On a Nonlinear Nonlocal Parabolic Problem

Abstract

We consider the Cauchy problem for a parabolic equation with a Ё-Laplacian or a general second-order quasilinear equation with boundary conditions of the Bitsadze–Samarskii type. We prove that at least one generalized solution of such problem exists.

Introduction

Nonlocal elliptic boundary value problems have been considered since the 1930s of the XXth century, see Carleman [1]. In the 50–60s of the XXth century, abstract nonlocal problems were studied by Vishik [2], Browder [3], etc. The theory of nonlocal boundary value problems has applications to physics, engineering, biology, e.a. In 1969, Bitsadze and Samarskii considered an elliptic equation with nonlocal conditions connecting the values of unknown function on a boundary with its values on some manifold inside a domain, see [4]. Since the 70s, linear nonlocal parabolic problems with the Bitsadze–Samarskii type of nonlocality have also been studied, see, for example, [5, 6] and the bibliography there. Note that, for a long time, only some special cases of the Bitsadze–Samarskii boundary value problems were considered. In 1980, Samarskii described the question of solvability for nonlocal elliptic problems as “unsolved one”, see [7]. A method for studying linear elliptic boundary value problems with such nonlocal conditions was developed in the 80-90s, see [8–12]. The nonlinear elliptic boundary value problems with such nonlocal conditions were studied in [13]. The linear parabolic boundary value problems with such nonlocal conditions were studied in [14]. In this paper, we consider nonlinear nonlocal parabolic problems. Note also that nonlocal parabolic problems have important applications to Feller’s semigroup theory, see [15–17].

1. Statement of the Problem

Let $Q\subset\mathbb{R}^n$ be a bounded domain with boundary $\partial Q\in{\mathbb C}^\infty$ or a cylinder $(0,d)\times G$, where $G\subset {\mathbb R}^{n-1}$ is a bounded domain (with boundary $\partial G\in{\mathbb C}^\infty$ if $n\ge 3$), $\Delta_p w=-\sum\limits_{1\le i\le n} \partial_i\left(|\partial_i w|^{p-2} \partial_i w\right)$. We assume that

$$2\le p<\infty,\quad 1/p+1/q=1,\quad f\in L_q(0,T;W^{-1}_q(Q)),\quad \psi\in L_2(Q).$$

All functions are real–valued. In the cylinder $\Omega_T=Q\times (0,T)$, we consider the differential equation

$$\partial_t w(x,t)+\Delta_p w(x,t)=f(x,t)\quad( (x,t)\in \Omega_T) $$

(1.1)

with initial condition

$$w(x,0)=\psi(x)\qquad(x\in Q) $$

(1.2)

and with nonlocal boundary conditions

$$ \left. \begin{array}{rll} w|_{\Gamma^T_{rl}}&=\sum\limits_{j=1}^{J_{0}} \gamma_{lj}^rw|_{\Gamma^T_{rj}} &\quad (r\in B,\;l=J_{0}+1,\ldots,J),\\ w|_{\Gamma^T_{rl}}&=0 &\quad (r\notin B,\;l=1,\ldots,J), \end{array} \right\}$$

(1.3)

where the set $\Gamma^T=\{\Gamma^T_{rl}\}$ is defined as follows.

Let $\cal M$ be a finite set of vectors $h\in{\mathbb Z}^n$, and let $\,M\,$ be the additive group generated by $\mathscr{M}$. Denote by $Q_{r}$ the open connected components of the set $Q\setminus(\bigcup\limits_{h\in M}(\partial Q+h))$. The set $Q_r$ is called a subdomain. The family $\mathscr{R}$ of all subdomains $Q_r$ ($r=1,2,\ldots$) is called a partition of the domain $Q$. It is easy to see that the set $\mathscr{R}$ is at most countable,

$$\bigcup\limits_r\partial Q_r=\Big(\bigcup\limits_{h\in M}(\partial Q+h) \Big)\cap \overline{Q},\quad\text{and}\quad\bigcup\limits_r\overline Q_r=\overline Q.$$

As is known, see Lemma 7.1 of [9, Ch.II,§7], for any subdomain $Q_{r_1}$ and an arbitrary vector $h\in M$, either there is a subdomain $Q_{r_2}$ such that $Q_{r_2}=Q_{r_1} +h$ or $Q_{r_1}+h\subset\mathbb{R}^n\setminus\overline Q$. Thus, the family $\mathscr{R}$ can be divided into disjoint classes as follows: subdomains $Q_{r_1},Q_{r_2}\in\mathscr{R}$ belong to the same class if $Q_{r_2}=Q_{r_1}+h$ for some $h\in M$. We denote the subdomains $Q_{r}$ by $Q_{sl}$, where $s$ is the class number and $l$ is the subdomain number in the $s$th class. Obviously, each class consists of a finite number $N=N(s)$ of subdomains $Q_{sl}$, and $N(s)\le([\mathrm{diam} Q]+1)^n$. The set of classes can be finite or countable (see examples in Section 7, Ch. II of [9]).

Introduce the set ${\cal K}$ given by the formula

$$ {\cal K}=\bigcup\limits_{h_1,h_2\in M} \left\{ \overline{Q}\cap(\partial Q+h_1)\cap \overline{[(\partial Q+h_2)\setminus(\partial Q+h_1)]}\right\},$$

(1.4)

Let $\Gamma_\rho$ denote the open connected (in the topology of $\partial Q$ ) components of the set $\partial Q\setminus{\cal K}$. The following result was obtained in [9, § 7].

Lemma 1.

If $(\Gamma_\rho+h)\cap \overline{Q}\not=\emptyset$ for some $h\in M$ , then either $\Gamma_\rho+h\subset Q$ or there exists a $\Gamma_r\subset \partial Q \setminus{\cal K}$ such that $\Gamma_\rho+h=\Gamma_r$ .

According to this property, the sets $\left\{\Gamma_\rho+h : \Gamma_\rho+h\subset \overline{Q},\, \rho=1,2,\dots,\, h\in M\right\}$ can be divided into classes. The sets $\Gamma_{\rho_1}+h_1$ and $\Gamma_{\rho_2}+h_2$ belong to the same class if

1) there is a vector $h\in M$ such that $\Gamma_{\rho_1}+h_1=\Gamma_{\rho_2}+h_2+h$;

2) for any $\Gamma_{\rho_1}+h_1, \Gamma_{\rho_2}+h_2 \subset \partial Q$ the normals to $\partial Q$ at the points $x\in \Gamma_{\rho_1}+h_1$ and $x-h\in \Gamma_{\rho_2}+h_2$ have the same direction.

Let the set $\Gamma_{\rho}+h$ be denoted by $\Gamma_{rj}$, where $r$ is the index of a class and $j$ is the index of an element in that class ($1\le j\le J=J(r)$). Without lost of generality, we assume that $\Gamma_{r1},\dots,\Gamma_{rJ_0}\subset Q$ and $\Gamma_{r,J_0+1},\dots,\Gamma_{rJ}\subset \partial Q$ ($0\leq J_0=J_0(r)<J(r)$).

It is well known (see [9, § 7]) that this partition has the following properties.

Lemma 2.

For any $\Gamma_{rj}\subset \partial Q$ , there exists a subdomain $Q_{sl}$ such that $\Gamma_{rj}\subset \partial Q_{sl}$ . Moreover, the inclusion $\Gamma_{rj}\subset \partial Q_{sl}$ implies $\Gamma_{rj}\cap \partial Q_{s_1l_1}=\emptyset$ if $(s_1,l_1)\not=(s,l)$ .

Lemma 3.

For every $r=1,2,\dots$ , there exists a unique index $s=s(r)$ such that $N(s)=J(r)$ and $\Gamma_{rl}\subset \partial Q_{sl}\, (l=1,\dots, N(s))$ $($ up to reindexing $)$ .

Write $\Gamma^T_{rl}:=\Gamma_{rl}\times (0,T)$.

Example 1.

We consider problem (1.1), (1.2) in the rectangular parallelepiped $\Omega_T=(0,2)\times(0,1)\times(0,T)$ with the Bitsadze–Samarskii nonlocal boundary conditions

$$ \left.\begin{array}{ll} w(x_1,0,t)=w(x_1,1,t)=0&(0\le x_1\le 2; 0<t<T),\\ w(x,t)|_{x_1=0}=\gamma_1w(x,t)|_{x_1=1},&(0< x_2< 1; 0<t<T)\\ w(x,t)|_{x_1=2}=\gamma_2w(x,t)|_{x_1=1}&(0< x_2< 1;0<t<T) \end{array}\right\}.$$

(1.5)

Here we have 4 classes of sets:

1) $\Gamma^T_{11}=\{1\}\times(0,1)\times(0,T)$, $\Gamma^T_{12}=\{0\}\times(0,1)\times(0,T)$; 2) $\Gamma^T_{21}=\{1\}\times(0,1)\times(0,T)$, $\Gamma^T_{22}=\{2\}\times(0,1)\times(0,T)$; 3) $\Gamma^T_{31}=(0,1)\times\{0\}\times(0,T)$, $\Gamma^T_{32}=(1,2)\times\{0\}\times(0,T)$; 4) $\Gamma^T_{41}=(0,1)\times\{1\}\times(0,T)$, $\Gamma^T_{42}=(1,2)\times\{1\}\times(0,T)$.

Fig. 1.

Fig. 1.

Our nonlocal boundary conditions (1.5) can be written as follows

$$ \left.\begin{array}{ll} w|_{\Gamma^T_{rl}}=0&(r=3,4; l=1,2),\\ w|_{\Gamma^T_{12}}=\gamma_1w|_{\Gamma^T_{11}},&\\ w|_{\Gamma^T_{22}}=\gamma_2w|_{\Gamma^T_{21}}& \end{array}\right\}.$$

(1.6)

2. Isomorphism of Spaces

We assume that the following conditions hold.

Condition 1.

The set ${\cal K}$ given by formula (1.4) satisfies condition

$$ {\rm mes}_{n-1}({\cal K}\cap\partial Q)=0.$$

(2.1)

Condition 2.

For any subdomain $Q_{sl}$ $(s=1,2,\ldots$, $l=1,\ldots,N(s))$ and an arbitrary $\varepsilon >0$, there exists some open set $G_{sl}\subset Q_{sl}$ with boundary $\partial G_{sl}\in C^1$ such that ${\rm mes}_n(Q_{sl}\setminus G_{sl})<\varepsilon$ and ${\rm mes}_{n-1}(\partial {G_{sl}}\Delta \partial {Q_{sl}})<\varepsilon$.

We consider our problem in the Sobolev space $L_p(0,T;W^1_p(Q))$. This is the set of functions $u\in L_p(\Omega_T)$ having all generalized derivatives $\partial_iu$ in $L_p(\Omega_T)$. It is well known that this space is reflexive and Banach. Let

$$\begin{gathered} \, L_p(0,T;W^1_{p,\gamma}(Q)):=\{w\in L_p(0,T;W^1_p(Q)): w\, \text{satisfies (1.3)}\}, \\ L_p(0,T; {\stackrel{\circ}{W}} _{p}^1(Q)):= \{u\in L_p(0,T;W^1_p(Q)): u|_{x\in\partial Q}=0\,\text{for almost all}\, t\in(0,T)\}, \end{gathered}$$

(2.2)

where $\|u\|^p_{L_p(0,T; {\stackrel{\circ}{W}} _{p}^1(Q))}:=\sum\limits_{1\le i\le n} \int\limits_0^T\int\limits_Q |\partial_i u(x,t)|^p\,dx\,dt$.

Consider a collection of real constant coefficients $\{a_h\in {\mathbb R}: h\in{\cal M}\}$. Define the difference operator $R:L_p(0,T;\mathbb{R}^n)\to L_p(0,T;\mathbb{R}^n)$ by the formula

$$ Ru(x,t)=\sum\limits_{h\in\mathscr{M}} a_h u(x+h,t).$$

(2.3)

We define the operator $R_Q=P_QRI_Q:L_p(\Omega_T)\to L_p(\Omega_T)$, where $I_Q:L_p(\Omega_T)\to L_p(\mathbb{R}^n\times (0,T))$ is the operator of extention for functions from $L_p(\Omega_T)$ by zero to $(\mathbb{R}^n \setminus Q)\times (0,T)$ and we denote by $P_Q:L_p(\mathbb{R}^n\times (0,T))\to L_p(\Omega_T)$ is the operator of restriction for functions from $L_p\left(\mathbb{R}^n\times (0,T)\right)$ to $\Omega_T$.

