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Controllability results for stochastic coupled systems of fourth- and second-order parabolic equations

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Abstract

In this paper, we study some controllability and observability problems for stochastic systems coupling fourth- and second-order parabolic equations. The main goal is to control both equations with only one controller localized on the drift of the fourth-order equation. We analyze two cases: on the one hand, we study the controllability of a linear backward system where the couplings are made only through first-order terms. The key point is to use suitable Carleman estimates for the heat equation and the fourth-order operator with the same weight to deduce an observability inequality for the adjoint system. On the other hand, we study the controllability of a simplified nonlinear coupled model of forward equations. This case, which is well known to be harder to solve, follows a methodology that has been introduced recently and relies on an adaptation of the well-known source term method in the stochastic setting together with a truncation procedure. This approach gives a new concept of controllability for stochastic systems.

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Appendices

Appendix A: Proof of Proposition 3.8

In the following, the constant \(C_T>0\) is always of the form \(\exp (C/T)\) where \(C>0\) only depends on \(\mathcal {D},\mathcal {D}_0, b_1,b_2\) and \(b_3\). In addition, \(C_T\) can vary from line to line.

We define \(T_k=T-\frac{T}{Q^{k}}\) for \(k\ge 0\). We can easily deduce the following relation between the weights defined in (75), (76) and (77)

$$\begin{aligned} \rho _0(T_{k+2})=\rho (T_{k})\gamma (T_{k+2}-T_{k+1}). \end{aligned}$$
(95)

We consider the equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {d}y_1+(y_{1,xxxx}+F)\text {d}t=(b_1y_1+ b_2 z_1)\, \text {d}W(t) &{}\text {in}\; \mathcal {D}\times (T_k,T_{k+1}), \\ \text {d}z_1- z_{1,xx}\,\text {d}t=y_1\,\text {d}t+b_3z_1\, \text {d}W(t) &{}\text {in}\; \mathcal {D}\times (T_k,T_{k+1}), \\ y_1=y_{1,xx}=0 &{}\text {on}\; \{0,1\}\times (T_k,T_{k+1}), \\ z_1=0 &{}\text {on }\{0,1\}\times (T_k,T_{k+1}), \\ y_1(T_k)=z_1(T_k)=0 &{}\text {in}\; \mathcal {D}. \end{array}\right. } \end{aligned}$$
(96)

We introduce the sequence of random variables \(\{(a_{k},b_{k})\}_{k\ge 0}\) such that

$$\begin{aligned} (a_0,b_0) = (y_0,z_0)\quad \text {and}\quad (a_{k+1},b_{k+1})=(y_1(T_{k+1}),z_1(T_{k+1})). \end{aligned}$$

We also consider the equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {d}y_2+y_{,2xxxx}\text {d}t=\chi _{\mathcal {D}_0}h_k\,\text {d}t+(b_1y_2+ b_2 z_2)\, \text {d}W(t) &{}\text {in}\; \mathcal {D}\times (T_k,T_{k+1}), \\ \text {d}z_2- z_{2,xx}\,\text {d}t=y_2\,\text {d}t+b_3z_2\, \text {d}W(t) &{}\text {in}\; \mathcal {D}\times (T_k,T_{k+1}), \\ y_2=y_{2,xx}=0 &{}\text {on}\; \{0,1\}\times (T_k,T_{k+1}), \\ z_2=0 &{}\text {on}\; \{0,1\}\times (T_k,T_{k+1}), \\ y_2(T_k)=a_k, \quad z(T_k)=b_k &{}\text {in}\; \mathcal {D}. \end{array}\right. } \end{aligned}$$
(97)

Observe that due to the regularity of the solution of (96), each \(a_{k},b_k\) with \(k\ge 0\), is \(\mathcal {F}_{T_k}\)-measurable and belongs to \(L^2(\Omega \times \mathcal {D})\). Therefore, thanks to Proposition B.3 the system (97) is well posed for each \(h_k\in L^2_{\mathcal {F}}(T_k,T_{k+1};L^2(\mathcal {D}))\).

According to Proposition 3.1, we can construct a control \(h_k\in L^2_{\mathcal {F}}(T_k,T_{k+1};L^2(\mathcal {D}))\) such that

$$\begin{aligned} y_2(T_{k+1})=z_2(T_{k+1})=0, \quad \text {a.s.} \end{aligned}$$

and the following estimates holds

$$\begin{aligned} {\mathbb {E}}\left( \int _{T_k}^{T_{k+1}}\!\!\!\!\int _{\mathcal {D}_0}|h_k(t,x)|^2\,\text {d}x\text {d}t\right) \le \gamma ^2(T_{k+1}-T_k){\mathbb {E}}\left( \Vert a_k\Vert ^2_{L^2(\mathcal {D})}+\Vert b_k\Vert ^2_{L^2(\mathcal {D})}\right) . \end{aligned}$$
(98)

On the other hand, applying Proposition B.3 to (96), we have

$$\begin{aligned} {\mathbb {E}}&\left( \Vert a_{k+1}\Vert _{L^2(\mathcal {D})}^2+\Vert b_{k+1}\Vert _{L^2(\mathcal {D})}^2\right) \le C_T{\mathbb {E}}\left( \int _{T_k}^{T_{k+1}} \Vert F(t)\Vert ^2_{L^2(\mathcal {D})}\text {d}t\right) . \end{aligned}$$
(99)

Then, combining estimates (98) and (99) we get

$$\begin{aligned} {\mathbb {E}}\left( \int _{T_{k+1}}^{T_{k+2}}\!\!\!\int _{\mathcal {D}_0}|h_{k+1}(t,x)|^2\,\text {d}x\text {d}t\right) \le C_T\gamma ^2\left( T_{k+2}-T_{k+1}\right) {\mathbb {E}}\left( \int _{T_k}^{T_{k+1}} \Vert F(t)\Vert ^2_{L^2(\mathcal {D})}\text {d}t\right) \end{aligned}$$

and using the fact that \(\rho \) is a non-increasing, deterministic function together with equality (95) yields

$$\begin{aligned} {\mathbb {E}}\left( \int _{T_{k+1}}^{T_{k+2}}\!\!\!\int _{\mathcal {D}_0}|h_{k+1}(t,x)|^2\,\text {d}x\text {d}t\right) \le C_T \rho _0^2\left( T_{k+2}\right) {\mathbb {E}}\left( \int _{T_k}^{T_{k+1}}\left\| \frac{F(t)}{\rho (t)}\right\| ^2_{L^2(\mathcal {D})}\text {d}t\right) . \end{aligned}$$

The fact that \(\rho _0\) is a non-increasing, deterministic function, implies

$$\begin{aligned} {\mathbb {E}}\left( \int _{T_{k+1}}^{T_{k+2}}\!\!\!\int _{\mathcal {D}_0}\left| \frac{h_{k+1}(t,x)}{\rho _0(t)}\right| ^2\,\text {d}x\text {d}t\right) \le C_T {\mathbb {E}}\left( \int _{T_k}^{T_{k+1}}\left\| \frac{F(t)}{\rho (t)}\right\| ^2_{L^2(\mathcal {D})}\text {d}t\right) . \end{aligned}$$
(100)

Let \(n\in \mathbb N^*\). From (100), we have

$$\begin{aligned}&{\mathbb {E}}\left( \int _{T_1}^{T}\!\int _{\mathcal {D}_0}\sum _{k=0}^{n}\mathbf {1}_{[T_{k+1},T_{k+2})}(t)\left| \frac{h_{k+1}}{\rho _0}\right| ^2\,\text {d}x\text {d}t\right) \nonumber \\&\quad \le C_T{\mathbb {E}}\left( \int _{0}^{T}\sum _{k=0}^n\mathbf {1}_{[T_{k},T_{k+1})}(t)\left\| \frac{F(t)}{\rho (t)}\right\| ^2_{L^2(\mathcal {D})}\text {d}t\right) . \end{aligned}$$
(101)

From (98) at \(k=0\), recalling that \((a_0,b_0)=(y_0,z_0)\) and using the properties of \(\rho _0\), we get

$$\begin{aligned} {\mathbb {E}}\left( \int _{0}^{T_1}\!\!\!\int _{\mathcal {D}_0}\left| \frac{h_0}{\rho _0}\right| ^2\,\text {d}x\text {d}t\right) \le \frac{\gamma ^2\left( T_1\right) }{\rho _0^2(T_1)}{\mathbb {E}}\left( \Vert y_0\Vert ^2_{L^2(\mathcal {D})}+\Vert z_0\Vert ^2_{L^2(\mathcal {D})}\right) . \end{aligned}$$
(102)

