Degenerate Nonlinear Semigroups of Operators and Their Applications

Abstract

In this paper, we construct the conditions for the existence of a degenerate nonlinear resolving semigroup of shift operators for a semilinear Sobolev type equation. Based on the phase space method, we find the conditions for the existence of solutions to the Cauchy problem for a semilinear Sobolev type equation. The solutions are defined only on a simple Banach manifold (quasistationary trajectories). Also, we find the conditions under which the solution can be continued in time. The obtained abstract results are illustrated by the Cauchy–Dirichlet problem for the generalized filtration Boussinesq equation.

1 Introduction

From the middle of the 20th century, the theory of (semi)groups of operators is one of the main research tools to study the Cauchy problem

$$\begin{aligned} u(0)=u_0 \end{aligned}$$

(1)

for abstract operator-differential equations

$$\begin{aligned} \dot{u}=A(u). \end{aligned}$$

(2)

One of the methods to find analytical solutions to problem (1), (2) is to construct resolving (semi)groups of operators [1]. In the case of linear equation (2), the theory of (semi)groups of operators was proposed and developed by E. Hille, R. S. Phillips, K. Yosida, W. Feller, and others, and was widely used to study various initial boundary-value problems for equations of mathematical physics [2,3,4,5,6,7,8].

The study of nonlinear semigroups of operators was initiated by the study of nonlinear parabolic equations. Among the first works in this field we note the papers [9,10,11,12] that made important contribution to the development of the theory. The papers [9, 10, 13] obtain the conditions for the existence of a solution to problem (1), (2) in the case of nonlinear dissipative operator A, acting in Hilbert space ${\mathbf{H}}$, when the image of operator ${I}-A$ coincides with whole initial space ${\mathbf{H}}$. Also, the papers [9, 10, 13] construct the semigroups of nonlinear shift operators. The main research method is the construction of approximate solutions to the initial problem as solutions to the Cauchy problem for the operator equation, where the operator in the right-hand side is dissipative and satisfies the Lipschitz condition. The work [9] was the first to obtain the necessary and sufficient conditions for the existence of a generator of resolving semigroup of nonlinear operators (an analogous of Hille–Yosida–Phillips theorem in the linear case) in the case of multi-valued nonlinear dissipative operators. Various applications to nonlinear evolution equations and quasilinear first order equations were investigated in the works [14,15,16].

In natural science and technology, an extensive class of processes and phenomena can be modelled by partial differential equations or systems of partial differential equations. Here the equations are not resolved with respect to the highest time derivative, and have the form

$$\begin{aligned} L\dot{u}=M(u), \ \mathrm{ker}\, L\ne \{0\}. \end{aligned}$$

(3)

Such equations are usually called Sobolev type equations [17,18,19,20]. For the first time, this term was introduced by  Showalter [21]. Problem (1), (3) is difficult to study. The main reason is that the problem is unsolvable in principle for an arbitrary initial value $u_0$, even if $u_0$ belongs to a dense lineal of the initial Banach space ${\mathscr {U}}$. Therefore, it is relevant to search and describe the structure of set of initial value condition (1), for which problem (1), (3) has a solution. One of the successful methods to study linear ($M\in Cl ({\mathscr {U}}; {\mathscr {F}})$) problem (1), (3) is the theory of degenerate resolving (semi)groups of operators, which was proposed in [22], and was developed in [23,24,25]. The papers [22, 23] construct analytical (semi)groups of resolving operators with kernels, and obtain the conditions under which the image of resolving (semi)group coincides with the phase space of the corresponding equation defined on the subspace. Other successful approaches to study degenerate equations are presented in [21, 26].

The paper [27] was the first to propose the phase space method, which is one of the effective methods to study the Cauchy problem for the semilinear Sobolev type equations ($M(u)=M_1u+M_2(u)$). A phase space is a closure of the set of all admissible initial values $u_0$ that are the vectors for which there exists the unique (local) solution to the Cauchy problem. Then, the vector $u_{0}$ is chosen such that the following two conditions hold in the neighborhood ${\mathscr {O}}_{u_{0}}$ of $u_{0}$. First, the phase space ${\mathbf{M}}$ is a smooth Banach manifold (phase manifold). Second, singular (i.e. with condition $\mathrm{ker}\, L \ne \{0\}$) Eq. (3) is reduced to the regular equation

$$\begin{aligned} \dot{u}=F(u), \end{aligned}$$

(4)

where F is a smooth section of the tangent bundle $T{\mathscr {O}}_{u_{0}}$. This method allows to find solutions to a wide class of problems [28,29,30]. Also, we note another important concept, which considers quasistationary (semi)trajectories [31] that pass through the point $u_0$, and pointwise belong to the phase space ${\mathbf{M}}$. Moreover, any stationary trajectory of Eq. (3) is quasistationary, but the converse is wrong. Note that the phase space of equation (3) can lie on a smooth manifold having singularities such as Whitney k-folds [32].

The review [33] gives the conditions under which the phase space of equation (3) is a simple Banach $C^{\infty }$-manifold, and there exists the unique quasistationary (semi)trajectory of Eq. (3) that passes through the point $u_0$. Recall that a Banach $C^{\infty }$-manifold is called simple if its atlas is equivalent to the atlas containing the unigue map. In particular, the sufficient conditions for the simplicity of the phase space of equation (3) are obtained in the case of the s-monotone, p-coercive operator $(-M_2)$ and the Fredholm operator L. Note that Eq. (3) with Showalter–Sidorov initial value condition can have several solutions, if the phase space is not simple [34].

The purpose of the article is to find the conditions for the existence of a non-local solution to Cauchy problem (1), (3). In Paragraph 1, we use the phase space method to construct the quasistationary semitrajectories, i.e. the solutions to Eq. (3) defined on the phase manifold ${\mathbf{M}}$. Also, we show that the solution can be continued in time on $(0, +\infty )$ in the case of the dissipative operator M. In Paragraph 2, we construct a non-linear resolving semigroup of shift operators of Eq. (3) defined on ${\mathbf{M}}$. The obtained abstract results are illustrated by the Cauchy–Dirichlet problem for the generalized filtration Boussinesq equation [35]. The paper [36] was the first to obtain the conditions for the existence of a local solution to the Showalter–Sidorov–Dirichlet problem. Also, the paper [36] shows that the phase space of the generalized filtration Boussinesq equation is a smooth Banach manifold. We obtain a priori estimates in order to prove that the solution can be continued in time on $(0, +\infty )$, and to construct the resolving semigroup of operators of Boussinesq equation.