As mentioned above, the difference operator $Ru(x)=\sum\limits_{h\in\mathscr{M}} a_h u(x+h)$ as well as the operator $R_Q=P_QRI_Q:L_p(Q)\to L_p(Q)$ were studied earlier, see [8, 9, 12] for $p=2$ and [19] for $1<p<\infty$. To simplify the record here and below, we will denote by $R_Q$ both the operators:

$$R_Q:L_p(\Omega_T)\to L_p(\Omega_T)\quad\text{ and}\quad R_Q:L_p(Q)\to L_p(Q).$$

Consequently, it will be clear from the context where we consider $R_Q:L_p(0,T; {\stackrel{\circ}{W}} ^1_p(Q))\to L_p(0,T;W_{p,\gamma}^1(Q))$ and where we consider $R_Q: {\stackrel{\circ}{W}} ^1_p(Q)\to W_{p,\gamma}^1(Q)$. Here we denote by ${\stackrel{\circ}{W}} ^1_p(Q)$ the space of elements from Sobolev space $u\in W^1_p(Q)$ that satisfy the Dirichlet boundary condition $u|_{x\in\partial Q}=0$; and we denote by $W_{p,\gamma}^1(Q)$ the elements from Sobolev space $u\in W^1_p(Q)$ that satisfy nonlocal boundary conditions

$$\left. \begin{array}{rll} w|_{\Gamma_{rl}}&=\sum\limits_{j=1}^{J_{0}} \gamma_{lj}^rw|_{\Gamma_{rj}} &\quad (r\in B,\;l=J_{0}+1,\ldots,J),\\ w|_{\Gamma_{rl}}&=0 &\quad (r\notin B,\;l=1,\ldots,J). \end{array} \right\}$$

Lemma 4.

The linear operators $R:L_p(\mathbb{R}^{n}\times(0,T))\to L_p(\mathbb{R}^{n}\times(0,T))$ and $R_Q:L_p(\Omega_T)\to L_p(\Omega_T)$ are bounded for $1<p<\infty$ .

This statement follows from Lemma 8.1 in [9] and Lemma 1 in [19].

Denote by $L_p\left(\bigcup\limits_lQ_{sl}\times (0,T)\right)$ the subspace of functions from $L_p(\Omega_T)$ that vanish for $x$ not belonging to $\bigcup\limits_lQ_{sl}$ ($l=1,\ldots,N(s)$). Introduce the bounded operator $P_s:L_p(\Omega_T)\to L_p\left(\bigcup\limits_lQ_{sl}\times (0,T)\right)$ by the formula $P_su(x,t)=u(x,t)$ $(x\in\bigcup\limits_l Q_{sl},\,t\in(0,T))$, $P_su(x,t)=0$ $(x\in Q\setminus\bigcup\limits_l Q_{sl},\,t\in(0,T))$. Obviuosly, $P_s$ is a projector to $L_p\left(\bigcup\limits_lQ_{sl}\times(0,T)\right)$. Since ${\rm mes}_n(\partial Q_{sl})=0$, we have

$$ L_p(Q)=\mathop{\dotplus}_s L_p\Biggl(\bigcup_lQ_{sl}\Biggr);\qquad L_p(\Omega_T)=\mathop{\dotplus}_s L_p\Biggl(\bigcup\limits_lQ_{sl}\times (0,T)\Biggr).$$

(2.4)

The following assertion is evident.

Lemma 5.

$L_p\left(\bigcup_lQ_{sl}\times (0,T)\right)$ is an invariant subspace of the operator $R_Q$ .

We define an isomorphism of reflexive Banach spaces

$$U_s:L_p\left(\bigcup\limits_l Q_{sl}\times (0,T)\right)\to L_p^N\biggl(Q_{s1}\times (0,T)\biggr)$$

by the formula

$$ (U_su)_l(x,t)=u(x,t+h_{sl})\qquad(x\in Q_{s1},\,t\in(0,T)),$$

(2.5)

where $l=1,\ldots, N=N(s)$ and the vector $h_{sl}$ is such that

$$Q_{s1}+h_{sl}=Q_{sl}\qquad (h_{s1}=0), \quad L_p^N(Q_{s1})=\prod\limits_lL_p(Q_{s1}), \quad L_p^N\biggl(Q_{s1}\times (0,T)\biggr)=\prod\limits_lL_p\biggl(Q_{s1}\times (0,T)\biggr).$$

We introduce the matrices $R_s=\{r^s_{ml}\}_{1\le m,l\le N(s)}$ by setting

$$ r_{ml}^s=\left\{\begin{array}{ll} a_h& (h=h_{sl}-h_{sm}\in \mathscr{M}),\\ 0 &(h_{sl}-h_{sm}\not\in\mathscr{M}).\end{array}\right.$$

(2.6)

The boundedness of $Q$ and formula (2.6) imply that a number of different matrices is finite. Let $n_1$ denote this number and let $R_{s_\nu}$ denote all different matrices $R_s$ $(\nu=1,\ldots,n_1)$. We define the operator $R_{Qs}:L_p^N\left(Q_{s1}\times (0,T)\right)\to L_p^N\left(Q_{s1}\times (0,T)\right)$ given by

$$ R_{Qs}=U_sR_QU_s^{-1}$$

(2.7)

By virtue of Lemma 8.6 from [9, Ch. II, § 8], $R_{Qs}$ is the operator of multiplication by the matrix $R_s$. Moreover, by virtue of Lemma 8.7 from [9, Ch. II, § 8], the spectrum of $R_Q$ is defined by the spectrums of $R_{s_\nu}$:

$$\sigma (R_Q)=\bigcup\limits_{1\le \nu\le n_1} \sigma (R_{s_\nu}).$$

Corollary 1.

Let $R_s$ $(s=s(r),r\in B)$ be nonsingular. Then the operator $R_Q$ has a bounded inverse operator $R_Q^{-1}:L_p(\Omega_T)\to L_p(\Omega_T)$ . Moreover, the operator $R_{Qs}^{-1}:L_p^N\left(Q_{s1}\times (0,T)\right)\to L_p^N\left(Q_{s1}\times (0,T)\right)$ given by

$$ R_{Qs}^{-1}=U_sR_Q^{-1}U_s^{-1}$$

(2.8)

is the operator of multiplication by the matrix $R_s^{-1}$ .

By Lemma 3, for every $r=1,2,\ldots$, there is a single index $s=s(r)$ such that $N(s)=J(r)$ and $\Gamma_{rl}\subset\partial Q_{sl}$, $l=1,\ldots,N(s)$, after reindexing the subdomains of the $s$th class. Let $R_{s(r)}$ denote the matrices obtained by renumbering the corresponding columns and rows in $R_s\, (s=s(r))$. Let $e^r_j\, (j=1,\dots,J(r))$ be the $j$th row of the $J\times J_0$ matrix obtained by deleting the last $J-J_0$ columns from $R_{s(r)}$. And let $R_{s0}$ denote the $J_0\times J_0$ matrix obtained by deleting the last $N(s)-J_0$ rows and columns from $R_s$.

Definition 1.

We say that the matrices $R_s$ correspond to the boundary conditions (1.3) if the following condition holds:

Condition 3.

There is a collection $\{a_h\in {\mathbb R}: h\in{\cal M}\}$ such that, for any $s=1,2,\dots$, the matrices $R_s$ are nonsingular and for all $r\in B$ and $s=s(r)$ it is true that

$$ e^r_{l}=\sum\limits_{1\le j\le J_0} \gamma^r_{lj} e^r_{j}\qquad (l=J_0+1,\dots, J).$$

(2.9)

Example 2.

We continue to consider problem (1.1), (1.2), (1.5) in the rectangular parallelepiped $\Omega_T=(0,2)\times(0,1)\times(0,T)$, see Example 1. By nonlocal conditions (1.5), we have the set of shifts ${\cal M}=\{(0,0);(1,0);(-1,0)\}$; the partition of the domain $Q=(0,2)\times(0,1)$ contains two subdomains $Q_{11}=(0,1)\times(0,1)$ and $Q_{12}=(1,2)\times(0,1)$ from the same class. The set ${\cal K}$ consists of 6 points ${\cal K}=\{(i,j):i=0,1,2; j=0,1\}$. Conditions 1 and 2 hold. We consider the difference operator

$$Ru(x,t)=u(x,t)+a_1 u(x_1+1,x_2,t)+a_{-1} u(x_1-1,x_2,t).$$

We define the operator $R_Q=P_QRI_Q:L_p(\Omega_T)\to L_p(\Omega_T)$. Clearly, here $R_1=\left( \begin{array}{cc} 1 & a_1 \\ a_{-1} & 1 \end{array}\right)$ and $R_{10}=1$. For any $u\in L_p(0,T; {\stackrel{\circ}{W}} _{p}^1(Q))$ and $w=R_Qu$, we have

$$\begin{aligned} \, w|_{x_1=0}&=R_Qu|_{x_1=0}=a_1u|_{x_1=1},\qquad w|_{x_1=2}=R_Qu|_{x_1=2}=a_{-1}u|_{x_1=1},\\ w|_{x_1=1}&=R_Qu|_{x_1=1}=u|_{x_1=1}.\end{aligned}$$

Thus, if $a_1=\gamma_1$, $a_{-1}=\gamma_2$, and $u\in L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))$, then $w=R_Qu$ satisfies the nonlocal boundary conditions (1.5), i.e. $R_Q(L_p(0,T; {\stackrel{\circ}{W}} _{p}^1(Q)))\subset L_p(0,T;W^1_{p,\gamma}(Q))$, where $\gamma=\{\gamma_1,\gamma_2\}$.

Theorem 1.

Let Conditions 1 – 3 hold and let the corresponding matrices $R_s$ and $R_{s0}$ $(s=s(r),r\in B)$ be nonsingular. Then there exists a set $\gamma=\{\gamma_{lj}^r\}$ such that the operator $R_Q$ is a continuous one-to-one mapping of $L_p(0,T; {\stackrel{\circ}{W}} ^1_p(Q))$ onto $L_p(0,T;W_{p,\gamma}^1(Q))$ .

Proof. Note that the operator $R_Q$ does not depend on $t$. Due to the nonsingularity of the matrices $R_s$ and $R_{s0}$, the operator $R_Q$ is a continuous one-to-one mapping of ${\stackrel{\circ}{W}} ^1_p(Q)$ onto $W_{p,\gamma}^1(Q)$, see Theorem 1.1 in [13] (for $p=2$, see Theorem 8.1 in [9] or Theorem 2.1 in [12]). Therefore, $R_Q$ is a continuous one-to-one mapping of $L_p(0,T; {\stackrel{\circ}{W}} ^1_p(Q))$ onto $L_p(0,T;W_{p,\gamma}^1(Q))$.

3. Operator Equation

Let us consider the generalized derivatives $\partial_t$ with respect to time variable. We assume that the unbounded operator $\partial_t:L_p(0,T;W^1_p(Q)) \supset{\mathscr D}(\partial_t)\to L_q(0,T;W^{-1}_q(Q))$ has the domain

$$ {\mathscr D}(\partial_t):= \{w\in L_p(0,T;W^1_p(Q)): \partial_tw\in L_q(0,T;W^{-1}_q(Q))\}.$$

(3.1)

The nonlinear operator $\Delta_p:L_p(0,T;W^1_{p,\gamma}(Q)) \to L_q(0,T;W^{-1}_q(Q))$ is given by the formula

$$\langle \Delta_p w,v\rangle=\sum\limits_{1\le i\le n}\, \int\limits_{\Omega_T} |\partial_i w|^{p-2}\partial_i w \, \partial_i v\,dx\,dt\qquad\forall v\in L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q)).$$

We introduce the space $W_\gamma$ by the formula

$$ W_\gamma:=\{w\in L_p(0,T;W_{p,\gamma}^1(Q)): \partial_t w\in L_q(0,T;W^{-1}_q(Q)) \}.$$

(3.2)

Definition 2.

The function $w\in W_\gamma$ is called a generalized solution of problem (1.1)–(1.3) if it satisfies the operator equation

$$ \partial_t w+ \Delta_p w =f, \qquad w\in W_\gamma$$

(3.3)

with initial condition (1.2).

We assume that Conditions 1–3 hold and the corresponding matrices $R_s$ and $R_{s0}$ are nonsingular. By virtue of Theorem 1, there exists an isomorphism $R_Q:L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))\to L_p(0,T;W_{p,\gamma}^1(Q))$. Thus, for any $w\in W_\gamma\subset L_p(0,T;W_{p,\gamma}^1(Q))$, there exists an unique element $u\in L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))$ such that $w=R_Qu$ and $u=R^{-1}_Qw$. We introduce the space

$$ W:=\{u\in L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q)): \partial_t u\in L_q(0,T;W^{-1}_q(Q)) \}.$$

(3.4)

Lemma 6.

For any $u\in W$ , $\partial_tR_Q u=R_Q\partial_tu\in L_q(0,T;W^{-1}_q(Q))$ .