Putting together (101) and (102) yields the existence of a constant \(C_T>0\) independent of n such that

$$\begin{aligned}&{\mathbb {E}}\left( \int _{0}^{T_1}\!\!\!\int _{\mathcal {D}_0}\left| \frac{h_0}{\rho _0}\right| ^2\,\text {d}x\text {d}t\right) +{\mathbb {E}}\left( \int _{T_1}^{T}\!\int _{\mathcal {D}_0}\sum _{k=0}^{n}\mathbf {1}_{[T_{k+1},T_{k+2})}(t)\left| \frac{h_{k+1}}{\rho _0}\right| ^2\,\text {d}x\text {d}t\right) \\&\quad \le C_T{\mathbb {E}}\left( \Vert y_0\Vert ^2_{L^2(\mathcal {D})}+\Vert z_0\Vert ^2_{L^2(\mathcal {D})}+\int _{0}^{T}\sum _{k=0}^n\mathbf {1}_{[T_{k},T_{k+1})}(t)\left\| \frac{F(t)}{\rho (t)}\right\| ^2_{L^2(\mathcal {D})}\text {d}t\right) . \end{aligned}$$

Finally, we set \(h:=\sum _{k=0}^\infty h_k\) thus, using Lebesgue’s convergence theorem, we obtain

$$\begin{aligned} {\mathbb {E}}\left( \int _{0}^{T}\!\!\!\int _{\mathcal {D}_0}\left| \frac{h}{\rho _0}\right| ^2\,\text {d}x\text {d}t\right) \le C_T{\mathbb {E}}\left( \Vert y_0\Vert ^2_{L^2(\mathcal {D})}+\Vert z_0\Vert ^2_{L^2(\mathcal {D})}+\int _{0}^{T} \left\| \frac{F(t)}{\rho (t)}\right\| ^2_{L^2(\mathcal {D})}\right) . \end{aligned}$$
(103)

Applying Itô’s rule to \(y:=y_1+y_2\) and \(z:=z_1+z_2\) for \(t\in [T_k,T_{k+1})\), we get

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {d}y+(y_{xxxx}+F)\text {d}t=\chi _{\mathcal {D}_0}h_k\,\text {d}t+(b_1y+ b_2 z)\, \text {d}W(t) &{}\text {in}\;\mathcal {D}\times (T_k,T_{k+1}), \\ \text {d}z- z_{xx}\,\text {d}t=y\,\text {d}t+b_3z\, \text {d}W(t) &{}\text {in}\; \mathcal {D}\times (T_k,T_{k+1}), \\ y=y_{xx}=0 &{}\text {on}\;\{0,1\}\times (0,T), \\ z=0 &{}\text {on}\;\{0,1\}\times (0,T), \\ y(T_k)=a_k, \quad z(T_k)=b_k &{}\text {in}\; \mathcal {D}. \end{array}\right. } \end{aligned}$$
(104)

Note that by construction (yz) is continuous at \(T_k\) a.s., for all \(k\ge 0\), therefore by using (104), (yz) is a solution to (78).

Moreover, applying Proposition B.3 to (104) for \(k\ge 1\), we have

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{T_k\le t\le T_{k+1}} \Vert y(t)\Vert _{L^2(\mathcal {D})}^2+\sup _{T_k\le t\le T_{k+1}} \Vert z(t)\Vert _{L^2(\mathcal {D})}^2\right) \\&\quad \le C_T{\mathbb {E}}\left( \Vert a_k\Vert ^2_{L^2(\mathcal {D})}+ \Vert b_k\Vert ^2_{L^2(\mathcal {D})} +\int _{T_{k}}^{T_{k+1}}\left[ \Vert \chi _{\mathcal {D}_0}h_{k}(t)\Vert ^2_{L^2(\mathcal {D})}+\Vert F(t)\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}t\right) . \end{aligned}$$

Then using inequalities (98) and (99) to estimate in the above equation yields

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{T_k\le t\le T_{k+1}} \Vert y(t)\Vert _{L^2(\mathcal {D})}^2+\sup _{T_k\le t\le T_{k+1}} \Vert z(t)\Vert _{L^2(\mathcal {D})}^2\right) \\&\quad \le C_T \gamma ^2(T_{k+1}-T_k) {\mathbb {E}}\left( \int _{T_{k-1}}^{T_{k+1}} \Vert F(t)\Vert ^2_{L^2(\mathcal {D})}\text {d}t\right) . \end{aligned}$$

The identity (95) allows us to get

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{T_k\le t\le T_{k+1}} \Vert y(t)\Vert _{L^2(\mathcal {D})}^2+\sup _{T_k\le t\le T_{k+1}} \Vert z(t)\Vert _{L^2(\mathcal {D})}^2\right) \\&\quad \le C_T\rho _0^2(T_{k+1}) {\mathbb {E}}\left( \int _{T_{k-1}}^{T_{k+1}} \left\| \frac{F(t)}{\rho (t)}\right\| ^2_{L^2(\mathcal {D})}\text {d}t\right) , \end{aligned}$$

so by using that \(\rho _0\) is non-increasing, we have

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{T_k\le t\le T_{k+1}} \left\| \frac{y(t)}{\rho _0(t)}\right\| _{L^2(\mathcal {D})}^2+\sup _{T_k\le t\le T_{k+1}} \left\| \frac{z(t)}{\rho _0(t)}\right\| _{L^2(\mathcal {D})}^2\right) \nonumber \\&\quad \le C_T {\mathbb {E}}\left( \int _{T_{k-1}}^{T_{k+1}} \left\| \frac{F(t)}{\rho (t)}\right\| ^2_{L^2(\mathcal {D})}\text {d}t\right) . \end{aligned}$$
(105)

Moreover, arguing as before, it is not difficult to establish that

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t\le T_1}\left\| \frac{y(t)}{\rho _0(t)}\right\| ^2_{L^2(\mathcal {D})}+\sup _{0\le t\le T_1}\left\| \frac{z(t)}{\rho _0(t)}\right\| ^2_{L^2(\mathcal {D})}\right) \nonumber \\&\quad \le C_T \left( \Vert y_0\Vert ^2_{L^2(\mathcal {D})}+\Vert z_0\Vert ^2_{L^2(\mathcal {D})}+\int _{0}^{T_1} \left\| \frac{F(t)}{\rho (t)}\right\| ^2_{L^2(\mathcal {D})}\text {d}t\right) . \end{aligned}$$
(106)

Let \(n\in \mathbb N^*\). From inequalities (105) and (106), we have

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t\le T_1}\left\| \frac{y(t)}{\rho _0(t)}\right\| ^2_{L^2(\mathcal {D})}+\sup _{0\le t\le T_1}\left\| \frac{z(t)}{\rho _0(t)}\right\| _{L^2(\mathcal {D})}^2\right) \nonumber \\&\qquad +\sum _{k=1}^{n}{\mathbb {E}}\left( \sup _{T_{k}\le t\le T_{k+1}}\left\| \frac{y(t)}{\rho (t)}\right\| ^2_{L^2(\mathcal {D})}+\sup _{T_{k}\le t\le T_{k+1}}\left\| \frac{z(t)}{\rho (t)}\right\| ^2_{L^2(\mathcal {D})}\right) \nonumber \\&\quad \le C_T{\mathbb {E}}\left( \Vert y_0\Vert ^2_{L^2(\mathcal {D})}+\Vert z_0\Vert ^2_{L^2(\mathcal {D})}+\sum _{k=1}^{n}\int _{0}^{T}\mathbf {1}_{[T_{k-1},T_{k+1})} \left\| \frac{F(t)}{\rho (t)}\right\| ^2_{L^2(\mathcal {D})} \right) \end{aligned}$$
(107)

where \(C_T>0\) is uniform with respect to n. Letting \(n\rightarrow \infty \) in (107) yields

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t\le T }\left\| \frac{y(t)}{\rho _0(t)}\right\| ^2_{L^2(\mathcal {D})}+\sup _{0\le t\le T }\left\| \frac{z(t)}{\rho _0(t)}\right\| ^2_{L^2(\mathcal {D})}\right) \nonumber \\&\quad \le C_T{\mathbb {E}}\left( \Vert y_0\Vert ^2_{L^2(\mathcal {D})}+\Vert z_0\Vert ^2_{L^2(\mathcal {D})}+\int _{0}^{T} \left\| \frac{F(t)}{\rho (t)}\right\| ^2_{L^2(\mathcal {D})}\text {d}t\right) . \end{aligned}$$
(108)

Finally, combining (103) and (108) gives the desired result. This concludes the proof.

Appendix B: Well-posedness results

We devote this section to present some general results and make some comments about the well-posedness of systems (2), (13) and systems (78), (64). For conciseness, we assume that the coefficients \(d_i\) and \(b_i\) have the same regularity as in Theorem 1.2 and Theorems 1.3 and 1.4, respectively.

1.1 Appendix B.1: Clamped boundary conditions

We present here the well-posedness results for the case of clamped boundary conditions for the fourth-order operator, i.e., \(y=y_x=0\) on \(\Sigma \).

We begin with the backward system.

