On One Boundary-Value Problem with Two Nonlocal Conditions for a Parabolic Equation

Abstract

This work is concerned with a boundary-value problem for a parabolic equation with nonlocal integral conditions of the second kind. Existence and uniqueness of a generalized solution are proved.

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19.1 Introduction

In recent years, nonlocal problems for PDEs have received a great deal of attention as a convenient way of description of different physical phenomena. These problems arise in a wide variety of applications, including heat conduction, processes in liquid plasma, dynamics of ground waters, thermo-elasticity and some technological processes.

In this paper, our main interest lies in the field of nonlocal problems with integral conditions that generalizes the discrete case. We mention the first papers in this area [6, 14] devoted to problems for parabolic equations. Then these results were extended [2, 7–11, 13, 16, 26, 28, 29]. For papers related to nonlocal problems for other evolution equations, we refer the reader to [1, 3–5, 12, 17, 19–25, 27].

In [25], the author studied two problems for the hyperbolic equation

$$\displaystyle{ u_{tt} - u_{xx} + c(x,t)u = f(x,t) }$$

with the initial condition

$$\displaystyle{ u(x,0) =\varphi (x),\quad u_{t}(x,0) =\psi (x) }$$

and two types of nonlocal conditions.

• Case 1:
  $$\displaystyle{ \int \limits _{0}^{l}K_{ i}(x)u(x,t)\,dx = 0,\quad i = 1,2. }$$
• Case 2:
  $$\displaystyle\begin{array}{rcl} u_{x}(0,t) -\int \limits _{0}^{l}K_{ 1}(x,t)u(x,t)\,dx& =& 0, {}\\ u_{x}(l,t) -\int \limits _{0}^{l}K_{ 2}(x,t)u(x,t)\,dx& =& 0. {}\\ \end{array}$$

Motivated by the ideas of Pulkina [25], in this paper we extend the results of Pulkina [25] to a special class of boundary-value problems with nonlocal integral conditions for parabolic equations. The proof of the main result is based on the method of energy estimates and the Faedo–Galerkin approximations.

19.2 Preliminaries

In the cylinder Q _T  = { (x, t):  x ∈ (0, l), t ∈ (0, T)} we consider the problem for the equation

$$\displaystyle{ u_{t} = u_{xx} + c(x,t)u }$$

(19.1)

with the initial condition

$$\displaystyle{ u(x,0) =\varphi (x) }$$

(19.2)

and the nonlocal conditions

$$\displaystyle\begin{array}{rcl} u_{x}(0,t) =\int \limits _{ 0}^{l}K_{ 1}(x,t)u_{x}(x,t)\,dx +\int \limits _{ 0}^{l}M_{ 1}(x,t)u(x,t)\,dx,& &{}\end{array}$$

(19.3)

$$\displaystyle\begin{array}{rcl} u_{x}(l,t) =\int \limits _{ 0}^{l}K_{ 2}(x,t)u_{x}(x,t)\,dx +\int \limits _{ 0}^{l}M_{ 2}(x,t)u(x,t)\,dx.& &{}\end{array}$$

(19.4)

In this paper, we shall assume that the following assumptions are satisfied.

• (A1) $c(x,t) \in C(\overline{Q}_{T})$, $\varphi (x) \in C^{1}[0,l]$;

• (A2) $K_{1}(x,t),\,K_{2}(x,t),\,M_{1}(x,t),\,M_{2}(x,t) \in C^{1}(\overline{Q}_{T})$.