Proof. The linear operator $R_Q:L_q(\Omega_T)\to L_q(\Omega_T)$ is bounded, see Lemma 4. For any $u\in W$ such that $\partial_t u\in L_q(\Omega_T)$, we have that $R_Q\partial_t u\in L_q(\Omega_T)$. Obviously, $\partial_tR_Q u=R_Q\partial_tu\in L_q(\Omega_T)$. Here we can use the density of inclution $L_q(\Omega_T)\subset L_q(0,T;W^{-1}_q(Q))$ and the closure of the graph of the bounded linear operator $R_Q$. I.e. for any $u\in W$, there exist the sequence $\{u_n\}\subset W$ such that $\partial_t u_n\in L_q(\Omega_T)$ and $\partial_t u_n\to\partial_t u$ in $L_q(0,T;W^{-1}_q(Q))$. Then $R_Q\partial_tu=\lim\limits_{n\to \infty}R_Q\partial_t u_n=\lim\limits_{n\to \infty}\partial_tR_Qu_n=\partial_tR_Qu.$

From Lemma 6 it follows that $w\in W_\gamma$ if $u\in W$, since $R_Q$ is an isomorphism. By virtue of Theorem 1.17 in [21, Ch.4,§ 1], $u\in C(0,T;L_2(Q))$ and $w\in C(0,T;L_2(Q))$. Thus,

$$\varphi(x):=u(x,0)=R_Q^{-1}w(x,0)=R_Q^{-1}\psi(x).$$

Therefore, $u|_{t=0}=\varphi\in L_2(Q)$ and $w|_{t=0}=\psi\in L_2(Q)$ are well defined. We obtain the following result.

Theorem 2.

Let Conditions 1–3 hold and let the corresponding matrices $R_s$ and $R_{s0}$ $(s=s(r),r\in B)$ be nonsingular. If $u\in W$ is a solution of the operator equation

$$ \partial_t R_Q u + \Delta_pR_Qu =f, \qquad u\in W,$$

(3.5)

with initial condition

$$ u(x,0)=\varphi(x)=R^{-1}_Q\psi(x),$$

(3.6)

then $w=R_Qu$ is a generalized solution of problem (1.1)–(1.3).

Example 3.

We continue to consider the problem (1.1), (1.2), (1.5) with Bitsadze–Samarskii nonlocal boundary conditions in the rectangular parallelepiped $\Omega_T=(0,2)\times(0,1)\times(0,T)$. As it was proved in the example 2, the corresponding operator $R_Q$ is defined by the matrix $R_1=\left( \begin{array}{cc} 1 & \gamma_1 \\ \gamma_2 & 1 \end{array}\right).$ This matrix is nonsingular if $\gamma_1\gamma_2\not=1$; the matrix $R_{10}=(1)$ is nonsingular also. Thus, in order to solve of the problem (1.1), (1.2), (1.5), we can study the following problem:

$$\partial_t R_Q u(x,t) +\Delta_p R_Qu(x,t)=f(x,t)\qquad ((x,t)\in\Omega_T),$$

(3.7)

$$u(0,x)=\varphi(x)=R_Q^{-1}\psi(x) \qquad (x\in Q),$$

(3.8)

$$u(x,t)=0 \qquad ((x,t)\in\partial Q\times(0,T)).$$

(3.9)

4. Properties of difference operators

Here we consider the properties of the operator $R_Q$ and its conjugate operator $R_Q^*$.

Lemma 7 cf. Lemma 8.2 in [9]..

The operator $R:L_p({\mathbb R}^n\times (0,T))\to L_p({\mathbb R}^n\times (0,T))$ is bounded, and

$$R^*u(x,t)=\sum\limits_{h\in\mathscr{M}} a_h u(x-h,t).$$

Let $G\subset {\mathbb R}^n$ be a domain such that $G\subset Q$, and let $G_T:=G\times (0,T)$. We denote by $W_p^{1,0}(G_T)$ the anisotropic Sobolev space consisting of functions $u\in L_p(G_T)$ which have all generalized derivatives $\partial_i u$ in $L_p(G_T)$ with the norm

$$\|u\|_{W_p^{1,0}(G_T)}=\left\{\sum\limits_{1\le i\le n} \int\limits_{G_T} |\partial_i u|^p\,dt\,dx\,+ \int\limits_{G_T} |u|^p\,dt\,dx\right\}^{1/p}.$$

The space $W_p^{1,0}(G_T)$ can be identified with the space $L_p(0,T;W^1_p(G))$. It is also clear that

$$L_p(0,T; {\stackrel{\circ}{W}} ^1_p(Q))=\left\{u\in W_p^{1,0}(\Omega_T): u|_{\partial Q\times (0,T)}=0\right\}.$$

As it was mentioned in Section 2, the operator $R_{Qs}:L_p^N\left(Q_{s1}\times (0,T)\right)\to L_p^N\left(Q_{s1}\times (0,T)\right)$ defined by the equality $R_{Qs}=U_sR_QU_s^{-1}$ is the operator of multiplication by the matrix $R_s=\{r^s_{ml}\}$. At the same time, the operator $R^*_{Qs}:L_p^N\left(\Omega_{s1}\right)\to L_p^N\left(\Omega_{s1}\right)$ defined by

$$ R^*_{Qs}=U_sR^*_QU_s^{-1},$$

(4.1)

is the operator of multiplication by the transposed matrix $R^*_s=\{r^s_{lm}\}$ to the matrix $R_s=\{r^s_{ml}\}$ of order $N(s)\times N(s)$.

From Lemmas 8.13, 8.14 in [9] and Lemma 5 in [19], we obtain

Lemma 8.

The operators $R_Q:L_p(0,T; {\stackrel{\circ}{W}} ^1_p(Q))\to L_p\left(0,T; W^1_p(Q)\right)$ and $R^*_Q:L_p(0,T; {\stackrel{\circ}{W}} ^1_p(Q))\to L_p\left(0,T; W^1_p(Q)\right)$ are continuous. Moreover,

$$ \partial_i(R_Qu)=R_Q(\partial_iu),\qquad \partial_i(R^*_Qu)=R^*_Q(\partial_iu).$$

(4.2)

From Lemma 8.15 in [9] and Lemma 6 in [19], we obtain

Lemma 9.

For all $u\in L_p\left(0,T; W^1_p(Q)\right)$ , we have $R_Qu\in L_p\left(0,T; W^1_p(Q_{sl})\right)$ and

$$R_{Q} =\sum\limits_{s} U_s^{-1}R_{s}U_{s} P_{s},$$

(4.3)

$$\|R_Qu\|_{L_p\left(0,T; W^1_p(Q_{sl})\right)} \le c_1\sum_{j=1}^{N(s)}\|u\|_{L_p\left(0,T; W^1_p(Q_{sj})\right)} \quad \left(s=1,2,\ldots;\,l=1,\ldots,N(s)\right).$$

(4.4)

If $\det R_{s_\nu}\ne 0$ $(\nu=1,\ldots,n_1)$ , then there exists an inverse operator $R_{Q}^{-1}:L_p(\Omega_T)\to L_p(\Omega_T)$ and $R_Q^{-1}w$ belongs to $L_p\left(0,T; W^1_p(Q_{sl})\right)$ for all $w\in L_p\left(0,T; W^1_p(Q)\right)$ ; moreover,

$$R_{Q}^{-1} =\sum\limits_{s} U_s^{-1}R_{s}^{-1}U_{s} P_{s},$$

(4.5)

$$\|R_Q^{-1}w\|_{ L_p\left(0,T; W^1_p(Q_{sl})\right)} \le c_2\sum\limits_{j=1}^{N(s)}\|w\|_{ L_p\left(0,T; W^1_p(Q_{sj})\right)} \quad \left(s=1,2,\ldots;\, l=1,\ldots,N(s)\right).$$

(4.6)

Here the constants $c_1,c_2>0$ are independent of $s$ , $u$ , and $w$ .

We also consider symmetric and skew-symmetric parts of the operator $R_Q$:

$$R^{sym}_Q:=\dfrac12(R_Q+R_Q^*),\qquad R^{sk}_Q:=\dfrac12(R_Q-R_Q^*).$$

Obviously, $R^{sym}_{Qs}$ and $R^{sk}_{Qs}$ are the operators of multiplication by the matrices

$$R^{sym}_{s}= \dfrac12(R_s+R_s^*)\qquad\text{and}\qquad R^{sk}_{s}= \dfrac12(R_s-R_s^*),$$

respectively. For these operators, the properties from Lemmas 8 and 9 hold.

In a standard way, we define scalar products in the spaces $L_2(Q)$ and $L_2(\Omega_T)$:

$$\left(u,v\right)_{L_2(Q)}:=\int\limits_Q u(x)\,v(x)\,dx,\qquad \left(u,v\right)_{L_2(\Omega_T)}:=\int\limits_{\Omega_T} u(x,t)\,v(x,t)\,dx\,dt.$$

Definition 3.

The operator $R_Q:L_2(Q)\to L_2(Q)$ is positive definite if there exists $c_3>0$ such that

$$\left(R_Qu,u\right)_{L_2(Q)}\ge c_3 \|u\|^2_{L_2(Q)} \qquad\qquad \forall u\in {L_2(Q)}.$$

Remark 1.

$R_Q$ and $R_Q^{sym}$ are positive definite if and only if $R_{s}^{sym}>0$ for any $s$, see Lemma 8.8 in [9] or Lemma 2.8 in [12]. Moreover, if $R_{s}^{sym}>0$, then $R_s$ and $R_{s0}$ are nonsingular.

Lemma 10.

If $R_s^{sym}>0$ for any $s$ , then there exists a $c_{3}>0$ such that

$$\begin{gathered} \, \left( R_Qu(t),u(t)\right)_{L_2(Q)}=\left( R_Q^{sym}u(t),u(t)\right)_{L_2(Q)}= \sum\limits_s \left\|\sqrt{R_{Qs}^{sym}}\,U_sP_su(t)\right\|^2_{L_2^N(Q_{s1})}\\ = \left\|\sqrt{R_{Q}^{sym}}\,u(t)\right\|^2_{L_2(Q)}\ge c_{3}\|u(t)\|^2_{L_2(Q)} \qquad\forall u(t) \in L_2(Q). \end{gathered}$$

(4.7)

Proof. Clearly,

$$\left( R_Qu(t),u(t)\right)_{L_2(Q)}= \left(u(t),R_Q^*u(t)\right)_{L_2(Q)} =\dfrac12\left( (R_Q+R_Q^*)u(t),u(t)\right)_{L_2(Q)} =\left( R_Q^{sym}u(t),u(t)\right)_{L_2(Q)}.$$

From here we obtain the first equality in (4.7). By virtue of the relation $R_s^{sym}>0$, a positive $\sqrt{R_s^{sym}}$ exists. From formulas (2.7), (4.1), and (4.3), we derive

$$\begin{gathered} \, \left(R_Q^{sym}u(t),u(t)\right)_{L_2(Q)}=\sum_s \left(R_s^{sym} U_sP_su(t), U_sP_su(t)\right)_{L_2^{N}(Q_{s1})} \\ =\sum\limits_s \, \left( \sqrt{R_s^{sym}} U_sP_su(t), \sqrt{R_s^{sym}}U_sP_su(t)\right)_{L_2^{N}(Q_{s1})} = \sum\limits_s \left\|\sqrt{R_{Qs}^{sym}}\,U_sP_su(t)\right\|^2_{L_2^{N}(Q_{s1})}. \end{gathered}$$

The symmetrical difference operator $R_Q^{sym}$ is positive definite if and only if $R_s^{sym}>0$ for any $s$, see Lemma 8.8 in [9]. Estimate (4.7) is proved. $\qquad\square$

Corollary 2.

If $R_s^{sym}>0$ for any $s$ , then there exists a $c_{4}>0$ such that

$$\begin{gathered} \, \left( R_Qu,u\right)_{L_2(\Omega_T)}=\left( R_Q^{sym}u,u\right)_{L_2(\Omega_T)}= \sum\limits_s \left\|\sqrt{R_{Qs}^{sym}}\,U_sP_su\right\|^2_{L_2^N(Q_{s1}\times(0,T))}\\ \ge c_{4}\|u\|^2_{L_2(\Omega_T)}\qquad \qquad \forall u\in L_2(\Omega_T). \end{gathered}$$

(4.8)

Definition 4.

A linear operator $\Lambda:L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q)) \supset{\mathscr D}(\Lambda)\to L_q(0,T;W^{-1}_q(Q))$ is monotone if

$$\langle \Lambda u,u\rangle\ge 0\qquad\forall u\in {\cal D} (\Lambda).$$

A linear densely defined monotone operator $\Lambda$ is maximally monotone if there is no linear monotone operator that is a strict extension of $\Lambda$.

As is known, in reflexive strictly convex spaces together with its conjugate, the maximal monotonicity of the operator is equivalent to the condition:

$$ \langle \Lambda u,u\rangle\ge 0\quad\forall u\in{\mathscr D}(\Lambda),\qquad \langle \Lambda^* v,v\rangle\ge 0\quad\forall v\in{\mathscr D}(\Lambda^*),$$

(4.9)

see Lemma 1.1 [18, Chapter 3]. It is well known that the operator $\Lambda=\partial_t$ with the domain

$$ {\cal D}(\Lambda)=\{u\in L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q)): \Lambda u\in L_q(0,T;W^{-1}_q(Q)), \, u|_{t=0}=0\},$$

(4.10)

is maximal monotone, $\partial_t^*=-\partial_t$, and ${\cal D}(\Lambda^*)= \{v\in L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q)): \partial_t v\in L_q(0,T;W^{-1}_q(Q)), \, v|_{t=T}=0\}$, see [18, Chapter 3].