We use the methods developed in Paragraph 1 and Paragraph 2 in order to find the solutions to the stochastic semilinear Sobolev type equation

$$\begin{aligned} L{\mathop {\eta }\limits ^{o}}= M(\eta ), \end{aligned}$$

(5)

endowed with the weakened (in the sense of S.G. Krein) Cauchy condition

$$\begin{aligned} \lim \limits _{t\rightarrow 0+}(\eta (t)-\eta _0)=0. \end{aligned}$$

(6)

Here $\eta =\eta (t)$ is the desired stochastic process, $\eta _0$ is a given random variable, and the symbol ${\mathop {\eta }\limits ^{o}}$ denotes the Nelson–Gliklikh derivative of the stochastic process $\eta =\eta (t)$ [37]. In order to study the Cauchy problem, we construct the spaces of K-“noises”, i.e. the spaces of stochastic K-processes that are almost surely differentiable in the sense of Nelson–Gliklikh. This approach is based on the paper [38]. Note that this approach allows to transfer on the stochastic case the methods of functional analysis that are applied in the deterministic case [20, 24, 25]. Further, we construct the quasistationary semitrajectories of Eq. (5) that pass through the point $\eta _0 (\omega )$ for each fixed $\omega \in \varOmega$, where $\varOmega \equiv (\varOmega , {\mathscr {A}}, \mathbf{P})$ is a complete probability space. We use the quasistationary semitrajectories in order to construct the stochastic nonlinear resolving semigroup of operators defined on the phase manifold ${\mathbf{M}}$.

2 Phase Space Quasistationary Semitrajectories

Let ${\mathbf{H}} = ({\mathbf{H}}; \left\langle \cdot , \cdot \right\rangle )$ be a real separable Hilbert space identified with its conjugate space; $({\mathscr {H}}, {\mathscr {H}}^*)$ and $({\mathscr {B}}, {\mathscr {B}}^*)$ be dual (with respect to the duality $\left\langle \cdot , \cdot \right\rangle$) pairs of reflexive Banach spaces such that the embeddings

$$\begin{aligned} {\mathscr {B}}\hookrightarrow {\mathscr {H}} \hookrightarrow {\mathbf{H}} \hookrightarrow {\mathscr {H}}^* \hookrightarrow {\mathscr {B}}^* \end{aligned}$$

(7)

are dense and continuous. Let $L \in {\mathscr {L}}({\mathscr {H}}; {\mathscr {H}}^*)$ be a linear, continuous, self-adjoint nonnegatively defined Fredholm operator, and the operator $M \in C^k({\mathscr {B}}; {\mathscr {B}}^{*}), k \ge 1$.

In view of the self-adjointness and Fredholm property of the operator L we identify ${\mathscr {H}} \supset \ker L \equiv \mathrm{coker}\, L \subset {\mathscr {H}}^*$, this implies ${\mathscr {H}}^* = \mathrm{coker}\, L \oplus \mathrm{im}\, L$. Denote by $\overline{\mathrm{im}\, L}$ closure $\mathrm{im}\, L$ in topology ${\mathscr {B}}^*$, then ${\mathscr {B}}^* = \mathrm{coker}\, L \oplus \overline{\mathrm{im}\, L}.$ Denote by Q projector ${\mathscr {B}}^*$ along $\mathrm{coker}\, L$ on $\overline{\mathrm{im}\, L}$. Then, if $u = u(t), t \in (0, T]$ is a solution of Eq. (3), then the solution belongs to the set

$$\begin{aligned} {\mathbf{M}} = \left\{ \begin{array}{c} \{u \in {\mathscr {B}}: ({ I} - Q) M(u) = 0 \},\quad \text{ if }\ \ker L \ne \{ 0 \}; \\ {\mathscr {B}}, \quad \text{ if }\quad \ker L = \{ 0 \}. \end{array} \right. \end{aligned}$$

(8)

Consider the set

$$\mathrm{coim}\, L = \{ u \in {\mathscr {H}}: \left\langle u, v \right\rangle = 0 ~~~~\forall v \in \mathrm{ker}\, L \backslash \{0\}\}.$$

In view of embeddings (7), the space ${\mathscr {B}}$ can be represented as a direct sum ${\mathscr {B}} = {\mathscr {B}}^0 \oplus {\mathscr {B}}^1,$ where $\ker L \equiv {\mathscr {B}}^0$, $\mathrm{coim} \, L \cap {\mathscr {B}} \equiv {\mathscr {B}}^1.$ Denote by P the projector on ${\mathscr {B}}^1$ along ${\mathscr {B}}^0.$

Definition 1

A vector-function $u\in C^k((0,T]; \mathscr {B})$, $k\in {\mathbf{N}}\cup \{\infty \},$ is called a solution to Eq. (3), satisfies the equation for some $T\in \mathbf {R}_{+}$. A solution $u=u(t)$ to Eq. (3) is called a solution to Cauchy problem (1), (3), if $u=u(t)$ also satisfies initial value condition (1).

Remark 1

A little unlike the standard [31], a vector function $u=u(t)$ is called a quasistationary semitrajectory of Eq. (3), if $u=u(t)$ is a solution to this equation and belongs to the set ${\mathbf{M}}$, i.e. $u(t)\in {\mathbf{M}}$ for all $t\in (0,T]$. If a quasistationary semitrajectory of Eq. (3) satisfies condition (1), then the semitrajectory is called a quasistationary semitrajectory of Eq. (3) that pass through the point $u_0$.

Definition 2

[28] Suppose that the point $u_0\in {\mathbf{M}}$ and $u_0^1 = Pu_0\in {\mathscr {B}}^1$. A set ${\mathbf{M}}$ is a Banach  $C^k$-manifold at the point $u_0$, if there exist the neighborhoods ${\mathscr {O}}_0^{\mathbf{M}}\subset {\mathbf{M}}$ and ${\mathscr {O}}_0^1\subset {\mathscr {B}}^1$ of the the points $u_0$ and $u_0^1$ respectively, and $C^k$-diffeomorphism $\delta : {\mathscr {O}}^1_0\rightarrow {\mathscr {O}}_0^{\mathbf{M}}$ such that $\delta ^{-1}$ is equal to the restriction of projector P on ${\mathscr {O}}_0^{\mathbf{M}}$. A set ${\mathbf{M}}$ is called Banach  $C^k$-manifold modelled by the subspace ${\mathscr {B}}^1$, if ${\mathbf{M}}$ is a Banach $C^k$-manifold at each of its points. A connected Banach $C^k$-manifold is called simple, if any atlas of the manifold is equivalent to the atlas containing the unique map.