Proposition B.1

Assume that \(y_T,z_T\in L^2(\Omega ,\mathcal {F}_T;L^2(\mathcal {D}))\) and \(F_i\in L^2_{\mathcal {F}}(0,T;L^2(\mathcal {D}))\), \(i=1,2\). Then, the system

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {d}y-(\gamma y_{xxxx}-y_{xxx}+y_{xx})\text {d}t=(z_x-d_1 Y + F_1)\text {d}t+ Y\text {d}W(t) &{}\text {in}\;Q, \\ \text {d}z+\Gamma z_{xx}\text {d}t=(z_x+y_x-d_2Z+F_2)\text {d}t+Z\text {d}W(t) &{}\text {in}\;Q, \\ y=y_x=0 &{}\text {on}\;\Sigma , \\ z=0 &{}\text {on}\;\Sigma , \\ y(T)=y_T, \quad z(T)=z_T &{}\text {in}\; \mathcal {D}, \end{array}\right. } \end{aligned}$$
(109)

has a unique solution \((y,z,Y,Z)\in \big [L^2_{\mathcal {F}}(0,T;H_0^2(\mathcal {D})\times H_0^{1}(\mathcal {D})) \bigcap L^2\big (\Omega ; C([0,T];L^2(\mathcal {D})^2)\big )\big ]\times L^2_{\mathcal {F}}(0,T;L^2(\mathcal {D})^2)\). Moreover, there exists some \(C>0\) only depending on T, \(\Gamma \), \(\gamma \) and \(d_i\), \(i=1,2\), such that

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t \le T}\Vert y(t)\Vert _{L^2(\mathcal {D})}^2+\sup _{0\le t \le T}\Vert z(t)\Vert ^2_{L^2(\mathcal {D})}\right) \nonumber \\&\qquad +{\mathbb {E}}\left( \int _{0}^{T}\Vert y(t)\Vert ^2_{H_0^2(\mathcal {D})}\text {d}t+\int _{0}^{T}\Vert z(t)\Vert ^2_{H_0^1(\mathcal {D})}\text {d}t\right) \nonumber \\&\qquad +{\mathbb {E}}\left( \int _{0}^T \Vert Y(t)\Vert _{L^2(\mathcal {D})}^2\text {d}t+\int _{0}^T \Vert Z(t)\Vert _{L^2(\mathcal {D})}^2\text {d}t\right) \nonumber \\&\quad \le C{\mathbb {E}}\left( \Vert y_T\Vert ^2_{L^2(\mathcal {D})}+\Vert z_T\Vert _{L^2(\mathcal {D})}^2+\int _{0}^{T}\Vert F_1(t)\Vert ^2_{L^2(\mathcal {D})}\text {d}t+\int _{0}^{T}\Vert F_2(t)\Vert ^2_{L^2(\mathcal {D})}\text {d}t\right) . \end{aligned}$$
(110)

Proof

The proof follows well-known and classical arguments, see, e.g., [18]. For readability we have divided it in three steps. We begin by presenting the following result. \(\square \)

Lemma B.2

[7, Lemma 2.3] There exists a set of positive real numbers \((\mu _j)_{j\ge 1}\) such that the corresponding solutions \((w_j)_{j\ge 1}\) of the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d^4}{dx^4}w_j(x)=\mu _j w_{j}(x), \qquad x\in \mathcal {D}, \\ w_j(0)=w_j^\prime (0)=w_j(1)=w_j^\prime (1)=0, \end{array}\right. } \end{aligned}$$

form a basis in \(H^4(\mathcal {D})\cap H_0^2(\mathcal {D})\), which is orthonormal in \(L^2(0,1)\).

Step 1: approximate solution. Let \((w_j)_{j\ge 1}\) be the basis provided by Lemma B.2 and let \(\Pi _{m}\) be the orthogonal projection from \(L^2(\mathcal {D})\) into \(V_m:=\text {span}\{w_1,w_2,\ldots ,w_m\}\). Without loss of generality, we assume that \(\gamma =\Gamma =1\).

Let \(m\in \mathbb N^*\) and consider the backward differential system

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {d}\mathsf {y}_i^{m}=\left( \sum _{j=1}^{m}[a_{ij}\mathsf {y}_j^m+b_{ij}\mathsf {z}_j^m - c_{ij} \mathsf {Y}_j^m ] +\mathsf {F}_{1,i}\right) \text {d}t+ \mathsf {Y}_i^m \text {d}{W}(t), \\ \text {d}\mathsf {z}_i^m=\left( \sum _{j=1}^{m}[ e_{ij}\mathsf {z}_j^m + f_{ij}\mathsf {y}_j^m-g_{ij}\mathsf {Z}_j^m]+\mathsf {F}_{2,i}\right) \text {d}t+\mathsf {Z}_i^m\text {d}{W}(t), \\ \mathsf {y}_i^m(T)=(\Pi _m y_T,w_i)_{L^2(\mathcal {D})}, \quad \mathsf {z}_i^m(T)=(\Pi _m z_T,w_i)_{L^2(\mathcal {D})}, \end{array}\right. } \end{aligned}$$
(111)

where

$$\begin{aligned}&a_{ij} = (w_{j,xxxx}-w_{j,xxx}+w_{j,xx},w_i)_{L^2(\mathcal {D})}, \quad b_{ij}=(w_{j,x},w_i)_{L^2(\mathcal {D})} , \\&c_{ij}= (d_1 w_{j},w_{i})_{L^2(\mathcal {D})}, \\&e_{ij}=(-w_{j,xx}+w_{j,x},w_i)_{L^2(\mathcal {D})}, \quad f_{ij}= (w_{j,x},w_i)_{L^2(\mathcal {D})} , \\&g_{ij}= (d_2w_j,w_i)_{L^2(\mathcal {D})} ,\quad \mathsf {F}_{k,i}=\left( F_k,w_i\right) , \quad k=1,2. \end{aligned}$$

We note that under the assumptions of \(d_i\) (\(i=1,2\)) and \(F_i\) (\(i=1,2\)), we have that \(c_{ij},g_{ij}\in L^\infty _{\mathcal {F}}(0,T;\mathbb R)\) (\(1\le i,j\le m\)) and \(\mathsf F_{k,i}\in L^2_{\mathcal {F}}(0,T;\mathbb R)\) (\(1\le i\le m\), \(k=1,2)\). We also note that since \(y_T,z_T\in L^2(\Omega ,\mathcal {F}_T;L^2(\mathcal {D}))\), then \(\mathsf y_i^m(T),\mathsf z_i^m(T)\in L^2(\Omega ,\mathcal {F}_T;\mathbb {R})\) (\(1\le i\le m\)).

We define the following vectors in \(\mathbb {R}^{2m}\)

$$\begin{aligned}&\mathbf {X}(t)=\left( \mathsf {y}_1^m(t),\ldots ,\mathsf {y}_m^m,\mathsf {z}_1^m(t),\ldots ,\mathsf {z}_m^m(t)\right) ^\mathsf {T}, \\&\mathbf {R}(t)=\left( \mathsf {Y}_{1}^m(t),\ldots , \mathsf {Y}_{m}^m(t),\mathsf {Z}_{1}^m(t),\ldots ,\mathsf {Z}_{m}^m(t)\right) ^\mathsf {T}. \end{aligned}$$

Hence, (111) can be written as

$$\begin{aligned} \text {d}\mathbf {X} = \left[ \underbrace{\left( \begin{array}{cc}A &{} B \\ F &{} E\end{array}\right) }_{:=\mathcal {A}}\mathbf {X}- \underbrace{\left( \begin{array}{cc}C &{} \mathbf{0} \\ \mathbf {0} &{} G\end{array}\right) }_{:=\mathcal {B}} \mathbf{R}+\mathbf{F}\right] \text {d}t+ \mathbf {R}\text {d}{W}(t) \end{aligned}$$
(112)

with terminal data \(\mathbf{X}(T)\in L^2(\Omega ,\mathcal {F}_T;\mathbb {R}^{2m})\), and where \(\mathbf{0}\) is the \(m\times m\) zero matrix and

$$\begin{aligned}&A=[(a_{ij})]_{1\le i,j\le m}, \quad B=[(b_{ij})]_{1\le i,j\le m}, \quad C=[(c_{ij})]_{1\le i,j\le m}, \\&E=[(e_{ij})]_{1\le i,j\le m},\quad F=[(f_{ij})]_{1\le i,j\le m}, \quad G=[(g_{ij})]_{1\le i,j\le m}, \\&\mathbf{F}=(F_{1,1},\ldots ,F_{1,m},F_{2,1},\ldots ,F_{2,m})^\mathsf {T}\in L^2_{\mathcal {F}}(0,T;\mathbb {R}^{2m}) \end{aligned}$$

Since \(\mathcal {A}\) is a constant matrix and \(\mathcal {B}\in L^\infty _{\mathcal {F}}(0,T;\mathbb {R}^{2m\times 2m})\), we can directly apply [38, Theorem 2.2] to get that (112) admits a unique adapted solution \((\mathbf{X},\mathbf{R})\in L^2_{\mathcal {F}}(\Omega ;C([0,T];\mathbb {R}^{2m}))\times L^2_{\mathcal {F}}(0,T;\mathbb {R}^{2m})\).