Theorem 3.

Let $R_s^{sym}>0$ for any $s$. Then the operator $\Lambda =\partial_t R_Q$ with the domain given by (4.10) is maximal monotone.

Proof. According to the rules of differentiation and by virtue of Lemma 6, we have

$$\begin{gathered} \, \partial_t \left(R_Qu(t),u(t)\right)_{L_2(Q)} = \left(\partial_t R_Qu(t),u(t)\right)_{L_2(Q)} + \left(R_Qu(t),\partial_t u(t)\right)_{L_2(Q)}\\ = \left( R_Q\partial_t u(t),u(t)\right)_{L_2(Q)} + \left(u(t),R_Q^*\partial_t u(t)\right)_{L_2(Q)} = 2 \left(\partial_t R_Q^{sym}u(t),u(t)\right)_{L_2(Q)}\end{gathered}$$

for any $u\in W$. Since $\langle \partial_t R_Q^{sk}u,u\rangle=0$, we obtain

$$\begin{gathered} \, \langle \partial_t R_Qu,u\rangle=\langle \partial_t R_Q^{sym}u,u\rangle= \int\limits_0^T \left(\partial_t R_Q^{sym}u(\tau), u(\tau)\right)_{L_2(Q)}\,d\tau\\ = \dfrac12\left(R_Q^{sym}u(T),u(T)\right)_{L_2(Q)}- \dfrac12\left(R_Q^{sym}u(0),u(0)\right)_{L_2(Q)}\qquad\forall u\in W. \end{gathered}$$

(4.11)

If $u\in {\cal D}(\Lambda)$, we can use estimate (4.7):

$$\langle \partial_t R_Qu,u\rangle=\dfrac12\left(R_Q^{sym}u(T),u(T)\right)_{L_2(Q)}\ge \dfrac{c_3}2 \|u(T)\|^2_{L_2(Q)}.$$

On the other hand, $\Lambda^*=-\partial_tR^*$ has the domain

$${\cal D}(\Lambda^*)=\{v\in L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q)): \partial_t R^* v\in L_q(0,T;W^{-1}_q(Q)), \, v|_{t=T}=0\},$$

and by (4.7) and (4.11), we get

$$\langle (\partial_t R_Q)^*v,v\rangle=- \langle \partial_t R_Q^* v,v\rangle= -\langle \partial_t R_Q^{sym}v,v\rangle\ge \dfrac{c_3}2 \|v(0)\|^2_{L_2(Q)}\ge 0$$

for any $v\in {\cal D}(\Lambda^*)$. Condition (4.9) is fulfilled.

5. Properties of the operator $\Delta_pR_Q$

Let us consider properties of the operator $\Delta_pR_Q:L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))\to L_q(0,T;W_q^{-1}(Q))$ given by

$$ \langle \Delta_pR_Qu,v\rangle=\sum\limits_{1\le i\le n}\, \int\limits_{\Omega_T}\, | \partial_i R_Qu|^{p-2} \partial_i R_Qu\, \partial_i v\, dx\,dt\qquad\forall u,v\in L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q)).$$

(5.1)

In order to state some definitions, we denote by $X$ a reflexive Banach space; $X^*$ is conjugate to $X$.

Definition 5.

An operator $A:X\to X^*$ is demicontinuous if it is continuous from the strong topology of $X$ to the weak topology of $X^*$.

A nonlinear analog of the notion of the nonnegative definiteness of an operator is accretivity. To define it, it is necessary to introduce the duality operator. Recall that a mapping $J:X\to X^*$ from a Banach space $X$ to the dual space $X^*$ is called the duality mapping with respect to a function $\Phi$ if

$$\langle J u, u\rangle_X=\|J u\|_{X^*}\|u\|_X\qquad\text{and} \qquad\|J u\|_{X^*}=\Phi(\|u\|_X)\qquad\text{for all}\,u\in X,$$

see [18, Sec. 2.2]. Note that the standard duality operator $J$ for the Lebesgue space $L_p(\Omega_T)$, which is given by $Ju = |u|^{p-2}u$, is the duality mapping with respect to the function $\Phi(r) = r^{p-1}$. By construction, $J$ is bounded. Moreover, $J$ is demicontinuous, see Proposition 2.4 in [18, Ch. 2, § 2].

Definition 6.

A linear operator $\hat R_Q:L_p(\Omega_T)\to L_p(\Omega_T)$ is said to be accretive if

$$\langle J u, \hat R_Q u\rangle := \int\limits_{\Omega_T} J u\, \hat R_Q u\, dx\,dt\,\ge 0\qquad \text{for any } u\in L_p(\Omega_T).$$

$\hat R_Q:L_p(\Omega_T)\to L_p(\Omega_T)$ is strongly accretive if there exists $c_5>0$ such that

$$\langle J u, \hat R_Q u\rangle \ge c_5\|u\|_{L_p(\Omega_T)}^p.$$

Further we assume that $\hat R_Q=R^{-1}_Q$. We denote $\hat R_s=R^{-1}_s$, $s=1,\ldots,n_1$.

We need the following auxiliary result.

Lemma 11.

Let $\lambda,a,b\in {\mathbb R}_+$, $p>2$. Then

$$ a^p\pm \lambda ab\left(a^{p-2}-b^{p-2}\right)+b^p\ge 0 \qquad\forall a,b\in {\mathbb R}_+$$

(5.2)

if $\lambda\in {\mathbb R}_+$ is such that

$$ \dfrac{\lambda^p}{\lambda+1}\le \dfrac{p^p}{(p-1)^{p-1}}.$$

(5.3)

In particular, estimate (5.2) is true if

$$ \lambda \le \dfrac p{p-1}\,\sqrt[p-1]p.$$

(5.4)

Proof. By virtue of the symmetry in formula (5.2), without loss of generality we can assume that $a\ge b$. Then we consider the function

$$a^p\pm \lambda ab\left(a^{p-2}-b^{p-2}\right)+b^p= a^p\left(1\pm \lambda \dfrac ba\mp \lambda \left(\dfrac ba\right)^{p-1}+ \left(\dfrac ba\right)^{p}\right).$$

Since $p>2$ and $b/a\le 1$, we obtain $b/a \ge \left(b/a\right)^{p-1}$ and

$$1+ \lambda \dfrac ba - \lambda \left(\dfrac ba\right)^{p-1}+ \left(\dfrac ba\right)^{p}\ge 1+ \left(\dfrac ba\right)^{p}>0.$$

Now let us consider the left part of (5.2) with the opposite signs before $\lambda$. Since $\left(b/a\right)^{p-1}\ge \left(b/a\right)^{p}$, we have

$$1- \lambda \dfrac ba + \lambda \left(\dfrac ba\right)^{p-1}+ \left(\dfrac ba\right)^{p}\ge 1- \lambda \dfrac ba + (\lambda+1)\left(\dfrac ba\right)^{p}.$$

Using the well-known formula $ab\le a^p/p+b^q/q$, we obtain

$$1- \lambda \dfrac ba + (\lambda+1)\left(\dfrac ba\right)^{p}\ge 1-\dfrac{\lambda^q}{q\varepsilon^q} -\dfrac{\varepsilon^p}p\left(\dfrac ba\right)^{p}+(\lambda+1)\left(\dfrac ba\right)^{p}\ge 0$$

if $1-{\lambda^q}/{q\varepsilon^q}\ge 0$ and $(\lambda+1)\ge {\varepsilon^p}/p$. We put $\varepsilon={\lambda}/{q^{1/q}}$. Then $(\lambda+1)\ge {\varepsilon^p}/p=\lambda^p p^{-1}q^{-p/q}$. Hence

$$\dfrac{\lambda^p}{\lambda+1}\le pq^{p/q}=\dfrac{p^p}{(p-1)^{p-1}}.$$

Therefore, (5.3) implies that (5.2) holds.

Obviously, if $\lambda \le \dfrac p{p-1}\,\sqrt[p-1]p$, then $\lambda^{p-1} \le p\left(\dfrac p{p-1}\right)^{p-1}$, i.e. $\dfrac{\lambda^p}{\lambda+1}\le \dfrac{p^p}{(p-1)^{p-1}}.$

Lemma 12.

Let the operator $\hat R_Q:L_p(\Omega_T)\to L_p(\Omega_T)$ be such that corresponding matrices $\hat R_s=\{\hat r_{ml}^s\}$ satisfy the following conditions: for any $s=1,\dots,n_1$ and for arbitrary $m=1,2,\dots, N(s)$,

$$ 2 \hat r^{s}_{mm}> \lambda^{-1}\sum\limits_{l\not=m}|\hat r^{s}_{ml}-\hat r^{s}_{lm}|+ \sum\limits_{l\not=m}|\hat r^{s}_{ml}+\hat r^{s}_{lm}|$$

(5.5)

where $\lambda$ satisfies estimate (5.3) or estimate (5.4). Then $\hat R_Q$ is strongly accretive.

Proof. By virtue of (4.5), we have the representation

$$\begin{gathered} \, \langle Ju,\hat R_Qu\rangle = \sum\limits_{s}\int\limits_{\Omega_{s}}\, \left|P_su\right|^{p-2}(P_s u)\, \left(U_s^{-1}\hat R_{s}U_{s}P_{s}u\right)\, dx\,dt\\ =\sum\limits_{s}\int\limits_{\Omega_{s1}} \biggl(U_s\left(|P_su|^{p-2}(P_su)\right), \hat R_{s} U_{s}P_{s}u\biggr)\,dx\,dt\\ =\sum\limits_{s}\sum\limits_{1\le m,l\le N(s)}\int\limits_{\Omega_{s1}} \hat r^{s}_{ml} |u(x+h_{sm},t)|^{p-2} u(x+h_{sm},t) u(x+h_{sl},t) \,dx\,dt. \end{gathered}$$

(5.6)

Let us evaluate the integrand in (5.6). Let $\xi\in {\mathbb R}^{N(s)}$ be an arbitrary vector. We denote by $\hat r^{s,sym}_{ml}=\dfrac12\bigl(\hat r^s_{ml} +\hat r^s_{lm}\bigr)$ and $\hat r^{s,sk}_{ml}=\dfrac12\bigl(\hat r^s_{ml} -\hat r^s_{lm}\bigr)$ the elements of the symmetric and skew-symmetric parts of $\hat R_s$. Then

$$\begin{gathered} \, \sum\limits_{1\le m,l\le N(s)} \hat r^{s}_{ml} |\xi_m|^{p-2}\xi_m\xi_l= \sum\limits_{1\le m,l\le N(s)} \hat r^{s,sk}_{ml} |\xi_m|^{p-2}\xi_m\xi_l +\sum\limits_{1\le m,l\le N(s)} \hat r^{s,sym}_{ml} |\xi_m|^{p-2}\xi_m\xi_l\\ \ge \lambda^{-1}\sum\limits_{1\le m<l\le N(s)} \bigl|\hat r^{s,sk}_{ml}\bigr| \bigl( |\xi_m|^p-\lambda |\xi_m\xi_l|\left||\xi_m|^{p-2}-|\xi_l|^{p-2}\right|+ |\xi_l|^p \bigr) \\+ \sum\limits_{1\le m<l\le N(s)} \bigl|\hat r^{s,sym}_{ml}\bigr| \bigl( |\xi_m|^p+{\rm sign}(\hat r^{s,sym}_{ml})|\xi_m|^{p-2}\xi_m\xi_l +{\rm sign}(\hat r^{s,sym}_{ml})|\xi_l|^{p-2}\xi_m\xi_l+ |\xi_l|^p \bigr) \\ +\sum\limits_{1\le m\le N(s)} \hat r^{s}_{mm} |\xi_m|^{p}- \lambda ^{-1} \sum\limits_{1\le m<l\le N(s)} \bigl|\hat r^{s,sk}_{ml}\bigr| \bigl( |\xi_m|^p+ |\xi_l|^p \bigr)\\ -\sum\limits_{1\le m<l\le N(s)} \bigl|\hat r^{s,sym}_{ml}\bigr| \bigl( |\xi_m|^p+ |\xi_l|^p \bigr). \end{gathered}$$

(5.7)

The first summand in the right part of (5.7) is nonnegative by virtue of estimate (5.2) in Lemma 11. The second summand in the right-hand side of (5.7) is nonnegative because

$$|\xi_m|^p\pm|\xi_m|^{p-2}\xi_m\xi_l \pm|\xi_l|^{p-2}\xi_m\xi_l+ |\xi_l|^p\ge (|\xi_m|^{p-1}-|\xi_l|^{p-1})(|\xi_m|-|\xi_l|)\ge 0.$$