Theorem 1

Suppose that the set ${\mathbf{M}}$ is a simple Banach $C^k$-manifold at the point $u_0$. Then there exists the unique quasistationary semitrajectory of Eq. (3) that pass through the point $u_0$.

Proof

In view of the representation of the space ${\mathscr {B}} = {\mathscr {B}}^0 \oplus {\mathscr {B}}^1$ and the existence of the projector P of the space ${\mathscr {B}}$ on ${\mathscr {B}}^1$ along ${\mathscr {B}}^0$, any vector $u=u^0+u^1$, where $({ I}-P)u=u^0\in {\mathscr {B}}^0$, and $Pu=u^1\in {\mathscr {B}}^1$. Denote by $L_1$ a restriction of the operator L on the subspace ${\mathscr {B}}^1$. Then, in view of the linearity of the operator L, we obtain

$$ Lu=L(u^0+u^1)=L_1u^1, $$

and Eq. (3) is equivalent to the system of equations

$$\begin{aligned} \begin{array}{c} \dot{u}^1=L_1^{-1}QM(\delta (u^1)), \quad u=\delta (u^1),\\ 0=({ I}-Q)M(u), \end{array} \end{aligned}$$

(9)

where $L_1^{-1}$ is inverse operator to the operator L.

Let ${\mathscr {O}}^{\mathbf{M}}_0$ and ${\mathscr {O}}^1_0$ be neighborhoods of the points $u_0\in {\mathbf{M}}$ and $u^1_0=Pu_0,$ respectively. Since the set ${\mathbf{M}}$ is a Banach $C^k$-manifold at the point $u_0$, then there exists a diffeomorphism $\delta \in C^k ({\mathscr {O}}^1_0; {\mathscr {O}}^{\mathbf{M}}_0)$ . Denote by $\delta '_{u^1}$ the Frechet derivative of the operator $\delta$ at the point $u^1\in {\mathscr {O}}^1_0$, then the operator $\delta '_{u^1}\in {\mathscr {L}} ({\mathscr {B}}^1; T_u {\mathbf{M}})$ for each fixed point $u^1\in {\mathscr {O}}^1_0$. Here $T_u {\mathbf{M}}$ is the tangent space at the point $u=\delta (u^1)$. Let us act on the left by the operator $\delta '_{u^1}$ on both sides of the first equation of system (9):

$$\begin{aligned} \delta '_{u^1}\dot{u}^1=\delta '_{u^1}L_1^{-1}QM(\delta (u^1)), \end{aligned}$$

(10)

as

$$\delta '_{u^1} \dot{u}^1 = \frac{d}{dt}\delta (u^1)=\dot{u},$$

then

$$ \dot{u}= F(u), $$

where operator $F= \delta '_{u^1}L_1^{-1}QM: u \rightarrow T_u {\mathbf{M}}$. Then the operator $F\in C^k ({\mathscr {O}}^{\mathbf{M}}_0; T_0 {\mathbf{M}})$, where $T_0 {\mathbf{M}}$ is a restriction of the tangent bundle $T {\mathbf{M}}$ on ${\mathscr {O}}^{\mathbf{M}}_0$. In view of the classical theorem of the existence and uniqueness of solution to the Cauchy problem [39], we obtain the simple local solvability of the problem

$$\begin{aligned} \dot{u} = F(u), \quad u(0) = u_0. \end{aligned}$$

(11)

Let us show that the solution to problem (11) is a quasistationary semitrajectory of Eq. (3). Let $u=u(t)$ be the solution to problem (11). Since $u=\delta (u^1)$, then $u(t)\in {\mathbf{M}}$ for each $t\in (0, T)$. In addition, the vector function $u^1(t) =Pu(t)$ satisfies Eq. (3), since $\delta '_{u^1}\in {\mathscr {L}} ({\mathscr {B}}^1; T_u {\mathbf{M}})$ is a toplinear isomorphism. So, if $u=u(t)$ is a solution of problem (11), then $u=u(t)$ is the quasistationary semitrajectory of Eq. (3) that pass through the point $u_0$. Thus, the Theorem is proved.

Definition 3

The operator $M:{\mathscr {B}}\rightarrow {\mathscr {B}}^*$ is called dissipative, if

$$ \left\langle M(u)-M(v), u-v\right\rangle \le 0 ~~~\forall \, u, v \in {\mathscr {B}}. $$

Note that the dissipativity of the operator M is equivalent to the monotonicity of the operator $(-M)$. According to the paper [33], the s-monotonicity (i.e. $\left\langle -M^{\prime }_{u} v, v\right\rangle \ge 0 \ \forall \, u, v \in {\mathscr {B}}$) of the operator $(-M)$ implies the strict monotonicity of the operator, and, consequently, the dissipativity of the operator.

Consider the stationary equation

$$\begin{aligned} M(v)=0 \end{aligned}$$

(12)

and suppose that there exists a solution to Eq. (12). Further, let the operator M be dissipative. Introduce the norm $\vert u \vert ^2 = \left\langle Lu, u \right\rangle$ in $\mathrm{coim}\, L$. In view of Courant principle, this norm is equivalent to the norm induced from the overspace ${\mathbf{H}}$. Then, it follows from Eq. (3) that

$$ \frac{1}{2}\frac{d}{dt}\vert u(t)\vert ^2=\frac{1}{2}\frac{d}{dt}\vert u(t)- v \vert ^2= \left\langle M(u)-M(v), u-v\right\rangle \le 0, $$

where v is a solution to Eq. (12), which in turn is a stationary solution to Eq. (3). Hence, in view of the uniqueness of solution to Cauchy problem (11), we obtain that the solution can be continued on the interval $(0, +\infty )$. Then the following theorem is correct.

Theorem 2

Suppose that the set $u_0$ set ${\mathbf{M}}$ is a simple Banach $C^k$-manifold at the point $u_0$, there exists a solution to Eq. (12), and the operator M is dissipative. Then there exists the unique quasistationary semitrajectory $u\in C^k((0,+\infty ); \mathscr {B})$ of Eq. (3) that pass through the point $u_0$.