Step 2. Energy estimates. Let us define

$$\begin{aligned} y^m=\sum _{i=1}^{m}\mathsf {y}_i^m w_i, \quad Y^m=\sum _{i=1}^{m}\mathsf {Y}_i^m w_i, \\ z^m=\sum _{i=1}^{m}\mathsf {z}_i^m w_i, \quad Z^m=\sum _{i=1}^{m}\mathsf {Z}_i^m w_i. \end{aligned}$$

From the above definitions, using (111) and the orthogonality of the basis \((w_j)_{j\ge 1}\), we readily obtain that \((y^m,z^m,Y^m,Z^m)\) verify the equations

$$\begin{aligned} \text {d}(y^m,w_k)_{L^2(\mathcal {D})}&=(y^m_{xxxx^m}-y_{xxx}^m+y^{m}_{xx},w_k)_{L^2(\mathcal {D})}\text {d}t+(z_x^m,w_k)_{L^2(\mathcal {D})} \text {d}t\\&\quad -(d_1Y^m,w_k)\text {d}t+ (F_1,w_k)_{L^2(\mathcal {D})}\text {d}t+(Y^m,w_k)\text {d}{W}(t), \\ \text {d}(z^m,w_k)_{L^2(\mathcal {D})}&= (-z^m_{xx}+z_x^m,w_k)_{L^2(\mathcal {D})}\text {d}t+(y_x^m,w_k)_{L^2(\mathcal {D})}\text {d}t-(d_2 Z^m,w_k)_{L^2(\mathcal {D})}\text {d}t\\&\quad + (F_2,w_k)\text {d}t+(Z^m,w_k)\text {d}{W}(t), \end{aligned}$$

for \(k=1,\ldots ,m\). Furthermore, from Itô’s formula

$$\begin{aligned}&\Vert y^m(t)\Vert ^2_{L^2(\mathcal {D})}+\int _{t}^{T}\left[ \left\Vert y^m_{xx}\right\Vert _{L^2(\mathcal {D})}^2+\left\Vert Y^m\right\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s} \nonumber \\&\quad = \Vert \Pi _{m}y_T\Vert _{L^2(\mathcal {D})}^2+2 \int _{t}^{T}\left( y_{xxx}^m-y_{xx}^m,y^m\right) _{L^2(\mathcal {D})}\text {d}{s} + 2\int _{t}^{T}\left( z_x^m,y^m\right) _{L^2(\mathcal {D})}\text {d}{s} \nonumber \\&\qquad +2\int _{t}^{T}\left( d_1Y^m,y^m\right) _{L^2(\mathcal {D})}\text {d}{s} + 2\int _{t}^{T}\left( F_1^m,y^m\right) _{L^2(\mathcal {D})}\text {d}{s}\nonumber \\&\qquad +2\int _{t}^{T}\left( Y^m,y^m\right) _{L^2(\mathcal {D})}\text {d}{W}(s) \end{aligned}$$
(113)

and

$$\begin{aligned}&\Vert z^m(t)\Vert ^2_{L^2(\mathcal {D})}+\int _{t}^{T}\left[ \Vert z_x\Vert ^2_{L^2(\mathcal {D})}+\left\Vert Z^m\right\Vert _{L^2(\mathcal {D})}^2\right] \text {d}{s} \nonumber \\&\quad =\Vert \Pi _{m}z_T\Vert ^2_{L^2(\mathcal {D})}-\int _{t}^{T}2(z_x^m,z^m)_{L^2(\mathcal {D})}^2 \text {d}{s}- 2 \int _{t}^{T}(y_x^m,z^m)_{L^2(\mathcal {D})}^2 \text {d}{s} \nonumber \\&\qquad + 2 \int _t^{T}(d_2 Z^m,z^m)_{L^2(\mathcal {D})} \text {d}{s} +2\int _t^{T}(F_2^m,z^m)_{L^2(\mathcal {D})} \text {d}{s}\nonumber \\&\qquad + 2\int _{t}^{T}(Z^m,z^m)_{L^2(\mathcal {D})}\text {d}{W}(s) \end{aligned}$$
(114)

where we have used that

$$\begin{aligned} (F_1,y^m)_{L^2(\mathcal {D})}=(F^m_1,y^m)_{L^2(\mathcal {D})}, \quad (F_2,z^m)_{L^2(\mathcal {D})}=(F^m_2,z^m)_{L^2(\mathcal {D})} \end{aligned}$$

with \(F_i^m=(F_i,w_i)_{L^2(\mathcal {D})} w_i\).

Therefore, taking conditional expectation, we get

$$\begin{aligned}&{\mathbb {E}}\left( \Vert y^m(t)\Vert ^2_{L^2(\mathcal {D})}+\int _{t}^{T}\left[ \left\Vert y^m_{xx}\right\Vert _{L^2(\mathcal {D})}^2+\left\Vert Y^m\right\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s} \Big | \mathcal {F}_r \right) \\&\quad = {\mathbb {E}}\left( \Vert \Pi _{m}y_T\Vert _{L^2(\mathcal {D})}^2+2 \int _{t}^{T}\left( y_{xxx}^m-y_{xx}^m,y^m\right) _{L^2(\mathcal {D})}\text {d}{s}\right. \nonumber \\&\qquad +\left. 2\int _{t}^{T}\left( z_x^m,y^m\right) _{L^2(\mathcal {D})}\text {d}{s} \Big | \mathcal {F}_r \right) \\&\qquad + {\mathbb {E}}\left( 2\int _{t}^{T}\left( d_1Y^m , y^m\right) _{L^2(\mathcal {D})}\text {d}{s} + 2\int _{t}^{T}\left( F_1^m,y^m\right) _{L^2(\mathcal {D})}\text {d}{s} \Big | \mathcal {F}_r \right) \end{aligned}$$

and

$$\begin{aligned}&{\mathbb {E}}\left( \Vert z^m(t)\Vert ^2_{L^2(\mathcal {D})}+\int _{t}^{T}\left( \Vert z_x\Vert ^2_{L^2(\mathcal {D})}+\left\Vert Z^m\right\Vert _{L^2(\mathcal {D})}^2\right) \text {d}{s} \Big | \mathcal {F}_r \right) \\&={\mathbb {E}}\left( \Vert \Pi _{m}z_T\Vert ^2_{L^2(\mathcal {D})}-\int _{t}^{T}2(z_x^m,z^m)_{L^2(\mathcal {D})}^2 \text {d}{s}- 2 \int _{t}^{T}(y_x^m,z^m)_{L^2(\mathcal {D})}^2 \text {d}{s} \Big | \mathcal {F}_r \right) \\&\quad + {\mathbb {E}}\left( 2 \int _0^{T}(d_2 Z^m,z^m)_{L^2(\mathcal {D})} \text {d}{s} +2\int _0^{T}(F_2^m,z^m)_{L^2(\mathcal {D})} \text {d}{s} \Big | \mathcal {F}_r \right) \end{aligned}$$

for \(0<r\le t\le T\).

Let us define \(\mathcal {E}^m(t):=\Vert y^m(t)\Vert ^2_{L^2(\mathcal {D})}+\Vert z^m(t)\Vert ^2_{L^2(\mathcal {D})}\). A long but straightforward computation using Cauchy–Schwarz, Young, and Poincaré inequalities yield

$$\begin{aligned}&{\mathbb {E}}\left( \mathcal {E}^m(t)+\int _{t}^{T}\left[ \left\Vert y^m_{xx}\right\Vert _{L^2(\mathcal {D})}^2+\left\Vert Y^m\right\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s}\right. \\&\qquad + \left. \int _t^T\left[ \Vert z_x\Vert ^2_{L^2(\mathcal {D})}+\left\Vert Z^m\right\Vert _{L^2(\mathcal {D})}^2 \right] \text {d}{s} \Big | \mathcal {F}_r \right) \\&\quad \le C {\mathbb {E}}\left( \Vert \Pi _{m}y_T\Vert ^2_{L^2(\mathcal {D})} + \Vert \Pi _{m}z_T\Vert ^2_{L^2(\mathcal {D})} \Big | \mathcal {F}_r \right) \\&\qquad + {\mathbb {E}}\left( \int _{t}^{T}\left[ \Vert F_1^m\Vert ^2_{L^2(\mathcal {D})}+\Vert F_2^m\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s}\Big | \mathcal {F}_r\right) \\&\qquad + C {\mathbb {E}}\left( \int _{t}^{T} \mathcal {E}^m(s) \text {d}{s} \Big | \mathcal {F}_r \right) \end{aligned}$$

where the constant \(C>0\) only depends on the norms of the coefficients \(d_i\) (\(i=1,2\)). Here, we have also used property (58) several times.