We note that presence of partial derivatives on the right-hand side of the nonlocal conditions (19.3), (19.4) can cause difficulties in constructing of a priori estimates. Therefore, to avoid this we integrate by parts in (19.3), (19.4) and obtain

$$\displaystyle\begin{array}{rcl} u_{x}(0,t) = K_{1}(l,t)u(l,t) - K_{1}(0,t)u(0,t) +\int \limits _{ 0}^{l}R_{ 1}(x,t)u(x,t)\,dx,& &{}\end{array}$$

(19.5)

$$\displaystyle\begin{array}{rcl} u_{x}(l,t) = K_{2}(l,t)u(l,t) - K_{2}(0,t)u(0,t) +\int \limits _{ 0}^{l}R_{ 2}(x,t)u(x,t)\,dx,& &{}\end{array}$$

(19.6)

where $R_{1}(x,t) = M_{1}(x,t) - (K_{1}(x,t))_{x}$, $R_{2}(x,t) = M_{2}(x,t) - (K_{2}(x,t))_{x}$.

Let W ₂ ^1, 0(Q _T ) be the usual Sobolev space. We define the space V ₂(Q _T ) which consists of elements of W ₂ ^1, 0(Q _T ) with the norm

$$\displaystyle{ \vert u\vert ^{2} = ess\sup _{ 0\leq t\leq T}\int \limits _{0}^{l}u^{2}(x,t)\,dt +\int \limits _{ Q_{T}}u_{x}^{2}(x,t)\,dx\,dt. }$$

Definition 19.1.

A function u(x, t) ∈ V ₂(Q _T ) is said to be a generalized solution to the problem (19.1), (19.2), (19.5), (19.6) provided for any function η(x, t) ∈ W ₂ ¹(Q _T ), η(x, T) = 0, the following integral identity holds:

$$\displaystyle\begin{array}{rcl} & & \int \limits _{Q_{T}}\left (-u\eta _{t} + u_{x}\eta _{x} - cu\eta \right )\,dx\,dt \\ & & \quad =\int \limits _{ 0}^{l}\varphi (x)\eta (x,0)\,dx +\int \limits _{ 0}^{T}\left (K_{ 1}(0,t)\eta (0,t) - K_{2}(0,t)\eta (l,t)\right )u(0,t)\,dt \\ & & \quad \quad +\int \limits _{ 0}^{T}\left (K_{ 2}(l,t)\eta (l,t) - K_{1}(l,t)\eta (0,t)\right )u(l,t)\,dt \\ & & \quad \quad +\int \limits _{Q_{T}}\left (R_{1}(x,t)\eta (l,t) - R_{2}(0,t)\eta (0,t)\right )u(x,t)\,dx\,dt. {}\end{array}$$

(19.7)

Lemma 19.2.

Let a function u(x,t) be a solution to the problem  (19.1) ,  (19.2) , (19.5) ,  (19.6) . Then the following identity holds:

$$\displaystyle\begin{array}{rcl} \frac{1} {2}\int \limits _{0}^{l}u^{2}(x,\tau )\,dx +\int \limits _{ Q_{T}}u_{x}^{2}\,dx\,dt& =& \frac{1} {2}\int _{0}^{l}\varphi ^{2}(x)\,dx +\int \limits _{ Q_{T}}cu^{2}\,dx\,dt {}\\ & & +\int \limits _{0}^{\tau }K_{ 2}(l,t)u^{2}(l,t)\,dt -\int \limits _{ 0}^{\tau }K_{ 1}(l,t)u(0,t)u(l,t)\,dt {}\\ & & +\int \limits _{0}^{\tau }K_{ 1}(0,t)u^{2}(0,t)\,dt -\int \limits _{ 0}^{\tau }K_{ 2}(0,t)u(0,t)u(l,t)\,dt {}\\ & & +\int \limits _{Q_{T}}R_{1}(x,t)u(x,t)\,dx\,u(l,t)\,dt {}\\ & & -\int \limits _{Q_{T}}R_{2}(x,t)u(x,t)\,dx\,u(0,t)\,dt {}\\ \end{array}$$

for a.e. τ ∈ [0,T].

Proof.