Thus, we obtain

$$ \sum\limits_{1\le m,l\le N(s)} \hat r^{s}_{ml} |\xi_m|^{p-2}\xi_m\xi_l \ge \sum\limits_{1\le m\le N(s)} \left( \hat r^{s}_{mm} -\lambda^{-1}\sum\limits_{l\not=m}\left|\hat r^{s,sk}_{ml}\right|- \sum\limits_{l\not=m}\left|\hat r^{s,sym}_{ml}\right|\right)|\xi_m|^{p}.$$

(5.8)

Substituting estimate (5.8) into (5.6), we obtain

$$\begin{gathered} \, \langle Ju,\hat R_Qu\rangle = \sum\limits_{s}\int\limits_{\Omega_{s}}\, \left|P_su\right|^{p-2}(P_s u)\, \left(U_s^{-1}\hat R_{s}U_{s}P_{s}u\right)\, dx\,dt\\ \ge \sum\limits_{s}\int\limits_{\Omega_{s1}} \sum\limits_{1\le m\le N(s)} \left( \hat r^{s}_{mm} -\lambda^{-1}\sum\limits_{l\not=m}\left|\hat r^{s,sk}_{ml}\right|- \sum\limits_{l\not=m}\left|\hat r^{s,sym}_{ml}\right|\right) |u(x+h_{sm},t)|^{p} \,dx\,dt\\ =\sum\limits_{s} \sum\limits_{1\le m\le N(s)} \left( \hat r^{s}_{mm} -\lambda^{-1}\sum\limits_{l\not=m}\left|\hat r^{s,sk}_{ml}\right|- \sum\limits_{l\not=m}\left|\hat r^{s,sym}_{ml}\right|\right) \|u\|^p_{L_p(\Omega_{sm})}, \end{gathered}$$

(5.9)

where $\Omega_{sm}=Q_{sm}\times (0,T)$, $Q_{sm}=Q_{s1}+h_{sm}$. Obviously, if condition (5.5) holds, the operator $\hat R_Q$ is strongly accretive with constant

$$ c_5= \min\limits_m\left\{ \hat r^{s}_{mm} -\lambda^{-1}\sum\limits_{l\not=m}|\hat r^{s,sk}_{ml}|- \sum\limits_{l\not=m}|\hat r^{s,sym}_{ml}|\right\}.$$

(5.10)

Remark 2.

Let $\lambda_{max}(p)$ be such that $\dfrac{\lambda_{max}(p)^p}{\lambda_{max}(p)+1}= \dfrac{p^p}{(p-1)^{p-1}}$. By virtue of inequality (5.3), the function $\lambda_{max}(p)$ monotonously decreases if $p$ increases, $\lambda_{max}(p)\in (1,2+2\sqrt2]$ for $p\ge 2$, and $\lambda_{max}(p)\to 1$ as $p\to \infty$.

Example 4.

Let $Ru(x,t)= u(x_1,x_2,t)-u(x_1+1,x_2,t)+u(x_1+2,x_2,t)$, $Q=(0,3)\times(0,1)$. Here the matrix

$$R_1=\left( \begin{array}{ccc} 1 & -1 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{array}\right)$$

corresponds to the operator $R_Q$. The inverse matrix is given by

$$R_1^{-1}=\left( \begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 1\\ 0 & 0 & 1 \end{array}\right).$$

For any $p\in (2,\infty)$, there exists a $\varepsilon>0$ such that $\lambda=1+\varepsilon$ satisfies inequality (5.3). Then

$$2\cdot 1>(1+\varepsilon)^{-1}|1-0|+|1+0|,\qquad 2\cdot 1>(1+\varepsilon)^{-1}|0-1|+|0+1|.$$

Condition (5.5) holds, i.e., the operator $R_Q^{-1}$ is strongly accretive for any $p\in(2,\infty)$.

Below we use the following properties of the duality operator and of accretive operators.

Lemma 13.

Let $u_m\rightharpoonup u$ weakly in $L_p(\Omega_T)$ . Then $J(u_m)\rightharpoonup J(u)$ weakly in $L_q(\Omega_T)$ and

$$ \mathop{\underline{\rm lim}}\nolimits \limits_{m\to\infty}\langle Ju_m, u_m\rangle\ge \langle Ju,u\rangle.$$

(5.11)

Moreover, for any strongly accretive operator $\hat R_Q$ ,

$$ \mathop{\underline{\rm lim}}\nolimits \limits_{m\to\infty}\langle Ju_m,\hat R_Q u_m\rangle\ge \langle Ju, \hat R_Qu\rangle.$$

(5.12)

The proof coincides with the proof of the Lemmas 5–7 in [22], since the conditions imposed in the present paper on the $\Omega_T$ coincide with the conditions imposed on the $Q$ in [22].

Definition 7.

An operator $A:X\to X^*$ is coercive if $\,\langle Au,u-\tilde u\rangle\ge c(\|u-\tilde u\|_X)\|u-\tilde u\|_{X}$ for some fixed $\tilde u\in X$, where $c:{\mathbb R}_+\to {\mathbb R}$ and $c(s)\to \infty$ as $s\to \infty.$

Definition 8.

$A:X\to X^*$ is pseudomonotone if for any $u_m\rightharpoonup u$ weakly in $X$ such that $\mathop{\overline{\rm lim}}\nolimits \limits_{m\to\infty}\langle Au_m,u_m-u\rangle\le 0$, we have

$$ \mathop{\underline{\rm lim}}\nolimits \limits_{m\to\infty}\langle Au_m,u_m-\xi\rangle\ge \langle Au,u-\xi\rangle \qquad\forall\xi\in X.$$

(5.13)

Lemma 14.

The operator $\Delta_p R_Q:L_p(0,T;R_Q( {\stackrel{\circ}{W}} _1^p(Q)) \to L_q(0,T;W^{-1}_q(Q))$ is bounded and demicontinuous.

Proof. The linear operator $R_Q:L_p(0,T;R_Q( {\stackrel{\circ}{W}} _1^p(Q)) \to L_p\left(0,T;W_{p,\gamma}^1(Q)\right)$ is bounded, see Lemmas 4 and 8; $\Delta_p:L_p\left(0,T;R_Q(W_{p,\gamma}^1(Q)\right)\to L_q(0,T;W^{-1}_q(Q))$ is bounded and demicontinuous, see, for example, [18, Ch. 2, § 1.1–1.2]. Thus, their composition is a bounded, demicontinuous operator.

Theorem 4.

Let $R_Q:L_p(\Omega_T)\to L_p(\Omega_T)$ be nonsingular, and let $R^{-1}_Q:L_p(\Omega_T)\to L_p(\Omega_T)$ be strongly accretive. Then $\Delta_pR_Q$ is pseudomonotone and coercive.

Proof. Since the difference operator $R_Q:L_p(\Omega_T)\to L_p(\Omega_T)$ is nonsingular, it follows that there exists an inverse operator $R_Q^{-1}:L_p(\Omega_T)\to L_p(\Omega_T)$. Moreover, by virtue of (5.1), for all $u,\xi\in L_p(0,T; {\stackrel{\circ}{W}} _p^{1}(Q))$ and $w,\zeta\in L_p(0,T;W_{p,\gamma}^{1}(Q))$ such that $w=R_Qu,\, \zeta=R_Q\xi$, we have

$$ \langle \Delta_pR_Q u,\xi\rangle =\sum\limits_{1\le i\le n} \langle J\partial_iR_Q u,\partial_i\xi\rangle=\sum\limits_{1\le i\le n} \langle J\partial_i w,R_Q^{-1}\partial_i\zeta\rangle.$$

(5.14)

Here we have taken into account the commutativity of the difference operator $R_Q$ with constant coefficients $a_h$ and the differential operator $\partial_i$ (see Lemma 10): $R_Q^{-1}\partial_i\zeta=R_Q^{-1}\partial_iR_Q\xi=R_Q^{-1}R_Q\partial_i\xi= \partial_i\xi$.

Suppose that $u_m\rightharpoonup u$ weakly in $L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))$ and

$$\mathop{\overline{\rm lim}}\nolimits \limits_{m\to\infty}\langle \Delta_pR_Qu_m,u_m-u\rangle \le 0.$$

Consider the sequence $\{w_m=R_Qu_m\}$. By virtue of equalities (4.2),

$$\begin{gathered} \, \lim\limits_{m\to\infty} \langle \partial_i w_m,\xi \rangle= \lim\limits_{m\to\infty} \langle \partial_i R_Qu_m,\xi\rangle= \lim\limits_{m\to\infty} \langle R_Q \partial_i u_m,\xi\rangle= \lim\limits_{m\to\infty} \langle \partial_i u_m,R^*_Q\xi \rangle \\ = \langle \partial_i u,R^*_Q\xi\rangle= \langle \partial_i R_Q u,\xi\rangle= \langle \partial_iw,\xi\rangle\qquad\forall \xi\in L_q(\Omega_T), \end{gathered}$$

(5.15)

i.e., $\partial_iw_m\rightharpoonup \partial_iw=R_Q \partial_iu$ weakly in $L_p(\Omega_T)$ for any $i=1,\dots,n$. By virtue of (5.14), we also have

$$ \mathop{\overline{\rm lim}}\nolimits \limits_{m\to\infty} \sum\limits_{1\le i\le n}\langle J\partial_iw_m,R_Q^{-1}\partial_i( w_m-w)\rangle= \mathop{\overline{\rm lim}}\nolimits \limits_{m\to\infty}\langle \Delta_pR_Qu_m,u_m-u\rangle\le 0.$$

(5.16)

Thus, there exists at least one index $i=i_1$ such that

$$ \mathop{\overline{\rm lim}}\nolimits \limits_{m\to\infty}\langle J\partial_{i}w_m,R_Q^{-1}\partial_{i}( w_m-w)\rangle \le 0.$$

(5.17)

i.e., using the weak convergence $J\partial_{i}w_m\rightharpoonup J\partial_{i}w$ in $L_q(\Omega)$, see Lemma 13, we have

$$\mathop{\overline{\rm lim}}\nolimits \limits_{m\to\infty}\langle J\partial_{i}w_m,R_Q^{-1}\partial_i w_m\rangle \le \lim\limits_{m\to\infty} \langle J\partial_{i}w_m,R_Q^{-1}\partial_i w\rangle= \langle J\partial_iw,R_Q^{-1}\partial_i w\rangle.$$

On the other hand, by virtue of strong accretivity of $R_Q^{-1}$, we get

$$\mathop{\underline{\rm lim}}\nolimits \limits_{m\to\infty}\langle J\partial_{i}w_m,R_Q^{-1}\partial_i w_m\rangle\ge \langle J\partial_{i}w,R_Q^{-1}\partial_i w\rangle, \qquad i=i_1,$$

see estimate (5.12) in Lemma 13. Thus, for $i=i_1$,

$$ \lim\limits_{m\to\infty}\langle J\partial_{i}w_m,R_Q^{-1}\partial_i w_m\rangle= \langle J\partial_{i}w,R_Q^{-1}\partial_i w\rangle.$$

(5.18)

According (5.16) and (5.18), we have

$$ \mathop{\overline{\rm lim}}\nolimits \limits_{m\to\infty}\sum\limits_{i\ne i_1}\langle J \partial_i w_m, R_Q^{-1}(\partial_iw_m-\partial_iw)\rangle\le 0.$$

(5.19)

Repeating the arguments used in the proof of (5.18), we see that (5.19) implies

$$\lim\limits_{m\to\infty}\langle J \partial_i w_m,R_Q^{-1}\partial_i w_m\rangle= \langle J \partial_i w,R_Q^{-1}\partial_iw\rangle\qquad\text{for some}\,i=i_2\not=i_1.$$

Thus, in a finite number of steps, we conclude that

$$\lim\limits_{m\to\infty}\langle J \partial_i w_m,R_Q^{-1}\partial_iw_m\rangle= \langle J \partial_i w,R_Q^{-1}\partial_iw\rangle\qquad\forall i=1,\dots,n.$$

Therefore,

$$ \lim\limits_{m\to\infty}\langle \Delta_pR_Qu_m,u_m\rangle= \lim\limits_{m\to\infty} \sum\limits_{1\le i\le n}\langle J\partial_iw_m,R_Q^{-1}\partial_i w_m\rangle =\sum\limits_{1\le i\le n}\langle J\partial_iw ,R_Q^{-1}\partial_i w\rangle= \langle \Delta_pR_Qu,u\rangle.$$

(5.20)

Pseudomonotonicity of the operator $\Delta_pR_Q$ is proved.

Let us show that the operator $\Delta_pR_Q$ is coercive. Since the operator $R_Q$ commutes with the $\partial_i$ and the operator $R_Q^{-1}$ is strongly accretive, we have

$$\langle \Delta_pR_Qu,u\rangle= \sum\limits_{1\le i\le n}\langle J\partial_iR_Qu,R_Q^{-1}\partial_iR_Qu\rangle =\sum\limits_{1\le i\le n}\langle J\partial_i w,R_Q^{-1}\partial_iw\rangle\\ \ge c_5\sum\limits_{1\le i\le n}\|\partial_iw\|^p_{L_p(\Omega_T)},$$

where $c_5$ is the coefficient corresponding to the strongly accretive operator $R_Q^{-1}$. By virtue of (4.4) and (4.6),

$$\|\partial_iw\|^p_{L_p(\Omega_T)}=\|\partial_iR_Qu\|^p_{L_p(\Omega_T)}\ge c_6 \|\partial_i u\|^p_{L_p(\Omega_T)},$$

i.e.,

$$ \langle \Delta_pR_Qu,u\rangle \ge c_5 c_6 \sum\limits_{1\le i\le n} \|\partial_iu\|^p_{L_p(\Omega_T)}= c_7\|u\|^p_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))},$$

(5.21)

for some $c_7>0$. The operator $\Delta_pR_Q$ is coercive with respect to $\tilde u=0$. $\qquad\square$

Remark 3.

