{p}} \bigl\Vert {u(s,x)} \bigr\Vert ^{p}}\,ds \\ &\le\frac{{{{\tilde{M}}^{p}}}}{{{\gamma^{p - 1}}}} \mathbf{E} { \int_{0}^{t} {{e^{ - \gamma(t - s)}} {{\bigl\vert { \alpha\bigl(r(s)\bigr)} \bigr\vert }^{p}} \bigl\Vert {u(s,x)} \bigr\Vert } ^{p}}\,ds \\ &\le\frac{{{{\tilde{M}}^{p}} {{\vert {{\alpha^{*}}} \vert }^{p}}}}{{{\gamma^{p - 1}}}} \mathbf{ E} { \int_{0}^{t} {{e^{ - \gamma(t - s)}} \bigl\Vert {u(s,x)} \bigr\Vert } ^{p}}\,ds \\ &\le\frac{{{{\tilde{M}}^{p}} {{\vert {{\alpha^{*}}} \vert }^{p}}}}{{{\gamma^{p - 1}}}} { \int_{0}^{t} {{e^{ - \gamma(t - s)}} {M^{*}}\mathbf{E} \Vert {{u_{0}}} \Vert } ^{p}} {e^{ - \eta t}}\,ds \\ &\le\frac{{{{\tilde{M}}^{p}} {M^{*}} {{\vert {{\alpha^{*}}} \vert }^{p}}}}{{{\gamma^{p - 1}}}} \mathbf{ E} {\Vert {{u_{0}}} \Vert ^{p}} \int _{0}^{t} {{e^{ - \gamma(t - s)}}} {e^{ - \eta t}}\,ds \\ &\le\frac{{{{\tilde{M}}^{p}} {M^{*}} {{\vert {{\alpha^{*}}} \vert }^{p}}}}{{{\gamma^{p - 1}} (\gamma- \eta)}} \mathbf{E} {\Vert {{u_{0}}} \Vert ^{p}} {e^{ - \eta t}}, \end{aligned}$$
(3.16)
$$\begin{aligned} \mathbf{E} {\bigl\Vert {{I_{3}}(t)} \bigr\Vert ^{p}} &= \mathbf{E} {\biggl\Vert { \int _{0}^{t} {{e^{(A - {A^{2}})(t - s)}}\beta\bigl(r(s) \bigr)u(s,x)\,dB(s)} } \biggr\Vert ^{p}} \\ &\le{\tilde{M}^{p}} {J_{p}} \mathbf{E} {\biggl({ \int_{0}^{t} {{e^{ - 2\gamma(t - s)}} {{\bigl\vert { \beta\bigl(r(s)\bigr)} \bigr\vert }^{2}} \bigl\Vert {u(s,x)} \bigr\Vert } ^{2}}\,ds\biggr)^{\frac{p}{2}}} \\ &\le{\tilde{M}^{p}} {J_{p}} \mathbf{ E} {\biggl({ \int_{0}^{t} {{e^{ - 2\gamma(t - s)(1 - \frac{2}{p})}} {e^{ - 2\gamma(t - s)\frac{2}{p}}} {{\bigl\vert {\beta \bigl(r(s)\bigr)} \bigr\vert }^{2}} \bigl\Vert {u(s,x)} \bigr\Vert } ^{2}}\,ds\biggr)^{\frac {p}{2}}} \\ &\le{\tilde{M}^{p}} {J_{p}} \mathbf{ E}\biggl( \int_{0}^{t} {e^{ - 2\gamma(t - s)}}\,ds \biggr){^{\frac{p}{2} - 1}} \int_{0}^{t} {{e^{ - 2\gamma(t - s)}}} {{\bigl\vert { \beta\bigl(r(s)\bigr)} \bigr\vert }^{p}} { \bigl\Vert {u(s,x)} \bigr\Vert } ^{p}\,ds \\ &\le\frac{{{{\tilde{M}}^{p}} {J_{p}}}}{{{{(2\gamma)}^{\frac{p}{2} - 1}}}} {\bigl\vert {{\beta^{*}}} \bigr\vert ^{p}} { \int_{0}^{t} {{e^{ - 2\gamma(t - s)}}\mathbf{E}\bigl\Vert {u(s,x)} \bigr\Vert } ^{p}}\,ds \\ &\le\frac{{{{\tilde{M}}^{p}} {J_{p}}}}{{{{(2\gamma)}^{\frac{p}{2} - 1}}}} {\bigl\vert {{\beta^{*}}} \bigr\vert ^{p}} {M^{*}} { \int_{0}^{t} {{e^{ - 2\gamma(t - s)}}\mathbf{E}\Vert {{u_{0}}} \Vert } ^{p}} {e^{ - \eta s}}\,ds \\ &\le\frac{{{{\tilde{M}}^{p}} {J_{p}}}}{{{{(2\gamma)}^{\frac{p}{2} - 1}}}} {\bigl\vert {{\beta^{*}}} \bigr\vert ^{p}} {M^{*}}\mathbf{E} {\Vert {{u_{0}}} \Vert ^{p}} \int_{0}^{t} {{e^{ - 2\gamma(t - s)}}} {e^{ - \eta s}}\,ds \\ &\le\frac{{{{\tilde{M}}^{p}}{M^{*}}{J_{p}}{{\vert {{\beta^{*}}} \vert }^{p}}}}{{{{(2\gamma)}^{\frac{p}{2} - 1}} (2\gamma- \eta )}}\mathbf{E} {\Vert {{u_{0}}} \Vert ^{p}} {e^{ - \eta t}}. \end{aligned}$$
(3.17)
From (3.15)-(3.17) it is easy to see that
$$ \mathbf{ E} {\bigl\Vert {\phi(u) (t)} \bigr\Vert ^{p}} \le k \mathbf{E} {\bigl\Vert {\phi(u) (0)} \bigr\Vert ^{p}} {e^{ - \eta t}}, $$
(3.18)
where \(k = {3^{p - 1}}{\tilde{M}^{p}}(1 + \frac{{{M^{*}}{{\vert {{\alpha ^{*}}} \vert }^{p}}}}{{{\gamma^{p - 1}}(\gamma- \eta)}} + \frac {{{M^{*}}{J_{P}}{{\vert {{\beta^{*}}} \vert }^{p}}}}{{(2\gamma- \eta ){{(2\gamma)}^{\frac{p}{2} - 1}}}}) \), which means that \(\phi(H) \subseteq H \).
Claim 3: ϕ is contractive for arbitrary \(u,v \in H \) with \(u(0,x) = v(0,x) = {u_{0}}(x) \).
Proof of Claim 3: Consider the following:
$$\begin{aligned} & \mathbf{E}\sup _{0 \le t < \infty} {\bigl\Vert {\phi (u) (t) - \phi(v) (t)} \bigr\Vert ^{p}} \\ &\quad\le{2^{p - 1}}\mathbf{E}\sup _{0 \le t < \infty} {\biggl\Vert { \int_{0}^{t} {{e^{(A - {A^{2}})(t - s)}}\alpha\bigl(r(s) \bigr) \bigl(u(s,x) - v(s,x)\bigr)\,ds} } \biggr\Vert ^{p}} \\ &\qquad{} + {2^{p - 1}}\mathbf{E}\sup _{0 \le t < \infty} {\biggl\Vert { \int_{0}^{t} {{e^{(A - {A^{2}})(t - s)}}\beta\bigl(r(s) \bigr) \bigl(u(s,x) - v(s,x)\bigr)\,dB(s)} } \biggr\Vert ^{p}} \\ &\quad\le{2^{p - 1}} {\tilde{M}^{p}} \mathbf{E}\sup _{0 \le t < \infty} {\biggl( \int_{0}^{t} {{e^{ - \gamma(t - s)}}} \bigl\vert { \alpha\bigl(r(s)\bigr)} \bigr\vert \bigl\Vert {u(s,x) - v(s,x)} \bigr\Vert \,ds\biggr)^{p}} \\ &\qquad{} + {2^{p - 1}} {\tilde{M}^{p}} {J_{p}} \mathbf{E} \sup _{0 \le t < \infty} {\biggl( \int_{0}^{t} {{e^{ - 2\gamma(t - s)}}} {\bigl\vert { \beta \bigl(r(s)\bigr)} \bigr\vert ^{2}} {\bigl\Vert {u(s,x) - v(s,x)} \bigr\Vert ^{2}}\,ds\biggr)^{\frac {p}{2}}} \\ &\quad\le{2^{p - 1}} \frac{{{{\tilde{M}}^{p}} {{\vert {{\alpha^{*}}} \vert }^{p}}}}{{{\gamma^{p - 1}}}}\mathbf{E}\sup _{0 \le t < \infty} \int_{0}^{t} {{e^{ - \gamma(t - s)}}} {\bigl\Vert {u(s,x) - v(s,x)} \bigr\Vert ^{p}}\,ds \\ &\qquad{} + {2^{p - 1}} \frac{{{{\tilde{M}}^{p}} {J_{p}} {{\vert {{\beta^{*}}} \vert }^{p}}}}{{{{(2\gamma)}^{\frac{p}{2} - 1}}}}\mathbf{E}\sup _{0 \le t < \infty} \int_{0}^{t} {{e^{ - 2\gamma(t - s)}}} {\bigl\Vert {u(s,x) - v(s,x)} \bigr\Vert ^{p}}\,ds \\ &\quad\le{2^{p - 1}} \frac{{{{\tilde{M}}^{p}} {{\vert {{\alpha^{*}}} \vert }^{p}}}}{{{\gamma^{p - 1}}}}\mathbf{E}\sup _{0 \le t < \infty} {\bigl\Vert {u(s,x) - v(s,x)} \bigr\Vert ^{p}} \int_{0}^{t} {{e^{ - \gamma (t - s)}}}\,ds \\ &\qquad{} + {2^{p - 1}} \frac{{{{\tilde{M}}^{p}} {J_{p}} {{\vert {{\beta^{*}}} \vert }^{p}}}}{{{{(2\gamma)}^{\frac{p}{2} - 1}}}}\mathbf{E}\sup _{0 \le t < \infty} {\bigl\Vert {u(s,x) - v(s,x)} \bigr\Vert ^{p}} \int _{0}^{t} {{e^{ - 2\gamma(t - s)}}}\,ds \\ &\quad\le{2^{p - 1}} \frac{{{{\tilde{M}}^{p}} {{\vert {{\alpha^{*}}} \vert }^{p}}}}{{{\gamma^{p}}}}\mathbf{E}\sup _{0 \le t < \infty } {\bigl\Vert {u(s,x) - v(s,x)} \bigr\Vert ^{p}} \\ &\qquad{} + {2^{p - 1}} \frac{{{{\tilde{M}}^{p}} {J_{p}} {{\vert {{\beta^{*}}} \vert }^{p}}}}{{{{(2\gamma)}^{\frac{p}{2}}}}} \mathbf{E}\sup _{0 \le t < \infty} {\bigl\Vert {u(s,x) - v(s,x)} \bigr\Vert ^{p}} \\ &\quad\le\tilde{k} \mathbf{E}\sup _{0 \le t < \infty} {\bigl\Vert {u(s,x) - v(s,x)} \bigr\Vert ^{p}}, \end{aligned}$$
(3.19)
where
$$ \tilde{k} = \frac{{{2^{p - 1}} {{\tilde{M}}^{p}} {{\vert {{\alpha^{*}}} \vert }^{p}}}}{{{\gamma^{p}}}} + \frac{{{2^{p - 1}} {{\tilde{M}}^{p}} J{}_{p} {{\vert {{\beta^{*}}} \vert }^{p}}}}{{{{(2\gamma)}^{\frac{p}{2}}}}}. $$
(3.20)
Recalling condition (3.1) and noting that \(\tilde{k} \in(0,1) \), we see that ϕ is a contractive 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 proofs of the two parts.
Therefore, the proof of the desired assertion in Theorem 1 is completed. □
Remark 1
If \(p = 2 \), then it is obvious that (2.1) is mean square exponentially stable.
Remark 2
In Theorem 1, we apply the fixed point theory to obtain the existence and uniqueness of the solution for a class of linear hybrid stochastic fourth-order parabolic equations. In fact, if we add some proper assumptions, then we can also get some good results for the nonlinear case. We leave this for the future work.