Let a function u(x, t) ∈ W ₂ ¹(Q _τ ) and satisfy the integral identity (19.7) for all functions η(x, t) ∈ W ₂ ¹(Q _T ), η(x, T) = 0. For an arbitrary τ ∈ [0, T], we take

$$\displaystyle{ \eta (x,t) = \left \{\begin{array}{cc} u(x,t),& 0 < t <\tau,\\ 0, &\tau \leq t < T. \end{array} \right. }$$

After integration by parts in (19.7) we obtain

$$\displaystyle\begin{array}{rcl} \frac{1} {2}\int \limits _{0}^{l}u^{2}(x,\tau )\,dx +\int \limits _{ Q_{\tau }}u_{x}^{2}\,dx\,dt& =& \frac{1} {2}\int \limits _{0}^{l}\varphi ^{2}(x)\,dx +\int \limits _{ Q_{\tau }}cu^{2}\,dx\,dt \\ & & +\int \limits _{0}^{\tau }K_{ 2}(l,t)u^{2}(l,t)\,dt -\int \limits _{ 0}^{\tau }K_{ 1}(l,t)u(0,t)u(l,t)\,dt \\ & & +\int \limits _{0}^{\tau }K_{ 1}(0,t)u^{2}(0,t)\,dt -\int \limits _{ 0}^{\tau }K_{ 2}(0,t)u(0,t)u(l,t)\,dt \\ & & +\int \limits _{Q_{\tau }}R_{1}(x,t)u(x,t)\,dx\,u(l,t)\,dt \\ & & -\int \limits _{Q_{\tau }}R_{2}(x,t)u(x,t)\,dx\,u(0,t)\,dt. {}\end{array}$$

(19.8)

We shall prove that a function u(x, t) ∈ V ₂(Q _T ) also satisfies (19.8). To this aim consider a sequence v ^m(x, t) ∈ W ₂ ¹(Q _T ) which satisfies the identity (19.7) and hence, (19.8), that is

$$\displaystyle\begin{array}{rcl} \frac{1} {2}\int \limits _{0}^{l}(v^{m})^{2}(x,\tau )\,dx +\int \limits _{ Q_{\tau }}(v^{2})_{ x}^{2}\,dx\,dt& =& \frac{1} {2}\int \limits _{0}^{l}\varphi ^{2}(x)\,dx +\int \limits _{ Q_{\tau }}c(v^{m})^{2}\,dx\,dt \\ & & +\int \limits _{0}^{\tau }K_{ 2}(l,t)(v^{m})^{2}(l,t)\,dt \\ & & -\int \limits _{0}^{\tau }K_{ 1}(l,t)v^{m}(0,t)v^{m}(l,t)\,dt \\ & & +\int \limits _{0}^{\tau }K_{ 1}(0,t)(v^{m})^{2}(0,t)\,dt \\ & & -\int \limits _{0}^{\tau }K_{ 2}(0,t)v^{m}(0,t)v^{m}(l,t)\,dt \\ & & +\int \limits _{Q_{\tau }}R_{1}(x,t)v^{m}(x,t)\,dx\,v^{m}(l,t)\,dt \\ & & -\int \limits _{Q_{\tau }}R_{2}(x,t)v^{m}(x,t)\,dx\,v^{m}(0,t)\,dt.{}\end{array}$$

(19.9)

Note that W ₂ ¹(Q _τ ) is dense in V ₂ ^1, 0(Q _T ) [15] and hence, in V ₂(Q _T ). Therefore, there exists a function u ^∗ ∈ V ₂(Q _T ) such that $\vert v^{m} - u^{{\ast}}\vert _{Q_{T}} \rightarrow 0$ as $m \rightarrow \infty$:

$$\displaystyle{ ess\sup _{0\leq t\leq T}\int \limits _{0}^{l}(v^{m} - u^{{\ast}})^{2}\,dx +\int \limits _{ Q_{T}}(v^{m} - u^{{\ast}})_{ x}^{2}\,dx\,dt \rightarrow 0. }$$