Whence, taking expectation

$$\begin{aligned}&{\mathbb {E}}\left( \mathcal {E}^m(t)+\int _{t}^{T}\left[ \left\Vert y^m_{xx}\right\Vert _{L^2(\mathcal {D})}^2+\left\Vert Y^m\right\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s}+ \int _t^T\left( \Vert z_x\Vert ^2_{L^2(\mathcal {D})}+\left\Vert Z^m\right\Vert _{L^2(\mathcal {D})}^2 \right) \text {d}{s} \right) \\&\quad \le C {\mathbb {E}}\left( \Vert \Pi _{m}y_T\Vert ^2_{L^2(\mathcal {D})} + \Vert \Pi _{m}z_T\Vert ^2_{L^2(\mathcal {D})} \right) \\&\qquad + C {\mathbb {E}}\left( \int _{t}^{T}\left[ \Vert F_1^m\Vert ^2_{L^2(\mathcal {D})}+\Vert F_2^m\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s} \right) + C {\mathbb {E}}\left( \int _{t}^{T} \mathcal {E}^m(s) \text {d}{s} \right) \end{aligned}$$

and using backward Gronwall inequality (see, e.g., [35, Lemma 3.1]) we obtain

$$\begin{aligned}&{\mathbb {E}}\left( \mathcal {E}^m(t)+\int _{t}^{T}\left[ \left\Vert y^m_{xx}\right\Vert _{L^2(\mathcal {D})}^2+\left\Vert Y^m\right\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s}+ \int _t^T\left( \Vert z_x\Vert ^2_{L^2(\mathcal {D})}+\left\Vert Z^m\right\Vert _{L^2(\mathcal {D})}^2 \right) \text {d}{s} \right) \\&\quad \le C {\mathbb {E}}\left( \Vert \Pi _{m}y_T\Vert ^2_{L^2(\mathcal {D})} + \Vert \Pi _{m}z_T\Vert ^2_{L^2(\mathcal {D})} \right) + C {\mathbb {E}}\left( \int _{t}^{T}\left[ \Vert F_1^m\Vert ^2_{L^2(\mathcal {D})}+\Vert F_2^m\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s} \right) \end{aligned}$$

for some \(C>0\) uniform with respect to m.

Thus,

$$\begin{aligned}&\sup _{0\le t\le T} {\mathbb {E}}\left( \mathcal {E}^m(t)\right) + {\mathbb {E}}\left( \int _{0}^{T}\left[ \left\Vert y^m_{xx}\right\Vert _{L^2(\mathcal {D})}^2+\left\Vert Y^m\right\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s} + \int _0^T\left[ \Vert z_x\Vert ^2_{L^2(\mathcal {D})}+\left\Vert Z^m\right\Vert _{L^2(\mathcal {D})}^2 \right] \text {d}{s} \right) \nonumber \\&\quad \le C {\mathbb {E}}\left( \Vert \Pi _{m}y_T\Vert ^2_{L^2(\mathcal {D})} + \Vert \Pi _{m}z_T\Vert ^2_{L^2(\mathcal {D})} \right) + C {\mathbb {E}}\left( \int _{0}^{T}\left[ \Vert F_1^m\Vert ^2_{L^2(\mathcal {D})}+\Vert F_2^m\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s} \right) \end{aligned}$$
(115)

Arguing as above, from (113) and (114), we obtain

$$\begin{aligned} \mathcal {E}^m(t)&\le \Vert \Pi _{m}y_T\Vert ^2_{L^2(\mathcal {D})} + \Vert \Pi _{m}z_T\Vert ^2_{L^2(\mathcal {D})} + \int _{t}^T\left[ \Vert F_1^m\Vert ^2_{L^2(\mathcal {D})}+\Vert F_2^m\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s} \\&\quad + 2 \left( \left|\int _0^t (y^m,Y^m)\text {d}{W}(s)\right|+\left|\int _0^T (y^m,Y^m)\text {d}{W}(s)\right|\right) \\&\quad + 2 \left( \left|\int _0^t (z^m,Z^m)\text {d}{W}(s)\right|+\left|\int _0^T (z^m,Z^m)\text {d}{W}(s)\right|\right) \end{aligned}$$

Taking supremum over t and then expectation

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t\le T}\mathcal {E}^m(t) \right) \le {\mathbb {E}}\left( \Vert \Pi _{m}y_T\Vert ^2_{L^2(\mathcal {D})} + \Vert \Pi _{m}z_T\Vert ^2_{L^2(\mathcal {D})}\right. \\&\qquad +\left. \int _{0}^T\left( \Vert F_1^m\Vert ^2_{L^2(\mathcal {D})}+\Vert F_2^m\Vert ^2_{L^2(\mathcal {D})}\right) \text {d}{s}\right) + 4 {\mathbb {E}}\left( \sup _{0\le t\le T}\left|\int _0^t (y^m,Y^m)\text {d}{W}(s)\right| \right) \\&\qquad + 4 {\mathbb {E}}\left( \sup _{0\le t\le T} \left|\int _0^t (z^m,Z^m)\text {d}{W}(s)\right|\right) \\&\quad \le {\mathbb {E}}\left( \Vert \Pi _{m}y_T\Vert ^2_{L^2(\mathcal {D})} + \Vert \Pi _{m}z_T\Vert ^2_{L^2(\mathcal {D})} + \int _{0}^T\left( \Vert F_1^m\Vert ^2_{L^2(\mathcal {D})}+\Vert F_2^m\Vert ^2_{L^2(\mathcal {D})}\right) \text {d}{s}\right) \\&\qquad + 4 C {\mathbb {E}}\left( \left|\int _0^T (y^m,Y^m)_{L^2(\mathcal {D})}^2\text {d}{s}\right|^{1/2} \right) + 4 C {\mathbb {E}}\left( \left|\int _0^T (z^m,Z^m)_{L^2(\mathcal {D})}^2\text {d}{s}\right|^{1/2} \right) , \end{aligned}$$

where we have used Burkholder–Davis–Gundy inequality. Using Hölder and Young inequalities

$$\begin{aligned} {\mathbb {E}}\left( \sup _{0\le t\le T}\mathcal {E}^m(t) \right)&\le {\mathbb {E}}\left( \Vert \Pi _{m}y_T\Vert ^2_{L^2(\mathcal {D})} + \Vert \Pi _{m}z_T\Vert ^2_{L^2(\mathcal {D})} + \int _{0}^T\left( \Vert F_1^m\Vert ^2_{L^2(\mathcal {D})}+\Vert F_2^m\Vert ^2_{L^2(\mathcal {D})}\right) \text {d}{s}\right) \nonumber \\&\quad + \frac{\varepsilon }{2} {\mathbb {E}}\left( \sup _{0\le t\le T}\mathcal {E}^m(t)\right) +C(\varepsilon ){\mathbb {E}}\left( \int _{0}^{T}\left[ \Vert Y^m\Vert ^2_{L^2(\mathcal {D})}+\Vert Z^m\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s}\right) \end{aligned}$$
(116)

for any \(\varepsilon >0\). Thus, combining (115), (116) and taking \(\varepsilon >0\) small enough, we get

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t\le T}\mathcal {E}^m(t)+\int _{0}^{T}\left[ \left\Vert y^m_{xx}\right\Vert _{L^2(\mathcal {D})}^2+\left\Vert Y^m\right\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s}+ \int _0^T\left[ \Vert z_x^m\Vert ^2_{L^2(\mathcal {D})}+\left\Vert Z^m\right\Vert _{L^2(\mathcal {D})}^2 \right] \text {d}{s} \right) \nonumber \\&\quad \le C {\mathbb {E}}\left( \Vert \Pi _{m}y_T\Vert ^2_{L^2(\mathcal {D})} + \Vert \Pi _{m}z_T\Vert ^2_{L^2(\mathcal {D})} \right) + C {\mathbb {E}}\left( \int _{0}^{T}\left[ \Vert F_1^m\Vert ^2_{L^2(\mathcal {D})}+\Vert F_2^m\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s} \right) . \end{aligned}$$
(117)

Step 3. Last arrangements and conclusion. Following the arguments above, for \(m\ge n\ge 1\) and the definition of the \(H_0^1(\mathcal {D})\) and \(H_0^2(\mathcal {D})\) norms, we can deduce that

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t\le T} \left[ \Vert y^m(t)-y^n(t)\Vert ^2_{L^2(\mathcal {D})}+\Vert z^m(t)-z^n(t)\Vert ^2_{L^2(\mathcal {D})}\right] \right) \\&\qquad +{\mathbb {E}}\left( \int _{0}^{T}\left[ \left\Vert y^m-y^n\right\Vert _{H_0^2(\mathcal {D})}^2+\left\Vert Y^m-Y^n\right\Vert ^2_{L^2(\mathcal {D})}\right] \text {d}{s}\right) \\&\qquad +{\mathbb {E}}\left( \int _0^T\left[ \Vert z^m-z^n\Vert ^2_{H_0^1(\mathcal {D})}+\left\Vert Z^m-Z^n\right\Vert _{L^2(\mathcal {D})}^2 \right] \text {d}{s} \right) \\&\quad \le C {\mathbb {E}}\left( \Vert \Pi _{m}y_T-\Pi _n y_T\Vert ^2_{L^2(\mathcal {D})} + \Vert \Pi _{m}z_T-\Pi _n z_{T}\Vert ^2_{L^2(\mathcal {D})} \right) \\&\qquad + C {\mathbb {E}}\left( \int _{0}^{T}\Vert F_1^m-F_1^n\Vert ^2_{L^2(\mathcal {D})}+\Vert F_2^m-F_2^n\Vert ^2_{L^2(\mathcal {D})} \right) . \end{aligned}$$

Observe that the right-hand side of the above expression converges to zero as \(n,m\rightarrow \infty \). Thus, it follows that \(\{(y^m,z^m,Y^m,Z^m)\}_{m\in \mathbb N}\) is a Cauchy sequence that converges strongly in \([L^2_{\mathcal {F}}(\Omega ;C([0,T];L^2(\mathcal {D})^2)\cap L^2_{\mathcal {F}}(0,T;H_0^2(\mathcal {D})\times H_0^1(\mathcal {D}))]\times L^2_{\mathcal {F}}(0,T;L^2(\mathcal {D})^2)\).