Definition 4

A set ${\mathbf{M}} \subset {\mathscr {B}}$ is called a phase space of Eq. (3), if

(i) any solution $u = u(t)$ to Eq. (3) belongs to the set ${\mathbf{M}}$ pointwise, i.e. $u(t)\in {\mathbf{M}}$, $t\in (0, +\infty )$;

(ii) ) for any $u_{0} \in {\mathbf{M}}$ there exists the unique solution to problem (1), (3).

Remark 2

If the set ${\mathbf{M}}$ is a simple Banach $C^{k}$-manifold, then ${\mathbf{M}}$ coincides with the phase space of equation (3).

3 Semigroups of Operators

Let $({\mathscr {V}}, <\cdot ,\cdot >)$ be a Hilbert space. Consider a family of (in the general case, nonlinear) operators $V^t(\cdot ): \mathrm{dom}\, V^t\subset {\mathscr {V}} \rightarrow {\mathscr {V}}$ $\forall t\ge 0$ with a common domain $D(V)=\mathrm{dom}\, V^t \subset {\mathscr {V}}$ $\forall t\ge 0$. Denote by $C ({\mathscr {V}})$ the set of continuous operators.

Definition 5

A mapping $V^{\bullet }\in C^k([0,+\infty ); C({\mathscr {V}}))$ is called a semigroup of operators, if

$$ V^{t+s}(u)=V^{t}(V^{s}(u)) \ \forall s, t \ge 0 \ \ \forall u\in {\mathscr {V}}. $$

We identify the semigroup $V^{\bullet }$ with its graph $\{V^t:\ t\in {\mathbf{R}}_+\}$.

Definition 6

A semigroup $\{V^t:\ t\in {\mathbf{R}}_+\}$ is called contractive, if

$$ \Vert V^{t}(u) -V^{t}(v)\Vert _{{\mathscr {V}}}\le \Vert u -v\Vert _{{\mathscr {V}}} \ \forall t \ge 0 \ \ \forall u, v\in {\mathscr {V}}. $$

Definition 7

An operator $A: \mathrm{dom}\, A\subset {\mathscr {V}} \rightarrow {\mathscr {V}}$ is called an infinitesimal generator of the semigroup $\{V^t:\ t\in {\mathbf{R}}_{+}\}$, if the condition

$$\begin{aligned} A(u)=\lim \limits _{t\rightarrow 0+}\frac{V^t(u)- u}{t}, \end{aligned}$$

(13)

holds, where $\mathrm{dom}\, A$ consists of all $u\in {\mathscr {V}}$ such that (13) is satisfied.

Let us construct a semigroup of resolving operators of Eq. (3). Define a shift operator $U^t(u_0)\equiv u(t)$, where u(t) is a solution to problem (11). Then $\{U^t:\ t\in {\mathbf{R}}_+\}$ forms a nonlinear semigroup of shift operators with common domain $D(U)={\mathbf{M}}\subset {\mathscr {B}}$. The infinitesimal generator of the semigroup of operators $\{U^t:\ t\in {\mathbf{R}}_+\}$ is operator $F= \delta '_{u^1}L_1^{-1}QM: u \rightarrow T_u {\mathbf{M}}$, defined in Theorem 1. Since the operator M is dissipative and the operator $L_1$ is positive defined, then the operator F is dissipative, and, therefore, the constructed semigroup is contractive. Indeed,

$$\begin{array}{c} \displaystyle<\frac{U^t(u)- u}{t}-\frac{U^t(v)- v}{t}, u-v>_{\mathbf{H}}=\frac{1}{t}<U^t(u)- U^t(v),u-v>_{\mathbf{H}} - \Vert u-v\Vert _{{\mathbf{H}}}^2\le \\ \displaystyle \le \frac{1}{t}\Vert u-v\Vert _{{\mathbf{H}}}\left( \Vert U^t(u)- U^t(v)\Vert _{{\mathbf{H}}} - \Vert u-v\Vert _{{\mathbf{H}}}\right) \le 0, \end{array} $$

then

$$ \Vert U^t(u)- U^t(v)\Vert _{{\mathbf{H}}} \le \Vert u-v\Vert _{{\mathbf{H}}}. $$

This completes the proof following Theorem.

Theorem 3

Suppose that the set ${\mathbf{M}}$ is a simple Banach $C^k$-manifold at point $u_0$, and there exists a solution to Eq. (12), and the operator M is dissipative. Then there exists a resolving semigroup of operators $\{U^t:\ t\in {\mathbf{R}}_+\}$ of Eq. (3), defined on the manifold ${\mathbf{M}}$.

Remark 3

Similarly to the linear case [23], a semigroup of operators $\{U^t:\ t\in {\mathbf{R}}_+\}$ of Eq. (3) is called degenerate, since the semigroup is given not on the whole space, but on some set ${\mathbf{M}}$ having the structure of a simple Banach manifold.

4 General Mathematical Filtration Boussinesq Model

Let $\varOmega \subset {\mathbf{R}}^n$ be a bounded domain with a smooth boundary of class $C^{\infty }$. In the cylinder $\varOmega \times {\mathbf{R}}_+$ consider Dirichlet problem

$$\begin{aligned} u(s, t) = 0, \, (s, t) \in \partial \varOmega \times {\mathbf{R}}_+ \end{aligned}$$

(14)

for the generalized filtration Boussinesq equation [35, 36]

$$\begin{aligned} (\lambda - \varDelta )u_{t} = \varDelta (\vert u \vert ^{p-2}u), \ p\ge 2. \end{aligned}$$

(15)

Equation (15) is the most interesting particular case of the equation obtained by Dzektser [35]. Here the desired function $u = u(s, t)$ corresponds to the potential of speed of movement of the free surface of the filtered liquid; the parameter $\lambda \in {\mathbf{R}}$ characterizes the medium, and this parameter $\lambda$ can take negative values.