It implies that $v^{m}(x,t) \rightarrow u^{{\ast}}$ strongly in L ₂(0, l) and $v_{x}^{m}(x,t) \rightarrow u_{x}^{{\ast}}$ strongly in L ₂(Q _T ). We also note that $v^{m} \rightarrow u^{{\ast}}$ in L ₂(Q _T ). Our next aim is to estimate terms on the right-hand side of (19.9). The assumptions (A1), (A2) imply that there exist positive numbers c ₁, k ₂ such that $\left \vert c(x,t)\right \vert \leq c_{1}$, $\left \vert K_{2}(x,t)\right \vert \leq k_{2}$. Applying $\varepsilon$-inequality [18]

$$\displaystyle{ \left.v^{2}\right \vert _{ x=0,\,x=l} \leq \int \limits _{0}^{l}\left (\varepsilon v_{ x}^{2} + C(\varepsilon )v^{2}\right )\,dx }$$

we obtain

$$\displaystyle\begin{array}{rcl} \left \vert \int \limits _{0}^{\tau }K_{ 2}(l,t)(v^{m}(l,t))^{2}\,dt\right \vert & \leq & k_{ 2}\int \limits _{0}^{\tau }(v^{m}(l,t))^{2}\,dt \\ & \leq & k_{2}\varepsilon \vert \vert v_{x}^{m}\vert \vert ^{2} + k_{ 2}C_{\varepsilon }\vert \vert v^{m}\vert \vert ^{2} \\ & \leq & H_{1}\left (\vert \vert v_{x}^{m}\vert \vert ^{2} + \vert \vert v^{m}\vert \vert ^{2}\right ),{}\end{array}$$

(19.10)

where $H_{1} =\max \{ k_{2}\varepsilon,\,k_{2}C_{\varepsilon }\}$. Similarly, we derive the estimates

$$\displaystyle\begin{array}{rcl} \left \vert \int \limits _{0}^{\tau }K_{ 1}(0,t)(v^{m}(0,t))^{2}\,dt\right \vert \leq H_{ 2}\left (\vert \vert v_{x}^{m}\vert \vert ^{2} + \vert \vert v^{m}\vert \vert ^{2}\right ),& &{}\end{array}$$

(19.11)

$$\displaystyle\begin{array}{rcl} \left \vert \int \limits _{0}^{\tau }(K_{ 1}(l,t) + K_{2}(0,t))v^{m}(0,t)v^{m}(l,t)\,dt\right \vert \leq H_{ 3}\left (\vert \vert v_{x}^{m}\vert \vert ^{2} + \vert \vert v^{m}\vert \vert ^{2}\right ).& &{}\end{array}$$

(19.12)

To obtain an estimate for the term

$$\displaystyle{ \int \limits _{Q_{\tau }}R_{1}(x,t)v^{m}(x,t)\,dx\,v^{m}(l,t)\,dt, }$$

we use Young’s inequality and the Cauchy–Schwartz inequality and then

$$\displaystyle\begin{array}{rcl} & & \left \vert \int \limits _{Q_{\tau }}R_{1}(x,t)v^{m}(x,t)\,dx\,v^{m}(l,t)\,dt\right \vert {}\\ & & \quad \leq \frac{l} {2}\int \limits _{0}^{\tau }(v^{m}(l,t))^{2}\,dt + \frac{r_{1}} {2} \int \limits _{Q_{\tau }}(v^{m}(x,t))^{2}\,dx\,\,dt. {}\\ \end{array}$$

Similarly,

$$\displaystyle{ \left \vert \int \limits _{Q_{\tau }}R_{1}(x,t)v^{m}(x,t)\,dx\,v^{m}(l,t)\,dt\right \vert \leq H_{ 4}\left (\vert \vert v_{x}^{m}\vert \vert ^{2} + \vert \vert v^{m}\vert \vert ^{2}\right ) }$$