Let us denote by (yzYZ) the limit. By a standard argument, we can check that (yzYZ) verify the first two equations of (109) together with their corresponding terminal and boundary data, and that (y(t), z(t), Y(t), Z(t)) is \(\mathbf {F}\)-adapted. By letting \(m\rightarrow \infty \) in (117), we obtained the desired estimate (110).

Finally, for the uniqueness, let \((y_1,z_1,Y_1,Z_1)\) and \((y_2,z_2,Y_2,Z_2)\) be two different solutions to (110). Defining \(y=y_1-y_2\), \(z=z_1-z_2\), \(Z=Z_1-Z_2\) and \(Y=Y_1-Y_2\), we have from (117)

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t \le T}\Vert y(t)\Vert _{L^2(\mathcal {D})}^2+\sup _{0\le t \le T}\Vert z(t)\Vert ^2_{L^2(\mathcal {D})}\right) \\&\quad +{\mathbb {E}}\left( \int _{0}^{T}\Vert y(t)\Vert ^2_{H_0^2(\mathcal {D})}\text {d}t+\int _{0}^{T}\Vert z(t)\Vert ^2_{H_0^1(\mathcal {D})}\text {d}t\right) \\&\quad +{\mathbb {E}}\left( \int _{0}^T \Vert Y(t)\Vert _{L^2(\mathcal {D})}^2\text {d}t+\int _{0}^T \Vert Z(t)\Vert _{L^2(\mathcal {D})}^2\text {d}t\right) \le 0 \end{aligned}$$

Thus, \(y(t)=z(t)=Y(t)=Z(t)= 0\) for any \(t\in [0,T]\), a.s. This proves the uniqueness and ends the proof. \(\square \)

Now, we present a general result for the forward system.

Proposition B.2

Assume that \(u_0,v_0\in L^2(\Omega ,\mathcal {F}_0;L^2(\mathcal {D}))\) and \(f_i,g_i\in L^2_{\mathcal {F}}(0,T;L^2(\mathcal {D}))\), \(i=1,2\). Then, the system

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {d}u+(\gamma u_{xxxx}+u_{xxx}+u_{xx})\text {d}t=(f_1+v_x )dt+(g_1+d_1 u)\, \text {d}W(t) &{}\text {in}\;Q, \\ \text {d}v-\Gamma v_{xx}\text {d}t=(f_2+v_x+u_x)\text {d}t+(g_2+d_2u+d_3 v) \text {d}W(t) &{}\text {in}\;Q, \\ u=u_x=0 &{}\text {on}\;\Sigma , \\ v=0 &{}\text {on}\;\Sigma , \\ u(0)=u_0, \quad v(0)=v_0 &{}\text {in}\; \mathcal {D}, \end{array}\right. } \end{aligned}$$
(118)

has a unique solution \((u,v)\in L^2_{\mathcal {F}}(0,T;H_0^2(\mathcal {D})\times H_0^{1}(\mathcal {D})) \bigcap L^2\big (\Omega ; C([0,T];L^2(\mathcal {D})^2)\big )\). Moreover, there exists some \(C>0\) only depending on T, \(\Gamma \), \(\gamma \) and \(d_i\), \(i=1,2,3\), such that

$$\begin{aligned} {\mathbb {E}}\left( \sup _{0\le t \le T} \Vert u\Vert _{L^2(\mathcal {D})}^2+ \sup _{0\le t \le T} \Vert v\Vert ^2_{L^2(\mathcal {D})}\right) +{\mathbb {E}}\left( \int _{0}^{T}\Vert u(t)\Vert ^2_{H_0^2(\mathcal {D})}\text {d}t+\int _{0}^{T}\Vert v(t)\Vert ^2_{H_0^1(\mathcal {D})}\text {d}t\right) \\ \le C{\mathbb {E}}\left( \Vert u_0\Vert ^2_{L^2(\mathcal {D})}+\Vert v_0\Vert _{L^2(\mathcal {D})}^2+\sum _{i=1}^{2}\left\{ \int _{0}^{T}\Vert f_i(t)\Vert ^2_{L^2(\mathcal {D})}\text {d}t+\int _{0}^{T}\Vert g_i(t)\Vert ^2_{L^2(\mathcal {D})}\text {d}t\right\} \right) . \end{aligned}$$

The proof of this result is analogous to that of Proposition B.1, but this time we can follow the methodology from [19] adapted to our coupled case. For brevity, we omit the details.

1.2 Appendix B.2: Hinged boundary conditions

Here, we present existence and uniqueness results for the case of systems with hinged boundary conditions, that is, when we impose \(y=y_{xx}=0\) on \(\Sigma \) to the fourth-order operator.

In what follows, we assume that \(a_{i}\in L^\infty _{\mathcal {F}}(0,T;\mathbb {R})\), \(i=1,\dots ,4\) and that \(T\in (0,1)\). We have the following result for the forward system.

Proposition B.3

  1. (a)

    Assume that \(y_0,z_0\in L^2(\Omega ,\mathcal {F}_0;L^2(\mathcal {D}))\) and \(f_i,g_i\in L^2_{\mathcal {F}}(0,T;L^2(\mathcal {D}))\), \(i=1,2\). Then, the system

    $$\begin{aligned} {\left\{ \begin{array}{ll} \text {d}y+y_{xxxx}\text {d}t=(f_1+a_1 y+ a_2 z)dt+(g_1+b_1 y+b_2 z)\, \text {d}W(t) &{}\text {in}\;Q, \\ \text {d}z- z_{xx}\text {d}t=(f_2+a_3y+a_4z)\text {d}t+(g_2+b_3 z) \text {d}W(t) &{}\text {in}\;Q, \\ y=y_{xx}=0 &{}\text {on}\;\Sigma , \\ z=0 &{}\text {on }\Sigma , \\ y(0)=y_0, \quad z(0)=z_0 &{}\text {in}\; \mathcal {D}, \end{array}\right. } \end{aligned}$$
    (119)

    has a unique solution \((y,z)\in L^2_{\mathcal {F}}(0,T;H^2(\mathcal {D})\times H_0^{1}(\mathcal {D})) \bigcap L^2\big (\Omega ; C([0,T];L^2(\mathcal {D})^2)\big )\). Moreover, there exists some \(C>0\) only depending on \(a_j\) and \(b_i\), such that

    $$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t \le T} \Vert y(t)\Vert _{L^2(\mathcal {D})}^2+\sup _{0\le t \le T}\Vert z(t)\Vert ^2_{L^2(\mathcal {D})}\right) +{\mathbb {E}}\left( \int _{0}^{T}\Vert y(t)\Vert ^2_{H^2(\mathcal {D})}\text {d}t+\int _{0}^{T}\Vert z(t)\Vert ^2_{H_0^1(\mathcal {D})}\text {d}t\right) \nonumber \\&\quad \le C{\mathbb {E}}\left( \Vert y_0\Vert ^2_{L^2(\mathcal {D})}+\Vert z_0\Vert _{L^2(\mathcal {D})}^2+\sum _{i=1}^{2}\left\{ \int _{0}^{T}\Vert f_i(t)\Vert ^2_{L^2(\mathcal {D})}\text {d}t+\int _{0}^{T}\Vert g_i(t)\Vert ^2_{L^2(\mathcal {D})}\text {d}t\right\} \right) . \end{aligned}$$
    (120)
  2. (b)

    Assume that \((y_0,z_0)\in L^2(\Omega ,\mathcal {F}_0;H^2(\mathcal {D})\cap H_0^1(\mathcal {D}))\times L^2(\Omega ,\mathcal {F}_0;H_0^1(\mathcal {D}))\) and \(f_i\in L^2_{\mathcal {F}}(0,T;L^2(\mathcal {D}))\), \(g_i\in L^2_{\mathcal {F}}(0,T; H^{3-i}(\mathcal {D}))\), \(i=1,2\). Then, the system (119) has a unique solution

    $$\begin{aligned} (y,z)\in L^2_{\mathcal {F}}(0,T;H^4(\mathcal {D})\times H^2(\mathcal {D})) \bigcap L^2\left( \Omega ; C([0,T];H^2(\mathcal {D})\times H_0^1(\mathcal {D}))\right) . \end{aligned}$$

    Moreover, there exists some \(C>0\) only depending on \(a_j\) and \(b_i\), such that

    $$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t \le T} \Vert y(t)\Vert _{H^2(\mathcal {D})}^2+\sup _{0\le t \le T}\Vert z(t)\Vert ^2_{H_0^1(\mathcal {D})}\right) +{\mathbb {E}}\left( \int _{0}^{T}\Vert y(t)\Vert ^2_{H^4(\mathcal {D})}\text {d}t+\int _{0}^{T}\Vert z(t)\Vert ^2_{H^2(\mathcal {D})}\text {d}t\right) \\&\quad \le C{\mathbb {E}}\left( \Vert y_0\Vert ^2_{H^2(\mathcal {D})\cap H_0^1(\mathcal {D})}+\Vert z_0\Vert _{H^1_0(\mathcal {D})}^2+\sum _{i=1}^{2}\left\{ \int _{0}^{T}\Vert f_i(t)\Vert ^2_{L^2(\mathcal {D})}\text {d}t+\int _{0}^{T}\Vert g_i(t)\Vert ^2_{H^{3-i}(\mathcal {D})}\text {d}t\right\} \right) . \end{aligned}$$

The proof of this result can be obtained in two steps. For a), we can adapt [19, Proposition 2.3] in the spirit of Proposition B.1 to get a well-posedness result for the coupled system (119). Actually, we just need to change the eigenvalue problem of Lemma B.2 for the Galerkin method. Point (b) comes by arguing similar to [16, Proposition 2.1] to deduce energy estimates with more regular data.