Let ${\mathbf{H}} = W_2^{-1}(\varOmega ),\, {\mathscr {H}} = L_2(\varOmega ),$ ${\mathscr {B}} = L_p(\varOmega )$ (all functional spaces are defined on domain $\varOmega$). Note that there exists the dense and continuous embedding ${\mathop {W}\limits ^{\circ }}{}\!_2^1(\varOmega ) \hookrightarrow L_q(\varOmega )$ for $p \ge \frac{2n}{n + 2}$, therefore $L_p(\varOmega ) \hookrightarrow W_2^{-1}(\varOmega )$, where $\frac{1}{p}+\frac{1}{q}=1$. In ${\mathbf{H}}$, define the scalar product by the formula

$$\begin{aligned} \left\langle x, y\right\rangle = \int \limits _{\varOmega } x \tilde{y}ds \ \ \forall x, y \in {\mathbf{H}}, \end{aligned}$$

(16)

where $\tilde{y}$ is the generalized solution to the homogeneous Dirichlet problem for Laplace operator $(-\varDelta )$ in the domain $\varOmega$. Let ${\mathscr {B}}^* = (L_p(\varOmega ))^*$ and ${\mathscr {H}}^* = (L_2(\varOmega ))^*$, where $(L_p(\varOmega ))^*$ is conjugate space with respect to duality (16). For thus defined ${\mathscr {H}}^*$ and ${\mathscr {B}}^*$ there exist dense and continuous embeddings (7). Define the operators L and M as follows:

$$ \langle Lu, v \rangle = \int \limits _{\varOmega }(\lambda u\tilde{v} + uv)ds,\quad u,\, v \in {\mathscr {H}}; $$

$$ \langle M(u), v \rangle = -\int \limits _{\varOmega } \vert u \vert ^{p-2}u v ds, \quad u,\, v \in {\mathscr {B}}. $$

Let $\{ \varphi _k \}$ be the sequence of eigenfunctions of the homogeneous Dirichlet problem for Laplace operator $(- \varDelta )$ in the domain $\varOmega$, and $\{ \lambda _k \}$ be the corresponding sequence of eigenvalues numbered in non-increasing order taking into account the multiplicity.

Lemma 1

(i) For all $\lambda \ge - \lambda _1$ the operator $L\in {\mathscr {L}}({\mathscr {H}}; {\mathscr {H}}^*)$ is self-adjoint, Fredholm, and non-negatively defined, and the orthonormal family $\{ \varphi _k \}$ of its functions is total in the space ${\mathscr {H}}$.

(ii) Operator $M \in C^{1}({\mathscr {B}}; {\mathscr {B}}^*)$ is dissipative and p-coercive.

Proof

Statement (i) is a classical result. As for statement (ii), the Frechet derivative of the operator M at the point $u \in {\mathscr {B}}$ is determined by the formula

$$ \vert \langle M^{\prime }_u v, w \rangle \vert = (p - 1) \vert \int \limits _{\varOmega } \vert u \vert ^{p - 2}vw ds\vert \le \mathrm{const} \Vert u \Vert _{L_p(\varOmega )}^{p -2}\Vert v \Vert _{L_p(\varOmega )}\Vert w \Vert _{L_p(\varOmega )}. $$

Hence, the operator $M \in C^{1}({\mathscr {B}}; {\mathscr {B}}^*)$. The dissipativity of the operator M is a consequence of the s-monotonicity of the operator $(-M)$ [36]:

$$ \langle -M^{\prime }_u v, v \rangle = (p - 1) \int \limits _{\varOmega } \vert u \vert ^{p - 2}v^2du > 0, \, u, v \in L_p(\varOmega ), \, u, v \ne 0. $$

In addition, the operator $(-M)$ is p-coercive (a.g. $<M(x),x>\ge C_{M}\Vert x\Vert ^{p}$ and $\Vert M(x)\Vert _{*}\le C^{M}\Vert x\Vert ^{p-1}~\forall x\in \mathscr {B})$ [36]:

$$ \langle -M(u), u \rangle = \int \limits _{\varOmega } \vert u \vert ^{p} ds = \Vert u \Vert ^p_{L_p(\varOmega )}, $$

$$ \vert \langle -M(u), v \rangle \vert \le \int \limits _{\varOmega } \vert u \vert ^{p -1} \vert v \vert ds \le \Vert u \Vert ^{p-1}_{L_p(\varOmega )}\Vert v \Vert _{L_p(\varOmega )}. $$

Hence, the operator $(-M)$ is strong coercive. So, Lemma is proved.

If $\lambda \ge - \lambda _1$

$$ \ker L = \left\{ \begin{array}{c} \{ 0\},\ if \ \lambda > -\lambda _1; \\ \mathrm{span} \{ \varphi _1\}, \ if \ \lambda = -\lambda _1. \end{array} \right. $$

Therefore

$$ \mathrm{im}\, L = \left\{ \begin{array}{c} {\mathscr {H}}^*,\ if \ \lambda > - \lambda _1; \\ \{ u \in {\mathscr {H}}^*: \left\langle u, \varphi _1\right\rangle = 0 \}, \ if \ \lambda = -\lambda _1, \end{array} \right. $$

$$ \mathrm{coim}\, L = \left\{ \begin{array}{c} {\mathscr {H}},\ if \ \lambda > - \lambda _1; \\ \{ u \in {\mathscr {H}}: \left\langle u, \varphi _1\right\rangle = 0 \}, \ if \ \lambda = -\lambda _1. \end{array} \right. $$

Hence, the projectors

$$ P = Q = \left\{ \begin{array}{c} {\mathbf{I}},\, \lambda > - \lambda _1; \\ {\mathbf{I}} - \left\langle \cdot , \varphi _1\right\rangle , \, \lambda = -\lambda _1. \end{array} \right. $$

Construct the set

$$ {\mathbf{M}} = \left\{ \begin{array}{c} {\mathscr {B}},\ if \ \lambda > - \lambda _1; \\ \{ u \in {\mathscr {B}}: \int \limits _{\varOmega } |u|^{p-2}u\varphi _1 \ ds = 0 \}, \ if \ \lambda = -\lambda _1. \end{array} \right. $$

Theorem 4

Suppose that $p \ge \frac{2n}{n + 2}, \,\lambda \ge - \lambda _1$. Then

(i) the set ${\mathbf{M}}$ is a simple Banach $C^{1}$-manifold modelled by the space $\mathrm{coim} \, L \cap {\mathscr {B}}$;

(ii) $\forall \, u_0\in {\mathbf{M}}$ there exists the unique solution $u\in C^k((0,+\infty ); {\mathbf{M}})$ to problem (1), (14), (15).

Proof

Statement (i) was obtained in [32]. Statement (ii) is a consequence of Theorem 2 and Lemma 1.