(19.13)

and

$$\displaystyle{ \left \vert \int \limits _{Q_{\tau }}R_{2}(x,t)v^{m}(x,t)\,dx\,v^{m}(0,t)\,dt\right \vert \leq H_{ 5}\left (\vert \vert v_{x}^{m}\vert \vert ^{2} + \vert \vert v^{m}\vert \vert ^{2}\right ), }$$

(19.14)

where the constants H _i , i = 2, 3, 4, 5 do not depend on m. Furthermore, we note that

$$\displaystyle{ \left \vert \int \limits _{Q_{\tau }}c(v^{m})^{2}\,dx\,dt\right \vert \leq c_{ 1}\vert \vert v^{m}\vert \vert ^{2}. }$$

(19.15)

Therefore, using strong convergence $v^{m}(x,t) \rightarrow u^{{\ast}}$ and the estimates (19.10)–(19.15) we pass to the limit as $m \rightarrow \infty$ in and obtain (19.8) for u(x, t) ∈ V ₂(Q _T ).

Lemma 19.3.

Let a function u(x,t) be a solution to the problem  (19.1) ,  (19.2) , (19.5) , (19.6) . Then there exists H > 0 such that $\vert u\vert _{Q_{T}} \leq H$ .

Proof.

By Lemma 19.2, the solution u(x, t) satisfies the integral identity (19.8). We shall estimate the right-hand side of (19.8). Note that for $\varepsilon _{1},\varepsilon _{2} > 0$

$$\displaystyle\begin{array}{rcl} u^{2}(0,t)& \leq & \int \limits _{ 0}^{l}\left (\varepsilon _{ 1}u_{x}^{2} + C(\varepsilon _{ 1})u^{2}\right )\,dx,\,u^{2}(l,t) {}\\ & \leq & \int \limits _{0}^{l}\left (\varepsilon _{ 2}u_{x}^{2} + C(\varepsilon _{ 2})u^{2}\right )\,dx {}\\ \end{array}$$

and hence,

$$\displaystyle{ \left \vert u(0,t)u(l,t)\right \vert \leq \frac{1} {2}\int \limits _{0}^{l}\left ((\varepsilon _{ 1} +\varepsilon _{2})u_{x}^{2} + (C(\varepsilon _{ 1}) + C(\varepsilon _{2}))u^{2}\right )\,dx. }$$

Therefore,

$$\displaystyle{ \left \vert \int \limits _{0}^{\tau }K_{ 2}(l,t)u^{2}(l,t)\,dt\right \vert \leq k_{ 2}\varepsilon \int \limits _{0}^{\tau }\int \limits _{ 0}^{l}u_{ x}^{2}\,dx\,dt + k_{ 2}C_{\varepsilon }\int \limits _{0}^{\tau }\int \limits _{ 0}^{l}u^{2}\,dx\,dt, }$$

(19.16)

$$\displaystyle\begin{array}{rcl} & & \left \vert \int \limits _{0}^{\tau }\left (K_{ 1}(l,t) + K_{2}(0,t)\right )u(0,t)u(l,t)\right \vert {}\\ & & \quad \leq \frac{(k_{1} + k_{2})} {2} (\varepsilon _{1} +\varepsilon _{2})\int \limits _{Q_{\tau }}u_{x}^{2}\,dx\,dt + {}\\ & & \quad \quad + \frac{(k_{1} + k_{2})} {2} (C_{\varepsilon _{1}} + C_{\varepsilon _{2}})\int \limits _{Q_{\tau }}u^{2}\,dx\,dt. {}\\ \end{array}$$