We conclude this section by giving the following result for the corresponding backward system.

Proposition B.4

Assume that \(u_T,v_T\in L^2(\Omega ,\mathcal {F}_T; L^2(\mathcal {D}))\) and \(F_i\in L^2_{\mathcal {F}}(0,T;L^2(\mathcal {D}))\). Then, the system

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {d}u-u_{xxxx}\text {d}t=-(a_1 u+a_3v+F_1+b_1\overline{u})\text {d}t+\overline{u}\,\text {d}W(t) &{}\text {in}\; Q, \\ \text {d}v+v_{xx}\text {d}t=-(a_2 u + a_4 v+F_2+b_2\overline{u}+b_3\overline{v})\text {d}t+\overline{v}\,\text {d}W(t) &{}\text {in}\; Q, \\ u=u_{xx}=0 &{}\text {on}\;\Sigma , \\ v=0 &{}\text {on}\; \Sigma , \\ u(T)=u_T, \quad v(T)=v_T &{}\text {in}\; \mathcal {D}, \end{array}\right. } \end{aligned}$$
(121)

has a unique solution

$$\begin{aligned} (u,v,\overline{u},\overline{v})\in & {} \left[ L^2_{\mathcal {F}}(0,T;H^2(\mathcal {D})\times H_0^{1}(\mathcal {D})) \bigcap L^2\left( \Omega ; C([0,T];L^2(\mathcal {D})^2)\right) \right] \\&\quad \times L^2_{\mathcal {F}}(0,T;L^2(\mathcal {D}))^2. \end{aligned}$$

Moreover, there exists some \(C>0\) only depending on \(a_j\) and \(b_i\), such that

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t \le T}\Vert u\Vert _{L^2(\mathcal {D})}^2+\sup _{0\le t \le T}\Vert v\Vert ^2_{L^2(\mathcal {D})}\right) +{\mathbb {E}}\left( \int _{0}^{T}\Vert u(t)\Vert ^2_{H^2(\mathcal {D})}\text {d}t+\int _{0}^{T}\Vert v(t)\Vert ^2_{H_0^1(\mathcal {D})}\text {d}t\right) \\&\qquad +{\mathbb {E}}\left( \int _{0}^T \Vert \overline{u}(t)\Vert _{L^2(\mathcal {D})}^2\text {d}t+\int _{0}^T \Vert \overline{v}(t)\Vert _{L^2(\mathcal {D})}^2\text {d}t\right) \\&\quad \le C{\mathbb {E}}\left( \Vert u_T\Vert ^2_{L^2(\mathcal {D})}+\Vert v_T\Vert _{L^2(\mathcal {D})}^2+\int _{0}^{T}\Vert F_1(t)\Vert ^2_{L^2(\mathcal {D})}\text {d}t+\int _{0}^{T}\Vert F_2(t)\Vert ^2_{L^2(\mathcal {D})}\text {d}t\right) . \end{aligned}$$

Appendix C: Sketch of the proof of Lemma 2.7

Since most of the arguments are similar to those in Step 3 of the proof of Theorem 2.5, we proceed briefly. Let us consider an open set \(\mathcal {D}_{2}\) such that \(\mathcal {D}_{3}\subset \subset \mathcal {D}_{2}\subset \subset \mathcal {D}_0\) and take \(\hat{\eta }\in C^\infty _0(\mathcal {D}_{2})\) satisfying \(\hat{\eta }\equiv 1\) in \(\mathcal {D}_3\). Using Itô’s formula, we compute \(\text {d}(\hat{\zeta }u u_{xx})\), where \(\hat{\zeta }:=\hat{\eta }\lambda ^5\phi _m^5\theta ^2\). After a long, but straightforward computation we get

$$\begin{aligned} 2\gamma {\mathbb {E}}\left( \int _{Q}\hat{\zeta }|u_{xxx}|^2\,\text {d}x\text {d}t\right) =&-\gamma {\mathbb {E}}\left( \int _Q\left[ 3\hat{\zeta }_{xx}u_xu_{xxx}+\hat{\zeta }_{xxx}uu_{xxx}-2\hat{\zeta }_{xx}|u_{xx}|^2\right] \,\text {d}x\text {d}t\right) \nonumber \\&\quad +{\mathbb {E}}\left( \int _{Q}\left[ 2\hat{\zeta }_xu_xu_{xxx}+\hat{\zeta }_{xx}uu_{xxx}+\hat{\zeta }_{xx}uu_{xx}-\hat{\zeta }_{xx}|u_x|^2\right] \,\text {d}x\text {d}t\right) \nonumber \\&\quad +{\mathbb {E}}\left( \int _{Q}\left[ \hat{\zeta }u_x v_{xx}+\hat{\zeta }_x u v_{xx}-\hat{\zeta }_t u u_{xx}-\hat{\zeta }_x|u_{xx}|^2\right] \,\text {d}x\text {d}t\right) \nonumber \\&\quad +{\mathbb {E}}\left( \int _{Q}\left[ 2\hat{\zeta }|u_{xx}|^2-\hat{\zeta }u_{xx}v_x\right] \,\text {d}x\text {d}t\right) \nonumber \\&\quad +{\mathbb {E}}\left( \int _{Q}\left[ \hat{\zeta }|(d_1u)_x|^2-\frac{1}{2}\hat{\zeta }_{xx}|d_1 u|^2\right] \,\text {d}x\text {d}t\right) . \end{aligned}$$
(122)

Using the definition of \(\phi _m\) and \(\theta \), we can see that

$$\begin{aligned}&|\partial _x^{i}(\theta ^2\phi _m^p)|\le C_i \lambda ^{i}\phi _m^{p+i}\theta ^2, \quad i=1,2,3, \\&|\partial _t(\theta ^2\phi _m^p)|\le C\lambda \phi _m^{p+1+\frac{1}{m}}\theta ^2, \quad \forall p\in \mathbb N^*, \end{aligned}$$

from which we deduce

$$\begin{aligned} \begin{aligned} |\hat{\zeta }_t|&\le C\lambda ^6\phi _m^{6+\frac{1}{m}}\theta ^2\eta , \\ |\partial _x^{i}(\hat{\zeta })|&\le C_i\lambda ^{5+i}\phi _m^{5+i}\theta ^2\sum _{j=0}^{i}(\partial _x^j\eta ), \quad i=1,2,3,\\ \end{aligned} \end{aligned}$$
(123)

for all \((x,t)\in \mathcal {D}_{2}\times (0,T)\).

Using Cauchy–Schwarz and Young inequalities together with (123) and taking into account the properties of the function \(\hat{\eta }\), we can obtain from (122) the following estimate

$$\begin{aligned}&{\mathbb {E}}\left( \int _{Q_{\mathcal {D}_3}}\lambda ^5\phi _m^5\theta ^2|u_{xxx}|^2\,\text {d}x\text {d}t\right) \\&\quad \le 4\epsilon {\mathbb {E}}\left( \int _{Q}\lambda \phi _m|u_{xxx}|^2\,\text {d}x\text {d}t\right) +2\delta {\mathbb {E}}\left( \int _{Q}\lambda \phi _m\theta ^2|v_{xx}|^2\,\text {d}x\text {d}t\right) \\&\qquad +\rho {\mathbb {E}}\left( \int _{Q}\lambda ^3\phi _m^3\theta ^2|v_{x}|^2\,\text {d}x\text {d}t\right) +C{\mathbb {E}}\left( \int _{Q_{\mathcal {D}_{2}}}\lambda ^{13}\phi _m^{13}\theta ^2|u_x|^2\,\text {d}x\text {d}t\right) \\&\qquad +C{\mathbb {E}}\left( \int _{Q_{\mathcal {D}_{2}}}\lambda ^{15}\phi _m^{15}\theta ^2|u|^2\,\text {d}x\text {d}t\right) +C{\mathbb {E}}\left( \int _{Q_{\mathcal {D}_{2}}}\lambda ^{7}\phi _m^{7}\theta ^2|u_{xx}|^2\,\text {d}x\text {d}t\right) , \end{aligned}$$

for any positive constants \(\epsilon ,\delta ,\rho \) and where \(C>0\) depends on \(\Vert d_1\Vert _{L^\infty _{\mathcal {F}}(0,T;W^{2,\infty }(\mathcal {D}))}\). This concludes the proof.