If the conditions of Theorem 4 hold, then the operator

$$\Delta_pR_Q(\cdot+\hat u):L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))\to L_q(0,T;W_q^{-1}(Q))$$

is demicontinuous, pseudomonotone, and coercive.

Proof. By construction, $\Delta_pR_Q(\cdot+\hat u)u:=\Delta_pR_Q(u+\hat u)$ for arbitrary $u\in L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))$. Thus, using (5.21), we have

$$\langle \Delta_pR_Q(u+\hat u),u+\hat u\rangle \ge c_7\|u+\hat u\|^p_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))},$$

for some $c_7>0$. $\Delta_pR_Q(\cdot+\hat u)$ is coercive with respect to $\tilde u=-\hat u$.

Let $u_m\rightharpoonup u$ weakly in $L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))$ and $\mathop{\overline{\rm lim}}\nolimits \limits_{m\to\infty}\langle \Delta_pR_Q(u_m+\hat u),u_m-u\rangle \le 0.$ Obviously, $\mathop{\overline{\rm lim}}\nolimits \limits_{m\to\infty}\langle \Delta_pR_Q(u_m+\hat u),(u_m+\hat u) -(u+\hat u)\rangle \le 0$ and $u_m+\hat u\rightharpoonup u+\hat u$ weakly in $L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))$ too. By virtue of (5.20),

$$\begin{gathered} \, \mathop{\overline{\rm lim}}\nolimits \limits_{m\to\infty}\langle \Delta_pR_Q(u_m+\hat u),u_m-\xi\rangle= \mathop{\overline{\rm lim}}\nolimits \limits_{m\to\infty}\langle \Delta_pR_Q(u_m+\hat u),u_m+\hat u-\xi-\hat u\rangle \\ =\langle \Delta_pR_Q(u+\hat u),u+\hat u-\xi-\hat u\rangle = \langle \Delta_pR_Q(u+\hat u),u-\xi\rangle \end{gathered}$$

(5.22)

for any $\xi\in L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))$. The operator $\Delta_pR_Q(\cdot+\hat u)$ is pseudomonotone. $\qquad\square$

6. Existence of Solution

Theorem 5.

Let the operator $R_Q^{sym}$ be positive definite, and let the inverse operator $R_Q^{-1}$ be strongly accretive. Then, for any $f\in L_q(0,T;W^{-1}_q(Q))$ and $\varphi\in L_2(Q)$, there exists at least one solution of problem (3.5), (3.6). Moreover, the set of such solutions is weakly compact in $L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))$, and the solutions satisfy the following estimates:

$$\|u(T)\|^2_{L_2(Q)} \le c_{8}\|f\|^q_{L_q(0,T;W^{-1}_q(Q))}+c_{9} \|\varphi\|^2_{L_2(Q)},$$

(6.1)

$$\|u\|^p_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))} \le c_{10}\|f\|^q_{L_q(0,T;W^{-1}_q(Q))}+ c_{11}\|\varphi\|^2_{L_2(Q)},$$

(6.2)

where $c_8, c_9, c_{10}, c_{11}>0$ do not depend on $u, f$, and $\varphi$.

Proof. First we consider the case $\varphi=0$. Then by virtue of Theorem 3, the operator $\partial_t R_Q$ is maximal monotone, and by virtue of Lemma 14 and Theorem 4 the operator $\Delta_pR_Q$ is demicontinuous, pseudomonotone, and coercive. Thus, the conditions of Theorem 1.1 [18, Ch.III, § 1] hold. Therefore problem (3.5), (3.6) has at least one solution.

If $\varphi\not=0$, we consider an auxiliary fixed element $\hat u\in W\subset C(0,T;L_2(Q))$. Since $C^1(0,T; {\stackrel{\circ}{W}} _p^1(Q))\cap W$ densely imbedded into $W$, see Lemma 1.12 in [21, Ch. IV], and $W$ continuously imbedded into $C(0,T;L_2(Q))$, see Theorem 1.17 in [21, Ch. IV], for any $\varphi\in L_2(Q)$, there exists $\hat u\in W\subset C(0,T;L_2(Q))$ such that $\hat u|_{t=0}=\varphi$. Substituting $u(x,t)=v(x,t)+\hat u(x,t)$, we obtain the equivalent equation

$$\partial_t R_Q v+\Delta_pR_Q(v+\hat u) = f - \partial_t R_Q\hat u:=\widehat{f}, \qquad 0<t<T,$$

(6.3)

$$v(0) =0.$$

(6.4)

By virtue of Remark 3, the operator $\Delta_pR_Q(\cdot+\hat u)$ is demicontinuous, pseudomonotone, and coercive. Thus, since the conditions of Theorem 1.1 in [18, Ch.III, § 1] hold, then problem (6.3), (6.4) has at least one solution. Consequently, problem (3.5), (3.6) has at least one solution too.

For a solution of problem (3.5), (3.6), we have

$$ \langle\partial_tR_Qu,u\rangle+\langle\Delta_pR_Qu,u\rangle= \langle f,u\rangle.$$

(6.5)

Repeating the arguments from (4.11), we obtain

$$\begin{gathered} \, \langle\partial_tR_Qu,u\rangle=\dfrac12 (R_Q^{sym}u(T),u(T))_{L_2(Q)} -\dfrac12 (R_Q^{sym}u(0),u(0))_{L_2(Q)}\\= \dfrac12\left\|\sqrt{R^{sym}_Q}u(T)\right\|^2_{L_2(Q)} -\dfrac12\left\|\sqrt{R^{sym}_Q}\varphi\right\|^2_{L_2(Q)}. \end{gathered}$$

The second summand in (6.5) was evaluated in (5.21). That is

$$\begin{gathered} \, \dfrac12\left\|\sqrt{R^{sym}_Q}u(T)\right\|^2_{L_2(Q)} -\dfrac12\left\|\sqrt{R^{sym}_Q}\varphi\right\|^2_{L_2(Q)} +c_7\|u\|^p_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))}\\ \le \langle\partial_tR_Qu,u\rangle+ \langle\Delta_pR_Qu,u\rangle = \langle f,u\rangle. \end{gathered}$$

(6.6)

Let’s estimate the right part using Hölder’s inequality and the well-known formula $ab\le a^p/p+b^q/q$. We obtain

$$\langle f,u\rangle \le \|f\|_{L_q(0,T;W^{-1}_q(Q))}\|u\|_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))}\le \dfrac 1{q\varepsilon^q} \|f\|^q_{L_q(0,T;W^{-1}_q(Q))}+ \dfrac{\varepsilon^p}{p}\|u\|^p_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))}.$$

Let $\varepsilon>0$ be such that $\varepsilon^p/p= c_7/2$. The first term of the left part of inequality (6.6) is nonnegative, thus

$$\dfrac{c_7}2 \|u\|^p_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))} \le \dfrac 1{q\varepsilon^q} \|f\|^q_{L_q(0,T;W^{-1}_q(Q))}+ \dfrac12 \left\|\sqrt{R^{sym}_Q}\,\varphi\right\|^2_{L_2(Q)},$$

i.e.

$$ \|u\|^p_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))} \le \dfrac1q \left(\dfrac2{pc_7}\right)^{q-1} \|f\|^q_{L_q(0,T;W^{-1}_q(Q))}+ \dfrac{1}{c_7} \left\|\sqrt{R^{sym}_Q}\,\varphi\right\|^2_{L_2(Q)}.$$

(6.7)

By virtue of boundedness of the linear operator $R^{sym}_Q$, we conclude that $\left\|\sqrt{R^{sym}_Q}\,\varphi\right\|^2_{L_2(Q)}\le c_{12} \left\|\varphi\right\|^2_{L_2(Q)}$ for some $c_{12}>0$. This estimate and inequality (6.7) prove that inequality (6.2) is true.

On the other hand, the third term of the left part of (6.6) is also nonnegative. Using estimates (4.7) and (6.2), we obtain

$$\begin{gathered} \, \dfrac{c_3}2 \left\|u(T)\right\|^2_{L_2(Q)}\le \dfrac12 \left\|\sqrt{R^{sym}_Q}\,u(T)\right\|^2_{L_2(Q)}\le \langle f,u\rangle+ \dfrac12 \left\|\sqrt{R^{sym}_Q}\,\varphi\right\|^2_{L_2(Q)}\\ \le \dfrac 1{q} \|f\|^q_{L_q(0,T;W^{-1}_q(Q))}+ \dfrac{1}{p}\|u\|^p_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))} + \dfrac{c_{12}}2 \left\|\varphi\right\|^2_{L_2(Q)}\\ \le \left(\dfrac1q+\dfrac{c_{11}}p\right) \|f\|^q_{L_q(0,T;W^{-1}_q(Q))}+ \left(\dfrac{c_{10}}p+\dfrac{c_{12}}2\right) \left\|\varphi\right\|^2_{L_2(Q)}. \end{gathered}$$

(6.8)

This proves estimate (6.1).

Now let us prove the weak compactness of the set of solutions. Let $\{u_m\}$ belong to the set of solutions of problem (3.5), (3.6) such that $u_m\rightharpoonup u$ in $L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))$. Since, $\left\{\partial_tR_Qu_m=f-\Delta_p R_Qu_m\right\}$ is a bounded set in $L_q(0,T;W^{-1}_q(Q))$, without loss of generality we can assume that $u_m\rightharpoonup u$ in $W$ (up to the subsequences), here $u\in W$. Obviously, $u|_{t=0}=u_m|_{t=0}=\varphi$. By virtue of (4.11) and (4.7), we have

$$\langle \partial_tR_Q u_m-\partial_tR_Q u,u_m-u\rangle = \dfrac12 \left\|\sqrt{R_Q^{sym}}\left(u_m(T)-u(T)\right) \right\|^2_{L_2(Q)} \ge 0\quad\forall m.$$

Moreover, by virtue of weak convergence of the sequence $\{u_m\}$, we obtain

$$\lim\limits_{m\to\infty}\langle f,u_m-u\rangle =0\qquad\text{and}\qquad \lim\limits_{m\to\infty}\langle \partial_tR_Q u,u_m-u\rangle =0.$$

Thus,

$$\mathop{\overline{\rm lim}}\nolimits \limits_{m\to\infty}\langle \Delta_pR_Qu_m,u_m-u\rangle= \mathop{\overline{\rm lim}}\nolimits \limits_{m\to\infty}\langle f-\partial_tR_Q u_m,u_m-u\rangle \le 0.$$

Using the pseudomonotonocity of $\Delta_pR_Q$, we get

$$\begin{gathered} \, \langle\Delta_pR_Qu,u-\xi\rangle\le \mathop{\underline{\rm lim}}\nolimits \limits_{m\to\infty} \langle \Delta_pR_Qu_m,u_m-\xi\rangle= \mathop{\underline{\rm lim}}\nolimits \limits_{m\to\infty}\langle f-\partial_tR_Q u_m,u_m-\xi\rangle\\ \le \langle f-\partial_tR_Q u,u-\xi\rangle \qquad\forall\xi\in L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q)). \end{gathered}$$

(6.9)

By virtue of (6.9), $u$ is a solution of operator equation (3.5). The set of solutions of problem (3.5), (3.6) is weakly compact in $L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))$.

Theorem 6.

Let Conditions 1–3 hold, $R_s^{sym}>0$, and let $R_Q^{-1}$ be strongly accretive operator. Then, for any $f\in L_q(0,T;W^{-1}_q(Q))$ and $\psi\in L_2(Q)$, there exists at least one generalized solution of problem (1.1)–(1.3). Moreover, the set of such solutions is weakly compact in $L_p(0,T;W_p^1(Q))$, solutions satisfy the following estimates:

$$\|w(T)\|^2_{L_2(Q)} \le c_{13} \|f\|^q_{L_q(0,T;W^{-1}_q(Q))} +c_{14} \|\psi\|^2_{L_2(Q)} ,$$

(6.10)

$$\|w\|^p_{L_p(0,T;W_p^1(Q))} \le c_{15} \|f\|^q_{L_q(0,T;W^{-1}_q(Q))}+ c_{16} \|\psi\|^2_{L_2(Q)}.$$

(6.11)

where $c_{13}, c_{14}, c_{15}, c_{16}>0$ do not depend on $w, f$, and $\psi$.