Moreover,

$$\displaystyle\begin{array}{rcl} & & \left \vert \int \limits _{Q_{\tau }}R_{1}(x,t)u(x,t)\,dx\,u(l,t)\,dt\right \vert {}\\ & & \quad \leq \frac{1} {2}(r_{1} + lC_{\varepsilon _{2}})\int \limits _{Q_{\tau }}(u(x,t))^{2}\,dx\,\,dt {}\\ & & \quad \quad +\, \frac{\varepsilon _{2}l} {2} \int \limits _{Q_{\tau }}(u_{x}(x,t))^{2}\,dx\,dt {}\\ \end{array}$$

and

$$\displaystyle\begin{array}{rcl} & & \left \vert \int \limits _{Q_{\tau }}R_{2}(x,t)u(x,t)\,dx\,u(0,t)\,dt\right \vert \\ & & \quad \leq \frac{1} {2}(r_{2} + lC_{\varepsilon _{1}})\int \limits _{Q_{\tau }}(u(x,t))^{2}\,dx\,\,dt \\ & & \quad \quad + \frac{\varepsilon _{1}l} {2} \int \limits _{Q_{\tau }}(u_{x}(x,t))^{2}\,dx\,dt. {}\end{array}$$

(19.17)

From the estimates (19.16)–(19.17) and the integral identity (19.8) it follows that

$$\displaystyle{ \int \limits _{0}^{l}u^{2}(x,\tau )\,dx +\int \limits _{ Q_{\tau }}u_{x}^{2}\,dx\,dt \leq \frac{1} {2}\int \limits _{0}^{l}\varphi ^{2}(x)\,dx + P\int \limits _{ Q_{\tau }}u^{2}\,dx\,dt. }$$

In particular,

$$\displaystyle{ \int \limits _{0}^{l}u^{2}(x,\tau )\,dx \leq \frac{1} {2}\int \limits _{0}^{l}\varphi ^{2}(x)\,dx + P\int \limits _{ Q_{\tau }}u^{2}\,dx\,dt. }$$

(19.18)

By Gronwall’s lemma we conclude that

$$\displaystyle{ \int \limits _{Q_{\tau }}u^{2}(x,t)\,dx\,dt \leq H_{ 1}\int \limits _{0}^{l}\varphi ^{2}(x)\,dx, }$$

(19.19)

and hence,

$$\displaystyle{ \int \limits _{Q_{\tau }}u_{x}^{2}(x,t)\,dx\,dt \leq H_{ 2}\int \limits _{0}^{l}\varphi ^{2}(x)\,dx. }$$

(19.20)

Therefore, from (19.19) and (19.20) we obtain

$$\displaystyle{ \vert u\vert _{Q_{T}} \leq H. }$$

19.3 The Main Result

In this section we shall prove existence and uniqueness theorem for the problem (19.1), (19.2), (19.5), (19.6).

Theorem 19.4.

Let the conditions (A1)–(A2) hold and

• (A3) $K_{1}(\xi _{1},0),\,K_{2}(\xi _{i},0) = 0,\,i = 1,2,\,\xi _{1} = 0,\,\xi _{2} = l,\,K_{1}(l,t) = K_{2}(0,t)$ ,

• (A4) $R_{1}^{2} + R_{2}^{2} \leq \frac{1} {2}$ .

Then there exists a unique generalized solution to the problem  (19.1) ,  (19.2) , (19.5) ,  (19.6) .

Proof.

The proof of the theorem is organized as follows. First, to prove the existence part we construct a sequence of Faedo–Galerkin approximations and show its convergence to the solution of the problem. Second, we prove uniqueness of the generalized solution. Let a system of functions $\{\varphi _{i}(x)\} \in C^{1}[0,l]$ be complete in W ₂ ¹ and

$$\displaystyle{ \left (\varphi _{i},\,\varphi _{j}\right )_{L_{2}(0,l)} = \left \{\begin{array}{cc} 1,&i = j,\\ 0, & i\neq j. \end{array} \right. }$$