Define the shift operator $U^t(u_0)\equiv u(t)$, where u(t) is a solution to problem (1), (14), (15). Then $\{U^t:\ t\in {\mathbf{R}}_+\}$ forms a nonlinear semigroup of operators with domain $D(U)={\mathbf{M}}$.

Theorem 5

Suppose that $p \ge \frac{2n}{n+2}, \,\lambda \ge - \lambda _1$. Then there exists a resolving semigroup of contractive operators $\{U^t:\ t\in {\mathbf{R}}_+\}$ of Eq. (15) defined on the manifold ${\mathbf{M}}$.

Proof

Obviously, $u\equiv 0$ is a stationary solution to Eq. (15). Statement of the theorem is a consequence of Theorem 3 and Lemma 1.

5 Stochastic K-Processes. Phase Space

Consider a complete probability space $\varOmega \equiv (\varOmega , \mathscr {A}, \mathbf{P})$ and the set of real numbers ${\mathbf{R}}$ endowed with a Borel $\sigma$-algebra. According to [38], a measurable mapping $\xi : \varOmega \rightarrow {\mathbf{R}}$ is called a random variable. The set of random variables having zero expectations (i.e. $\mathbf{E} \xi =0$) and finite variances (i.e. $\mathbf{D} \xi <+\infty$) forms Hilbert space $\mathbf{L_2}$ with the scalar product $(\xi _1, \xi _2) = \mathbf{E}\xi _1\xi _2,$ where $\mathbf{E}$, $\mathbf{D}$ is the expectation and variance of the random variable, respectively. Denote by ${\mathscr {A}}_0$ the $\sigma$-subalgebra of $\sigma$-algebra ${\mathscr {A}}$. Construct the space $\mathbf{L}_2^0$ of random variables that are measurable with respect to ${\mathscr {A}}_0,$ then $\mathbf{L}_2^0$ is a subspace of the space $\mathbf{L_2}.$ Suppose that $\varPi : \mathbf{L}_2 \rightarrow \mathbf{L}_2^0$ is the orthoprojector, and $\xi \in \mathbf{L}_2$, then $\varPi \xi$ is called the conditional expectation of the random variable $\xi$ and is denoted by $\mathbf{E}(\xi |{\mathscr {A}}_0).$

Let $\mathscr {I}\subset \mathbf{R}$ be a set. Consider two mappings: $f: \mathscr {I}\rightarrow \mathbf{L_2}$, that each $t\in \mathscr {I},$ associates with a random variable $\xi \in \mathbf{L_2}$, and $g: \mathbf{L}_2\times \varOmega \rightarrow {\mathbf{R}}$, that each pair $(\xi ,\omega )$ associates with a point $\xi (\omega )\in {\mathbf{R}}$. A mapping $\eta : \mathscr {I}\times \varOmega \rightarrow {\mathbf{R}}$, of the form $\eta = \eta (t, \omega )=g(f(t),\omega )$ is called an (one-dimensional) random process. According to [38], a random process $\eta$ is called continuous, if almost surely all its trajectories are continuous. Denote by $\mathbf{C L_2}$ the set of continuous random processes, which forms a Banach space. An example of the continuous process is one-dimensional Wiener process $\beta =\beta (t)$, which can be represented as [38]

$$\begin{aligned} \beta (t)=\sum _{k=0}^\infty \xi _k \sin \frac{\pi }{2}(2k+1)t, \end{aligned}$$

(17)

where $\xi _k$ are uncorrelated Gaussian variables, $\mathbf{E}\xi _k=0,\, \mathbf{D}\xi _k=[\frac{\pi }{2}(2k+1)]^{-2}.$

Fix $\eta \in \mathbf{CL_2}$ and $t\in \mathscr {I}$ and denote by ${\mathscr {N}}^\eta _t$ the $\sigma$-algebra generated by the random variable $\eta (t).$ Denote $\mathbf{E}^{\eta }_t=\mathbf{E}(\cdot |{\mathscr {N}}^\eta _t).$

Definition 8

(i) Suppose that $\eta \in \mathbf{CL_2}$. A random variable

$$\begin{aligned} D\eta \left( t,\cdot \right) ={\mathop {\lim }_{\triangle t\rightarrow 0+} E^{\eta }_t\ }\left( \frac{\eta \left( t+\triangle t,\cdot \right) -\eta (t,\cdot )}{\triangle t}\right) \end{aligned}$$

$$\begin{aligned} \left( D_*\eta \left( t,\cdot \right) ={\mathop {\lim }_{\triangle t\rightarrow 0+} E^{\eta }_t\ }\left( \frac{\eta \left( t,\cdot \right) -\eta \left( t-\triangle t,\cdot \right) }{\triangle t}\right) \right) \end{aligned}$$

is called a mean derivative on the right $D\eta (t,\cdot )$  (on the left $D_*\eta (t,\cdot )$) of a random process $\eta$ at the point $t\in (\varepsilon ,\tau ),$ if the limit exists in the sense of a uniform metric on ${\mathbf{R}}$. A random process $\eta$ is called mean differentiable on the right (on the left) on $\mathscr {I}$, if there exists the mean derivative on the right (on the left) at each point $t\in \mathscr {I}$.

(ii) Let the random process $\eta \in \mathbf{CL_2}$ be mean differentiable on the right and on the left on $\mathscr {I}$. The derivative ${\mathop {\eta }\limits ^{o}}=D_S\eta ={{\frac{1}{2}}}\left( D+D_*\right) \eta$ is called the symmetric mean derivative.

Remark 4

Futher, the symmetric mean derivative is called the Nelson–Gliklikh derivative. Denote the lth Nelson–Gliklikh derivative of the random process $\eta$ by ${\mathop {\eta }\limits ^{o}}^{(l)},$ $l\in {\mathbf{N}}$. Note that the Nelson–Gliklikh derivative coincides with the classical derivative, if $\eta (t)$ is a deterministic function. In the case of the one-dimensional Wiener process $\beta =\beta (t)$, the following statements are correct [37]:

(i) ${\mathop {\beta }\limits ^{o}}(t)=\frac{\beta (t)}{2t}$ for all $t\in {\mathbf{R}}_{+}$;

(ii) ${\mathop {\beta }\limits ^{o}}^{(l)}(t)={\left( -1\right) }^{l-1}\cdot {\prod \limits _{i=1}^{l-1}(2i-1)}\cdot \frac{\beta (t)}{(2t)^{l}}, \ \ l\in { \mathbf{N}}, \ \ l\ge 2.$

Consider the space of “noises” $\mathbf{C}^l\mathbf{L}_2,$ $l\in {\mathbf{N}},$ i.e. the space of random processes from $\mathbf{C}{} \mathbf{L}_2,$ whose trajectories are almost surely differentiable by Nelson–Gliklikh on $\mathscr {I}$ up to the order l inclusive.