Proof. From Conditions 1–3 and Theorem 1 it follows that there exists a set $\gamma=\{\gamma^r_{lj}\}$ such that the operator $R_Q$ is an isomorphism of $L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))$ onto $L_p(0,T;W_{p,\gamma}^1(Q))$. Therefore, by virtue of Theorem 5 and Theorem 2, there exists a generalized solution of nonlocal boundary value problem (1.1)–(1.3) with the above mentioned set $\gamma$. It remains to prove estimates (6.10) and (6.11). Let us prove estimate (6.10). From inequality (6.1) and boundedness of operators $R_Q:L_2(Q)\to L_2(Q)$ and $R_Q^{-1}:L_2(Q)\to L_2(Q)$ it follows that

$$\begin{gathered} \, \|w(T)\|^2_{L_2(Q)}= \|R_Qu\|^2_{C(0,T;L_2(Q))}\le c_{17}\|u(T)\|^2_{L_2(Q)}\\ \le c_{17}\left(c_{8}\|f\|^q_{L_q(0,T;W^{-1}_q(Q))} +c_{9}\|\varphi\|^2_{L_2(Q)}\right) = c_{17}\left(c_{8} \|f\|^q_{L_q(0,T;W^{-1}_q(Q))}+c_{9}\|R^{-1}_Q\psi\|^2_{L_2(Q)}\right)\\ \le c_{17}\left(c_{8} \|f\|^q_{L_q(0,T;W^{-1}_q(Q))}+c_{9} c_{18}\|\psi\|^2_{L_2(Q)}\right). \end{gathered}$$

In the same way, from inequality (6.2) and boundedness of operators $R_Q:L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))\to L_q(0,T;W^{-1}_q(Q))$ and $R_Q^{-1}:L_2(Q)\to L_2(Q)$ it follows that

$$\begin{gathered} \, \|w\|^p_{L_p(0,T;W_p^1(Q))}= \|R_Qu\|^p_{L_p(0,T;W_p^1(Q))} \le c_{19}\|u\|^p_{L_p(0,T;W_p^1(Q))}\\ \le c_{19}\left(c_{10}\|f\|^q_{L_q(0,T;W^{-1}_q(Q))}+ c_{11}\|\varphi\|^2_{L_2(Q)}\right)\\ \le c_{19}\left(c_{10}\|f\|^q_{L_q(0,T;W^{-1}_q(Q))}+ c_{11}c_{18}\|\psi\|^2_{L_2(Q)}\right). \end{gathered}$$

Now we prove the weak compactness of the set of solutions. Let $\{w_m\}$ belong to the set of generalized solutions of problem (1.1)–(1.3) such that $w_m\rightharpoonup w$ in $L_p(0,T;W_p^1(Q))$. Obviously, $w|_{t=0}=w_m|_{t=0}=\psi$. Then there exists a sequence $\{u_m\}$ such that $w_m=R_Qu_m$ and $u_m$ is a solution of (3.5), (3.6). As it was proved in (5.15), $u_m\rightharpoonup u$ in $W$ if and only if $w_m\rightharpoonup w$ in $W_{\gamma}$. But in the previous theorem it is proved that in this case $u$ is a solution of (3.5), (3.6). Therefore, $w=R_Qu$ is a generalized solution of problem (1.1)–(1.3).

7. Examples

Example 5.

We continue to consider problem (1.1), (1.2) in the rectangular parallelepiped $\Omega_T=(0,2)\times(0,1)\times(0,T)$ with Bitsadze–Samarskii nonlocal boundary conditions (1.5). As was proved in Example 2,

$$R_1=\left( \begin{array}{cc} 1 & \gamma_1 \\ \gamma_2 & 1 \end{array}\right),$$

i.e., $R_s^{sym}>0$ if

$$ |\gamma_1+\gamma_2|<2.$$

(7.1)

At the same time,

$$R_1^{-1}=\dfrac1{1-\gamma_1\gamma_2}\left( \begin{array}{cc} 1 & -\gamma_1 \\ -\gamma_2 & 1 \end{array}\right).$$

Thus, $R_Q^{-1}$ is strongly accretive if

$$ 2>|\gamma_1+\gamma_2|+\lambda^{-1}|\gamma_1-\gamma_2|,$$

(7.2)

where $\lambda$ satisfies estimate (5.3) or estimate (5.4), see Lemma 12. Obviously, if condition (7.2) holds, then condition (7.1) is also satisfied, i.e., by virtue of Theorem 6, problem (1.1), (1.2), (1.5) has at least one generalized solution.

Example 6.

We consider problem (1.1), (1.2) in the rectangular domain $\Omega_T=(0,3)\times(0,1)\times(0,T)$ with boundary conditions

$$ \left.\begin{array}{ll} w(x_1,0,t)=w(x_1,1,t)=0&(0\le x_1\le 2; 0<t<T),\\ w(0,x_2,t)=\gamma_1 w(1,x_2,t)+\gamma_2 w(2,x_2,t) &(0< x_2< 1; 0<t<T),\\ w(3,x_2,t)=\gamma_2 w(1,x_2,t)+\gamma_1 w(2,x_2,t)&(0< x_2< 1; 0<t<T) \end{array}\right\}$$

(7.3)

for $\gamma_1={39}/{28}$ and $\gamma_2=-6/7$.

In this case, the difference operator is given by $Ru(x)=\sum\limits_{-2\le k\le 2} a_ku(x_1+k,x_2,t)$. We shall find a nonsingular martix

$$R_1=\left( \begin{array}{ccc} a_0 & a_1 & a_2 \\ a_1 & a_0 & a_1 \\ a_2 & a_1 & a_0 \end{array}\right)$$

such that the relations

$$ \left\{ \begin{array}{ll} a_1=\gamma_1 a_0+\gamma_2 a_1,\\ a_2=\gamma_2 a_0 +\gamma_1 a_1 \end{array}\right.$$

(7.4)

hold. This means that Condition 3 is fulfilled if $a_0=1$, $a_1=3/4$, and $a_2=3/16$. It is easy to show that the symmetric matrices $R_1$ and $R_{10}=\left( \begin{array}{cc} 1 & 3/4 \\ 3/4 & 1 \end{array}\right)$ are positive definite. However, the inverse matrix

$$R_1^{-1}=\left( \begin{array}{ccc} 8.615 & -12 & 7.385 \\ -12 & 19 & -12 \\ 7.385 & -12 & 8.615 \end{array}\right)$$

does not satisfy the conditions of Lemma 12: $8.615< 12+7.385$. We cannot prove that $R_Q^{-1}$ is strongly accretive, and we cannot guarantee that problem (1.1), (1.2), (7.3) has at least one generalized solution.

Note that if $p=2$, then problem (1.1), (1.2), (7.3) is linear and has a unique generalized solution, see [14].

Example 7.

We consider problem (1.1), (1.2), (7.3) with $\gamma_1=1/2$ and $\gamma_2=0$.

As in Example 6, the difference operator is given by $Ru(x)=\sum\limits_{-2\le k\le 2} a_ku(x_1+k,x_2,t)$. We shall find a nonsingular martix $R_1$ such that the relations (7.4) hold. This means that Condition 3 is fulfilled if

$$R_1=\left( \begin{array}{ccc} 1 & 1/2 & 1/4 \\ 1/2 & 1 & 1/2 \\ 1/4 & 1/2 & 1 \end{array}\right) \qquad\text{and}\qquad R_{10}=\left( \begin{array}{cc} 1 & 1/2 \\ 1/2 & 1 \end{array}\right).$$

It is easy to show that the symmetric matrices $R_1$ and $R_{10}$ are positive definite. The inverse matrix

$$R_1^{-1}=\left( \begin{array}{ccc} 4/3 & -2/3 & 0 \\ -2/3 & 5/3 & -2/3 \\ 0 & -2/3 & 4/3 \end{array}\right)$$

satisfies condition (7.1), i.e., the operator $R_Q^{-1}$ is strongly accretive. Therefore, by virtue of Theorem 6, problem (1.1), (1.2), (7.3) with $\gamma_1=1/2$ and $\gamma_2=0$ has at least one generalized solution for any $p\in(2,\infty)$.

8. Solvability of quasilinear nonlocal parabolic problems

The above results can be applied not only to the equation with the $p$–Laplacian. In this section, we will consider nonlocal quasilinear parabolic problems. We will use the properties of quasilinear differential–difference operator that was studied in [23].

In the cylinder $\Omega_T=Q\times (0,T)$, we consider the differential equation

$$\partial_t w(x,t)-\sum\limits_{1\le i\le n} \partial_i A_i(x,t,w,\nabla w)+A_0(x,t,w,\nabla w) =f(x,t)\quad( (x,t)\in \Omega_T) $$

(8.1)

with initial condition

$$w(x,0)=\psi(x)\qquad(x\in Q) $$

(8.2)

and with nonlocal boundary conditions

$$ \left. \begin{array}{rll} w|_{\Gamma^T_{rl}}&=\sum\limits_{j=1}^{J_{0}} \gamma_{lj}^rw|_{\Gamma^T_{rj}} &\quad (r\in B,\;l=J_{0}+1,\ldots,J),\\ w|_{\Gamma^T_{rl}}&=0 &\quad (r\notin B,\;l=1,\ldots,J), \end{array} \right\}$$

(8.3)

see Section 1. Assume that $f\in L_q(0,T;W^{-1}_q(Q))$ and $\psi\in L_2(Q)$. Introduce the nonlinear operator $A:L_p(0,T;W^1_{p,\gamma}(Q)) \to L_q(0,T;W^{-1}_q(Q))$ given by the formula

$$ \langle A w,v\rangle=\sum\limits_{0\le i\le n}\, \int\limits_{\Omega_T} A_i(x,t,w,\nabla w) \, \partial_i v\,dx\,dt\qquad\forall v\in L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q)).$$

(8.4)

Here and below we write $\partial_0 w:=w$. Also we assume that for some $c_{20}>0$ and $g_0\in L_Q(\Omega_T)$,

$$\left|A_{i}(x,t,\xi)\right|\le g_0(x,t)+c_{20}\sum\limits_{0\le i\le n}|\xi_{i}|^{p-1}\qquad (i=0,1,\dots,n). $$

(8.5)

Definition 9.

A function $w\in W_\gamma$ is called a generalized solution of problem (8.1)–(8.3) if it satisfies the operator equation

$$ \partial_t w+ A w =f, \qquad w\in W_\gamma$$

(8.6)

and initial condition (8.2).

Moreover, we suppose that Conditions 1–3 hold. If $R_s^{sym}>0$, we can consider equation

$$ \partial_t R_Qu+ A R_Qu =f, \qquad u\in W,$$

(8.7)

with initial condition

$$ u(x,0)=R^{-1}_Q\psi(x)=\varphi(x)\qquad(x\in Q)$$

(8.8)

instead of (8.6), (8.2).

Theorem 7.

Let Conditions 1–3 hold, and let $R_s^{sym}>0$ $(s=s(r),r\in B)$. Suppose that the operator $A:L_p(0,T;W^1_p(Q)) \to L_q(0,T;W^{-1}_q(Q))$ is given by the formula (8.4), where the functions $A_i(x,t,\xi)$ are measurable in $(x,t)\in \Omega_T$ and differentiable in $\xi_j\in{\mathbb R}\, (j=0,1,\dots,n)$. Moreover, assume that

$$\sum\limits_{1\le m,l\le N(s)}\sum\limits_{0\le i,j\le n} r^s_{ml}A_{ij}\left(x+h_{sm},t,\zeta_{m\cdot}\right)\, \eta_{lj}\,\eta_{mi}\geq c_{21} \sum\limits_{1\le m\le N(s)}\sum\limits_{1\le i\le n} \left|\zeta_{mi}\right|^{p-2}\left|\eta_{mi}\right|^2,$$

(8.9)

$$\left|A_{ij}(x,t,\xi)\right|\le g_1(x,t)+c_{22}\sum\limits_{0\le i\le n}|\xi_{i}|^{p-2}\qquad (i,j=0,1,\dots,n),$$

(8.10)

where $A_{ij}(x,t,\xi)=\dfrac{\partial A_i(x,t,\xi)}{\partial \xi_j}$, $g_1\in L_{p/(p-2)}(\Omega_T)$, and $c_{21},c_{22}>0$ do not depend on $(x,t)\in \Omega_T$, $\xi\in {\mathbb R}^{n+1}$, $\zeta,\eta\in {\mathbb R}^{N(s)\times (n+1)}$, $\zeta_{m\cdot}=\{\zeta_{m0},\zeta_{m1},\dots,\zeta_{mn}\}$, $m=1,\dots,N(s)$.