Appendix D: A regularity lemma

Lemma D.1

Let g(t) be given by (29) and let \((u,v)\in L^2_{\mathcal {F}}(0,T;H_0^2(\mathcal {D})\times H_0^1(\mathcal {D}))\). Then, there exists a unique solution to (30) such that \(\widetilde{v}\in L^2_{\mathcal {F}}(0,T;H^3(\mathcal {D}))\) and such that

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t\le T}\Vert \widetilde{v}(t)\Vert ^2_{H^2(\mathcal {D})}\right) +{\mathbb {E}}\left( \int _{0}^{T}\Vert \widetilde{v}(t)\Vert ^2_{H^3(\mathcal {D})}\text {d}t\right) \nonumber \\&\quad \le C{\mathbb {E}}\left( \int _{0}^{T}\Vert g_t v+g u_x\Vert _{H^1(\mathcal {D})}^2\text {d}t\right) . \end{aligned}$$
(124)

for some constant \(C>0\) only depending on \(\mathcal {D}\), \(d_2\) and \(\Gamma \).

Proof

The proof of existence is a direct application of [12, Theorem 4.1] with \(n=1\). For this, we have to verify conditions i) to iv) in such result, but since in our case we have homogeneous boundary conditions we will see that some of them simplify straightforwardly.

Condition (i) and (iii) are direct since our initial/boundary conditions are zero. The compatibility conditions (iv) are fulfilled since \(d_2\) only depends on time and \(u(\cdot ,t,\omega )\in H_0^2(\mathcal {D})\).

To check condition (ii), let us introduce the functional space \(X(\mathcal {D}):=\{f\in L^2_{\mathcal {F}}(0,T;L^2(\mathcal {D})): Az \in L^2_{\mathcal {F}}(0,T;H^1(\mathcal {D}))\cap L^2_{\mathcal {F}}(C([0,T];L^2(\mathcal {D})))\}\) where z solves the PDE

$$\begin{aligned} {\left\{ \begin{array}{ll} z_t-Az=f &{} \text {in}\;Q, \\ z=0 &{}\text {on}\; \Sigma , \\ z(0)=0 &{}\text {in}\;\mathcal {D}\end{array}\right. } \end{aligned}$$

and \(Au:=\Gamma u_{xx}+u_{x}\) with \(D(A)=H^2(\mathcal {D})\cap H_0^1(\mathcal {D})\). Thus, we must verify that \(f:= g_t v+ g u_x\in X(\mathcal {D})\). Define \(\widetilde{z}=A z\), then \(\widetilde{z}\) verifies the equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \widetilde{z}_t-A \widetilde{z}= \Gamma (g_t v_{x}+g u_{xx})_x + (g_t v_x+ g u_{xx})&{} \text {in}\;Q, \\ \widetilde{z}=0 &{}\text {on}\; \Sigma , \\ \widetilde{z}(0)=0 &{}\text {in}\;\mathcal {D}\end{array}\right. } \end{aligned}$$
(125)

Observe that \(\widetilde{z}=0\) at the boundary since v and \(u_x\) are also zero at the boundary.

Note that since \((u,v)\in L^2_{\mathcal {F}}(0,T;H_0^2(\mathcal {D})\times H_0^1(\mathcal {D}))\), then \(\Gamma (g_t v_{x}+g u_{xx})_x \in L^2_{\mathcal {F}}(0,T; H^{-1}(\mathcal {D}))\) and \(\Gamma (g_t v_{x}+g u_{xx}) \in L^2_{\mathcal {F}}(0,T; L^2(\mathcal {D}))\). From classical well-posedness result for parabolic equations (see, e.g., [9, Section 3, p. 509]), there exists a unique solution \(\widetilde{z}\) to (125) in \(L^2_{\mathcal {F}}(0,T; H_0^1(\mathcal {D}))\cap L^2_{\mathcal {F}}(\Omega ; C([0,T];L^2(\mathcal {D}))\) with

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t\le T}\Vert \widetilde{z}(t)\Vert _{L^2(\mathcal {D})}^2\right) + {\mathbb {E}}\left( \int _{0}^{T}\Vert \widetilde{z}\Vert ^2_{H^1_0(\mathcal {D})}\text {d}{t}\right) \nonumber \\&\quad \le C{\mathbb {E}}\left( \int _{0}^{T}\Vert (\Gamma g_t v_{x}+g u_{xx})_x\Vert ^2_{H^{-1}(\mathcal {D})} \text {d}t+ \int _{0}^{T}\Vert \Gamma g_t v_{x}+g u_{xx}\Vert ^2_{H^{-1}(\mathcal {D})} \text {d}t\right) \end{aligned}$$
(126)

for some \(C>0\) only depending on \(\mathcal {D}\), \(\Gamma \) and g. Thus, \(f\in X(\mathcal {D})\) and (ii) is verified, and the existence of a unique solution \(\widetilde{v}\in L^2_{\mathcal {F}}(0,T;H^2(\mathcal {D}))\cap L^2_{\mathcal {F}}(\Omega ; C([0,T];H^3(\mathcal {D})))\) is verified.

To prove the energy estimate (124), we begin by noting that from (126)

$$\begin{aligned}&{\mathbb {E}}\left( \sup _{0\le t\le T}\Vert \widetilde{z}(t)\Vert _{L^2(\mathcal {D})}^2\right) + {\mathbb {E}}\left( \int _{0}^{T}\Vert \widetilde{z}\Vert ^2_{H^1_0(\mathcal {D})}\text {d}{t}\right) \nonumber \\&\quad \le C{\mathbb {E}}\left( \int _{0}^{T}\Vert \Gamma g_t v_{x}+g u_{xx}\Vert ^2_{L^2(\mathcal {D})} \text {d}t+ \int _{0}^{T}\Vert \Gamma g_t v_{x}+g u_{xx}\Vert ^2_{L^2(\mathcal {D})} \text {d}t\right) \nonumber \\&\quad \le C{\mathbb {E}}\left( \int _{0}^{T}\Vert g_t v+g u_{x}\Vert ^2_{H^1(\mathcal {D})} \text {d}t\right) \end{aligned}$$
(127)

where we recall that g is a function only depending on time.

As in the proof of [12, Theorem 4.1], we have that \(\widetilde{v}(t)=A^{-1}Z(t)\), where Z solves the equation

$$\begin{aligned} Z(t)=A\int _0^{t}e^{(t-s)A}(g_s(s) v(s)+g(s) u_x(s))\text {d}{s} +\int _{0}^{t}e^{(t-s)A}\left[ d_2(s) Z(s))\right] \text {d}{W}(s) \end{aligned}$$

We rewrite the above as

$$\begin{aligned} Z(t)=\phi (t)+\int _{0}^{t}e^{(t-s)A}\left[ d_2(s) Z(s))\right] \text {d}{W}(s). \end{aligned}$$

where

$$\begin{aligned} \phi (t):= A\int _0^{t}e^{(t-s)A}(g_s(s) v(s)+g(s) u_x(s))\text {d}{s} \end{aligned}$$

Thus, we shall prove that \(\phi \in L^2_{\mathcal {F}}(0,T;H_0^1(\mathcal {D}))\cap L^2_{\mathcal {F}}(\Omega ;C([0,T];L^2(\mathcal {D})))\), but this is direct since \(\phi =\widetilde{z}\), which in turn satisfies (127). Whence, a direct application of [13, Corollary 3.3] yields that Z has a unique solution in \(L^2_{\mathcal {F}}(0,T;H_0^1(\mathcal {D}))\cap L^2_{\mathcal {F}}(\Omega ;C([0,T];L^2(\mathcal {D})))\) and there exists some \(C>0\) such that

$$\begin{aligned} {\mathbb {E}}\left( \sup _{0\le t\le T}\Vert Z(t)\Vert _{L^2(\mathcal {D})}^2\right) + {\mathbb {E}}\left( \int _{0}^{T}\Vert Z\Vert ^2_{H^1(\mathcal {D})}\text {d}{t}\right)&\le C\Vert \phi \Vert ^2_{L^2_{\mathcal {F}}(0,T;H_0^1(\mathcal {D}))\cap L^2_{\mathcal {F}}(\Omega ;C([0,T];L^2(\mathcal {D})))} \nonumber \\&\le C{\mathbb {E}}\left( \int _{0}^{T}\Vert g_t v+g u_{x}\Vert ^2_{H^1(\mathcal {D})} \text {d}t\right) \end{aligned}$$
(128)

where we have used (127) and equivalency of norms. Arguing as in the end of [12, Theorem 4.1], we use that \(A^{-k}\) is a bounded linear operator from \(H^{s}(\mathcal {D})\) to \(H^{s+2k}(\mathcal {D})\) for every \(k\ge 1\) and \(s\ge 0\) and (128) yields the desired result. This ends the proof. \(\square \)

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Hernández-Santamaría, V., Peralta, L. Controllability results for stochastic coupled systems of fourth- and second-order parabolic equations. J. Evol. Equ. 22, 23 (2022). https://doi.org/10.1007/s00028-022-00758-x

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