Consider a real separable Hilbert space $({\mathscr {V}}, <\cdot ,\cdot >)$ with orthonormal basis $\{ \varphi _k\}$. Each element $u\in {\mathscr {V}}$ can be represented as $u=\sum \limits _{k=1}^\infty u_k \varphi _k=\sum \limits _{k=1}^\infty <u,\varphi _k>\varphi _k$. Next, choose a monotonely decreasing numerical sequence $K=\{\mu _k \}$ such that $\sum \limits _{k=1}^{\infty }\mu ^2_k<+\infty .$ Consider a sequence of random variables $\{\xi _k\}\subset \mathbf{L}_2$, such that $\sum \limits _{k=1}^\infty {\mu ^2_k} \mathbf{D}\xi _k<+\infty .$ Denote by ${{\mathscr {V}}}_K\mathbf{L}_2$ the Hilbert space of random K-variables having the form $\xi =\sum \limits _{k=1}^\infty {\mu _k} \xi _k\varphi _k.$ Moreover, a random K-variable $\xi \in {{\mathscr {V}}}_K\mathbf{L}_2$ exists, if, for example, $\mathbf{D}\xi _k <\mathrm{const}\ \forall k$. Note that space ${{\mathscr {V}}}_K\mathbf{L}_2$ is a Hilbert space with scalar product $(\xi ^1, \xi ^2)=\sum \limits _{k=1}^\infty \mu _k^2 \mathbf{E}\xi ^1_k\xi ^2_k.$ Consider a sequence of random processes $\{\eta _k\}\subset \mathbf{C}{} \mathbf{L}_2$ and define ${\mathscr {V}}$-valued continuous stochastic K-process

$$\begin{aligned} \eta (t)=\sum _{k=1}^\infty \mu _k \eta _k(t)\varphi _k \end{aligned}$$

(18)

if series (18) converges uniformly by the norm ${{{\mathscr {V}}}}_K\mathbf{L}_2$ on any compact set in $\mathscr {I}$. Consider the Nelson–Gliklikh derivatives of random K-process 

$${\mathop {\eta }\limits ^{o}}^{(l)}(t)=\sum \limits _{k=1}^\infty \mu _k {\mathop {\eta }\limits ^{o}}_k^{(l)}(t)\varphi _k $$

on the assumption that there exist the Nelson–Gliklikh derivatives up to the order l inclusive in the right-hand side, and all series converge uniformly according to the norm ${{\mathscr {V}}}_K\mathbf{L}_2$ on any compact from $\mathscr {I}$. Next, consider the space $\mathbf{C}(\mathscr {I};{{\mathscr {V}}}_K\mathbf{L}_2)$ of continuous stochastic K-processes and the space $\mathbf{C}^l (\mathscr {I};{{\mathscr {V}} }_K\mathbf{L}_2)$ of stochastic K-processes whose trajectories are almost surely continuously differentiable by Nelson–Gliklikh up to the order $l\in {\mathbf{N}}$ inclusive. An example of a K-process from the space $\mathbf{C}^l (\mathscr {I};{{\mathscr {V}}}_K\mathbf{L}_2)$ is a Wiener K-process [38] $W_K(t)=\sum \limits _{k=1}^\infty {\mu _k} \beta _k(t)\varphi _k,$ where $\{\beta _k\}\subset \mathbf{C}^l\mathbf{L}_2$ is a sequence of one-dimensional Wiener processes (Brownian motions on $\mathscr {I}$). Note that the space $\mathbf{C}^l (\mathscr {I};{{\mathscr {V}}}_K\mathbf{L}_2)$ will call the space of K-“noises”.

Let us extend the constructions from Paragraph 1 to the stochastic case. By analogy with Paragraph 1, consider a real separable Hilbert space ${\mathbf{H}} = ({\mathbf{H}}; \left\langle \cdot , \cdot \right\rangle )$ identified with its conjugate space, and dual pairs of reflexive Banach spaces $({\mathscr {H}}, {\mathscr {H}}^*)$ and $({\mathscr {B}}, {\mathscr {B}}^*)$, such that embeddings (7) are dense and continuous. Let an operator $L \in {\mathscr {L}}({\mathscr {H}}; {\mathscr {H}}^*)$ be linear, continuous, self-adjoint, non-negative defined Fredholm operator, and an operator $M \in C^k({\mathscr {B}}; {\mathscr {B}}^{*}), k \ge 1,$ be dissipative. In space ${\mathbf{H}}$ choose an orthonormal basis $\{ \varphi _k\}$ so that $\mathrm{span} \{ \varphi _1,$ $\varphi _2,\ldots ,$ $\varphi _l \} = \ker L, \dim \ker L = l$ and the following condition holds:

$$\begin{aligned} \{\varphi _k\}\subset {\mathscr {B}}. \end{aligned}$$

(19)

We use the space ${\mathbf{H}}$ in order to construct the space of ${\mathbf{H}}$-valued random K-variables ${\mathbf{H}}_K\mathbf{L}_2$ as a completion of the linear span of random K-variables $\xi =\sum \limits _{k=1}^\infty {\mu _k} \xi _k\varphi _k$. Taking into account that the operator L is self-adjoint and Fredholm, we identify ${\mathbf{H}} \supset \ker L \equiv \mathrm{coker}\, L \subset {\mathbf{H}}^*$ and, similarly, construct the space ${\mathbf{H}}^*_K\mathbf{L}_2$ according to the corresponding orthonormal basis. We use the subspace $\ker L$ in order to construct the subspace $[\ker L]_K\mathbf{L}_2 \subset {\mathbf{H}}_K\mathbf{L}_2$ and, similarly, the subspace $[\mathrm coker\, L]_K\mathbf{L}_2 \subset {\mathbf{H}}^*_K\mathbf{L}_2$. Taking into account that embeddings (7) are continuous and dense, we construct the spaces ${\mathscr {H}}^*_K\mathbf{L}_2 = [\mathrm{coker}\,L]_K\mathbf{L}_2 \oplus [{{\mathrm{im}\, L}]_K\mathbf{L}_2}$ and ${\mathscr {B}}^*_K\mathbf{L}_2 = [\mathrm{coker}\,L]_K\mathbf{L}_2 \oplus [{\overline{\mathrm{im}\, L}]_K\mathbf{L}_2}.$