Then, for any $f\in L_q(0,T;W^{-1}_q(Q))$ and $\psi\in L_2(Q)$, there exists a unique generalized solution of problem (8.1)–(8.3). Moreover, the following estimates hold:

$$\|w_1(T)-w_2(T)\|_{L_2(Q)} \le c_{23}\|f_1-f_2\|^{q/2}_{L_q(0,T;W^{-1}_q(Q))} +c_{24}\|\psi_1-\psi_2\|_{L_2(Q)},$$

(8.11)

$$\|w_1-w_2\|_{L_p(0,T;W_p^1(Q))} \le c_{25} \|f_1-f_2\|^{q/p}_{L_q(0,T;W^{-1}_q(Q))} + c_{26}\|\psi_1-\psi_2\|^{2/p}_{L_2(Q)},$$

(8.12)

here $w_1$ and $w_2$ are generalized solutions of problem (8.1)—(8.3) with the right parts $f_1$ and $f_2$ and with initial conditions $\psi_1$ and $\psi_2$, respectively; the positive constant $c_{23},c_{24},c_{25},c_{26}$ do not depend on $w_k, f_k$, and $\psi_k$.

Proof. First we consider the case $\psi=0$, i.e., $\varphi=0$. By virtue of (8.9)–(8.10), the operator $AR_Q:L_p(0,T; {\stackrel{\circ}{W}} ^1_p(Q)) \to L_q(0,T;W^{-1}_q(Q))$ is demicontinuous, monotone, and coercive. This follows from Theorem 1 in [23]. Moreover, inequalities (8.9) and (8.10) imply that there exists $c_{27}>0$ such that

$$ \langle AR_Qu-AR_Qy,u-y\rangle\ge c_{27} \|u-y\|^p_{L_p(0,T; {\stackrel{\circ}{W}} ^1_p(Q))},$$

(8.13)

see Theorem 1 in [23]. Thus, the conditions of Theorem 1.1 in [18, Ch.III, § 1] hold, problem (8.7), (8.8) has at least one solution. Moreover, since $AR_Q$ is monotone, this solution is unique. Since $R_Q$ is an isomorphism, there exists a unique generalized solution of (8.1)–(8.3).

If $\varphi\not=0$, we consider an auxiliary fixed element $\hat u\in W\subset C(0,T;L_2(Q))$. Since $C^1(0,T; {\stackrel{\circ}{W}} _p^1(Q))\cap W$ densely imbedded into $W$, see Lemma 1.12 in [21, Ch. IV], and $W$ continuously imbedded into $C(0,T;L_2(Q))$, see Theorem 1.17 in [21, Ch. IV], for any $\varphi\in L_2(Q)$, there exists $\hat u\in W\subset C(0,T;L_2(Q))$ such that $\hat u|_{t=0}=\varphi$. Substituting $u(x,t)=v(x,t)+\hat u(x,t)$, we obtain the equivalent equation

$$\partial_t R_Q v+AR_Q(v+\hat u) = f - \partial_t R_Q\hat u:=\widehat{f},$$

(8.14)

$$v(0) =0.$$

(8.15)

Clearly, $\hat f\in L_q(0,T;W^{-1}_q(Q))$. By virtue of construction of the operator $AR_Q(\cdot+\hat u)$, we obtain estimates that are similar to estimates (8.5)–(8.10). Therefore, the operator

$$AR_Q(\cdot+\hat u):L_p(0,T; {\stackrel{\circ}{W}} ^1_p(Q)) \to L_q(0,T;W^{-1}_q(Q))$$

is demicontinuous, monotone, and coercive too for any fixed $\hat u\in L_p(0,T; {\stackrel{\circ}{W}} ^1_p(Q))$. Thus, problem (8.14), (8.15) has a unique solution, see Theorem 1.1 in [18, Ch.III, § 1]. Therefore problem (8.1)–(8.3) has a unique generalized solution.

Let $w_1\in W_{\gamma}$ and $w_2\in W_{\gamma}$ be solutions of (8.1)–(8.3) with the right parts $f_1\in L_q(0,T;W^{-1}_q(Q))$ and $f_2\in L_q(0,T;W^{-1}_q(Q))$ and with initial conditions $\psi_1\in L_2(Q)$ and $\psi_2\in L_2(Q)$, respectively. We denote $u_i=R^{-1}_Qw_i$ and $\varphi_i=R^{-1}_Q\psi_i$. Since $u_i$ is a solution of (8.7), (8.8) with the right parts $f_i$ and with initial conditions $\varphi_i$, we have

$$\langle \partial_tR_Qu_1- \partial_tR_Qu_2,u_1-u_2\rangle +\langle AR_Qu_1-AR_Qu_2,u_1-u_2\rangle=\langle f_1-f_2,u_1-u_2\rangle.$$

Using equality (4.11), we obtain

$$\begin{gathered} \, \dfrac12 \left\|\sqrt{R^{sym}_Q}\left(u_1(T)-u_2(T)\right)\right\|^2_{L_2(Q)} +\langle AR_Qu_1-AR_Qu_2,u_1-u_2\rangle\\ =\langle f_1-f_2,u_1-u_2\rangle+\dfrac12\left\|\sqrt{R^{sym}_Q}(\varphi_1 -\varphi_2)\right\|^2_{L_2(Q)}. \end{gathered}$$

(8.16)

Let also $u_3\in W$ be a generalized solution of the problem (8.7), (8.8) with the initial condition $\varphi=\varphi_1$ and the right-hand sides of equation $f=f_2$. Then,

$$\dfrac12\left\|\sqrt{R^{sym}_Q}\left(u_1(T)-u_3(T)\right)\right\|^2_{L_2(Q)} +\langle AR_Qu_1-AR_Qu_3,u_1-u_3\rangle =\langle f_1-f_2,u_1-u_3\rangle,$$

(8.17)

$$\dfrac12\left\|\sqrt{R^{sym}_Q}\left(u_3(T)-u_2(T)\right)\right\|^2_{L_2(Q)} +\langle AR_Qu_3-AR_Qu_2,u_3-u_2\rangle =\dfrac12\left\|\sqrt{R^{sym}_Q}R_Q^{-1}(\psi_1-\psi_2)\right\|^2_{L_2(Q)}.$$

(8.18)

By virtue of estimate (8.13) and equality (8.17), we have

$$\begin{aligned} \, c_{27}\|u_1-u_3\|^p_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))} & \le \langle AR_Qu_1-AR_Qu_3,u_1-u_3\rangle\notag\\ &=\langle f_1-f_2,u_1-u_3\rangle -\dfrac12\left\|\sqrt{R^{sym}_Q}\left(u_1(T)-u_3(T)\right)\right\|^2_{L_2(Q)}\notag\\ &\le \langle f_1-f_2,u_1-u_3\rangle \le \|f_1-f_2\|_{L_q(0,T;W^{-1}_q(Q))}\|u_1-u_3\|_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))}\notag\\ &\le \dfrac1{\varepsilon^q q}\|f_1-f_2\|^q_{L_q(0,T;W^{-1}_q(Q))} +\dfrac{\varepsilon^p}p\|u_1-u_3\|^p_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))}, \end{aligned}$$

(8.19)

i.e., for ${\varepsilon^p}/p\le {c_{27}}/2$, we get

$$\|u_1-u_3\|^{p}_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))} \le c_{28} \|f_1-f_2\|^q_{L_q(0,T;W^{-1}_q(Q))},$$

(8.20)

$$\langle f_1-f_2,u_1-u_3\rangle \le c_{29} \|f_1-f_2\|^q_{L_q(0,T;W^{-1}_q(Q))}.$$

(8.21)

On the other hand, the second summand in the left-hand side of (8.17) is nonnegative, too, because $AR_Q$ is monotone. Thus, substituting (8.21), we obtain

$$ \dfrac12\left\|\sqrt{R^{sym}_Q}\left(u_1(T)-u_3(T)\right)\right\|^2_{L_2(Q)}\le \langle f_1-f_2,u_1-u_3\rangle\le c_{29} \|f_1-f_2\|^q_{L_q(0,T;W^{-1}_q(Q))}.$$

(8.22)

Substituting estimate (4.7) into (8.22), we obtain

$$ c_3 \left\|u_1(T)-u_3(T)\right\|^2_{L_2(Q)} \le \left\|\sqrt{R^{sym}_Q}\left(u_1(T)-u_3(T)\right)\right\|^2_{L_2(Q)} \le 2{c_{29}} \|f_1-f_2\|^q_{L_q(0,T;W^{-1}_q(Q))}.$$

(8.23)

Nonnegativity of the second summand in the left part of (8.18) implies that

$$\left\|\sqrt{R^{sym}_Q}(u_3(T)-u_2(T))\right\|^2_{L_2(Q)}\le \left\|\sqrt{R^{sym}_Q}(\varphi_1-\varphi_2)\right\|^2_{L_2(Q)}. $$

(8.24)

Using estimate (4.7) and the boundedness of the operators $\sqrt{R^{sym}_Q}$ and $R^{-1}_Q$, from (8.24), we obtain

$$\begin{gathered} \, c_3\left\|(u_3(T)-u_2(T))\right\|^2_{L_2(Q)}\le \left\|\sqrt{R^{sym}_Q}(u_3(T)-u_2(T))\right\|^2_{L_2(Q)}\\ \le \left\|\sqrt{R^{sym}_Q}(\varphi_1-\varphi_2)\right\|^2_{L_2(Q)} = \left\|\sqrt{R^{sym}_Q}R^{-1}_Q(\psi_1-\psi_2)\right\|^2_{L_2(Q)}\\ \le c_{12}c_{18}\left\|\psi_1-\psi_2\right\|^2_{L_2(Q)}. \end{gathered}$$

(8.25)

Using the triangle inequality and (8.23) and (8.25), we can write

$$\begin{gathered} \, \|u_1(T)-u_2(T)\|_{L_2(Q)}\le \|u_1(T)-u_3(T)\|_{L_2(Q)}+\|u_3(T)-u_2(T)\|_{L_2(Q)}\\ \le \sqrt{\dfrac{2c_{29}}{c_3}}\|f_1-f_2\|^{q/2}_{L_q(0,T;W^{-1}_q(Q))} +\sqrt{\dfrac{c_{12}c_{18}}{c_3}}\|\psi_1-\psi_2\|_{L_2(Q)}. \end{gathered}$$

Therefore, by virtue of boundedness of operator $R_Q:L_2(Q)\to L_2(Q)$, we obtain

$$\begin{gathered} \, \|w_1(T)-w_2(T)\|_{L_2(Q)}= \|R_Q(u_1(T)-u_2(T))\|_{L_2(Q)} \le c_{17} \|u_1(T)-u_2(T)\|_{L_2(Q)}\\ \le c_{17}\left(\sqrt{\dfrac{2c_{29}}{c_3}}\|f_1-f_2\|^{q/2}_{L_q(0,T;W^{-1}_q(Q))} +\sqrt{\dfrac{c_{12}c_{18}}{c_3}}\|\psi_1-\psi_2\|_{L_2(Q)}\right). \end{gathered}$$

Estimate (8.11) is proved.

On the other hand, (8.13), (8.18), and (8.25) imply

$$\begin{gathered} \, c_{27}\|u_3-u_2\|^p_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))}\le\langle AR_Qu_3-AR_Qu_2,u_3-u_2\rangle\\ \le \dfrac12\left\|\sqrt{R^{sym}_Q}(u_3(T)-u_2(T))\right\|^2_{L_2(Q)} +\langle AR_Qu_3-AR_Qu_2,u_3-u_2\rangle\\ =\dfrac12\left\|\sqrt{R^{sym}_Q}R^{-1}_Q(\psi_1-\psi_2)\right\|^2_{L_2(Q)} \le \dfrac{c_{12}c_{18}}2\left\|\psi_1-\psi_2\right\|^2_{L_2(Q)}, \end{gathered}$$

(8.26)

i.e. $\|u_3-u_2\|^p_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))} \le \dfrac{c_{12}c_{18}}{2c_{27}}\left\|\psi_1-\psi_2\right\|^2_{L_2(Q)}$. Hence, from (8.20) and (8.26) it follows that

$$\begin{gathered} \, \|u_1-u_2\|_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))}\le \|u_1-u_3\|_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))}+ \|u_3-u_2\|_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))}\\ \le c_{28}^{1/p}\|f_1-f_2\|^{q/p}_{L_q(0,T;W^{-1}_q(Q))} + \left(\dfrac{c_{12}c_{18}}{2c_{27}}\right)^{1/p} \|\psi_1-\psi_2\|^{2/p}_{L_2(Q)}. \end{gathered}$$

Thus, by virtue of boundedness of the operator $R_Q:L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))\to L_q(0,T;W^{-1}_q(Q))$, we derive the following estimate

$$\begin{gathered} \, \|w_1-w_2\|_{L_p(0,T;W_p^1(Q))}= \|R_Q(u_1-u_2)\|_{L_p(0,T;W_p^1(Q))} \le c_{19} \|u_1-u_2\|_{L_p(0,T; {\stackrel{\circ}{W}} _p^1(Q))}\\ \le c_{19} \left( c_{28}^{1/p}\|f_1-f_2\|^{q/p}_{L_q(0,T;W^{-1}_q(Q))} + \left(\dfrac{c_{12}c_{18}}{2c_{27}}\right)^{1/p} \|\psi_1-\psi_2\|^{2/p}_{L_2(Q)}\right). \end{gathered}$$