We use the subspace $\mathrm{coim}\,L\subset {\mathscr {H}}$ in order to construct the subspace $[\mathrm{coim}\,L]_K\mathbf{L}_2$ such that the space ${\mathscr {H}}_K\mathbf{L}_2=[\ker L]_K\mathbf{L}_2\oplus [\mathrm{coim}\,L]_K\mathbf{L}_2$. Denote $[\ker L]_K\mathbf{L}_2 \equiv {\mathscr {B}}^0_K\mathbf{L}_2$ such that the space $\mathrm{coim}\,L \cap {\mathscr {B}}$ in order to construct the set ${\mathscr {B}}^1_K\mathbf{L}_2$, then ${\mathscr {B}}_K\mathbf{L}_2 = {\mathscr {B}}^0_K\mathbf{L}_2 \oplus {\mathscr {B}}^1_K\mathbf{L}_2.$ The following lemma is correct, since the operator L is self-adjoint and Fredholm.

Lemma 2

(i) Let operator $L \in {\mathscr {L}}({\mathscr {H}}; {\mathscr {H}}^*)$ be a linear, continuous, self-adjoint, non-negatively defined Fredholm operator, then the operator $L \in {\mathscr {L}}({\mathscr {H}}_K\mathbf{L}_2;{\mathscr {H}}^*_K\mathbf{L}_2)$, and

$$ {\mathbf{H}}_K\mathbf{L}_2 \supset [\ker L]_K\mathbf{L}_2 \equiv [\mathrm{coker}\, L]_K\mathbf{L}_2 \subset {\mathbf{H}}^*_K\mathbf{L}_2 $$

if

$$ {\mathbf{H}}\supset \ker L \equiv \mathrm{coker}\, L \subset {\mathbf{H}}^*. $$

(ii) There exists a projector Q of the space ${\mathscr {B}}^*_K\mathbf{L}_2$ on $[\overline{\mathrm{im}\, L}]_K\mathbf{L}_2$ along $[\mathrm{coker}\, L]_K\mathbf{L}_2$.

(iii) There exists a projector P of the space ${\mathscr {B}}_K\mathbf{L}_2$ on ${\mathscr {B}}^1_K\mathbf{L}_2$ along ${\mathscr {B}}^0_K\mathbf{L}_2.$

Suppose that ${\mathscr {I}}\equiv (0, +\infty )$. We use the space ${\mathbf{H}}$ in order to construct the spaces of K-“noises” spaces $\mathbf{C}^k({\mathscr {I}};{\mathbf{H}}_K\mathbf{L}_2)$ and $\mathbf{C}^k({\mathscr {I}};{\mathscr {B}}_K\mathbf{L}_2), \ k\in {\mathbf{N}}$. Next, we extend the constructions from Paragraph 1 to the stochastic case. Consider the stochastic Sobolev type equation

$$\begin{aligned} L{\mathop {\eta }\limits ^{o}}=M (\eta ). \end{aligned}$$

(20)

A solution to Eq. (20) is a stochastic K-process. Stochastic K-processes $\eta =\eta (t)$ and $\zeta =\zeta (t)$ are considered to be equal, if almost surely each trajectory of one of the processes coincides with a trajectory of other process.

Definition 9

A stochastic K-process $\eta \in { C}^1({\mathscr {I}};{\mathscr {B}}_K\mathbf{L}_2)$ is called a solution to Eq. (20), if almost surely all trajectories of $\eta$ satisfy Eq. (20) for all $t\in \mathscr {I}$. A solution $\eta =\eta (t)$ to Eq. (20) that satisfies the initial value condition

$$\begin{aligned} \lim \limits _{t\rightarrow 0+}(\eta (t)-\eta _0)=0 \end{aligned}$$

(21)

is called a solution to Cauchy problem (20), (21), if the solution satisfies condition (21) for some random K-variable $\eta _0 \in {\mathscr {B}}_{K} \mathbf{L}_{2}$.

Fix $\omega \in \varOmega$. Let $\eta = \eta (t), t \in \mathscr {I}$ be a solution to Eq. (20), then $\eta$ belongs to the set

$$\begin{aligned} {\mathbf{M}} = \left\{ \begin{array}{c} \{\eta \in {\mathscr {B}}_K\mathbf{L}_2: ({\mathbf{I}} - Q) M(\eta ) = 0 \},\quad \text{ if }\ \ker L \ne \{ 0 \}; \\ {\mathscr {B}}_K\mathbf{L}_2, \quad \text{ if }\quad \ker L = \{ 0 \}. \end{array} \right. \end{aligned}$$

(22)

Theorem 6

Suppose that the set ${\mathbf{M}}$ is a simple Banach $C^k$-manifold at the point $\eta _0 \in {\mathscr {B}}_{K} \mathbf{L}_{2}$, there exists a solution to the equation

$$ M (\eta )=0, $$

and the operator M is dissipative. Then there exists a resolving semigroup of operators $\{\varXi ^t:\ t\in {\mathbf{R}}_+\}$ of Eq. (20) defined on the manifold ${\mathbf{M}}$.

Next, we consider the Dirichlet problem

$$\begin{aligned} \eta (s, t) = 0, \, (s, t) \in \partial \varOmega \times {\mathbf{R}}_+ \end{aligned}$$

(23)

for the stochastic Boussinesq equation

$$\begin{aligned} (\lambda - \varDelta ){\mathop {\eta }\limits ^{o}} = \varDelta (\vert \eta \vert ^{p-2}\eta ), \ p\ge 2. \end{aligned}$$

(24)

Define the shift operator $\varXi ^t(\eta _0)\equiv \eta (t)$, where $\eta (t)$ is the solution of problem (21), (23), (24). Then $\{\varXi ^t:\ t\in {\mathbf{R}}_+\}$ forms a nonlinear semigroup of operators with domain $D(\varXi )={\mathbf{M}}$.

Theorem 7

Suppose that $p \ge \frac{2n}{n + 2}, \,\lambda \ge - \lambda _1$. Exists a resolving semigroup of compressing operators $\{\varXi ^t:\ t\in {\mathbf{R}}_+\}$ of Eq. (24) defined on the manifold ${\mathbf{M}}$.