We define for each $N \in \mathbb{N}$ the approximate solution in the following form

$$\displaystyle{ u^{N}(x,t) =\sum _{ k=1}^{N}c_{ k}^{N}(t)\varphi _{ k}(x), }$$

where the functions c _k (t) are unknown for the moment. We shall consider c _k (t) which are solutions to the Cauchy problem

$$\displaystyle\begin{array}{rcl} & & \int \limits _{0}^{l}u_{ t}^{N}\varphi _{ i}\,dx +\int \limits _{ 0}^{l}u_{ x}^{N}\varphi _{ i}^{{\prime}}\,dx -\int \limits _{ 0}^{l}c(x,t)u^{N}\varphi _{ i}\,dx \\ & & \quad = K_{1}(0,t)u^{N}(0,t)\varphi _{ i}(0) - K_{1}(l,t)u^{N}(l,t)\varphi _{ i}(0) \\ & & \qquad +\,\int \limits _{ 0}^{l}R_{ 1}(x,t)u^{N}\,dx\varphi _{ i}(0) -\int \limits _{0}^{l}R_{ 2}(x,t)u^{N}\,dx\varphi _{ i}(l) \\ & & \qquad -\, K_{2}(0,t)u^{N}(0,t)\varphi _{ i}(l) + K_{2}(l,t)u^{N}(l,t)\varphi _{ i}(l),\quad {}\end{array}$$

(19.21)

$$\displaystyle{ c_{i}^{N}(0) = \left (\varphi,\,\varphi _{ i}\right ), }$$

(19.22)

$i = \overline{1,\,N}$. We write the Cauchy problem (19.21)–(19.22) such that

$$\displaystyle{ \frac{d} {dt}c_{i}^{N}(t) +\sum _{ k=1}^{N}c_{ k}^{N}(t)A_{ k,i}(t) = 0,i = \overline{1,\,N}, }$$

(19.23)

where

$$\displaystyle\begin{array}{rcl} A_{k,i}(t)& =& \int \limits _{0}^{l}\varphi _{ k}^{{\prime}}(x)\varphi _{ i}^{{\prime}}(x)\,dx -\int \limits _{ 0}^{l}c(x,t)\varphi _{ k}(x)\varphi _{i}(x)\,dx {}\\ & & -\,\varphi _{i}(0)\left (K_{1}(l,t)\varphi _{k}(l) - K_{1}(0,t)\varphi _{k}(0) +\int \limits _{ 0}^{l}R_{ 1}(x,t)\varphi _{k}(x)\,dx\right ) {}\\ & & +\,\varphi _{i}(l)\left (K_{2}(l,t)\varphi _{k}(l) - K_{2}(0,t)\varphi _{k}(0) +\int \limits _{ 0}^{l}R_{ 2}(x,t)\varphi _{k}(x)\,dx\right ). {}\\ \end{array}$$

We estimate the coefficients A _k, i as follows:

$$\displaystyle\begin{array}{rcl} \left \vert A_{k,i}(t)\right \vert & \leq & \frac{1} {2}\int \limits _{0}^{l}\varphi _{ k}^{{\prime}2}(x)\,dx + \frac{1} {2}\int \limits _{0}^{l}\varphi _{ i}^{{\prime}2}(x)\,dx {}\\ & & +\,\frac{\overline{c}} {2}\int \limits _{0}^{l}\varphi _{ k}^{2}(x)\,dx + \frac{\overline{c}} {2}\int \limits _{0}^{l}\varphi _{ i}^{2}(x)\,dx {}\\ & & +\left \vert \varphi _{i}(0)\right \vert \left (k_{1}\left (\varphi _{k}(l)\vert + \left \vert \varphi _{k}(0)\right \vert \right ) + \frac{r_{1}} {2} + \frac{1} {2}\int \limits _{0}^{l}\varphi _{ k}^{2}(x)\,dx\right ) {}\\ & & +\,\left \vert \varphi _{i}(0)\right \vert \left (k_{2}\left (\varphi _{k}(l)\vert + \left \vert \varphi _{k}(0)\right \vert \right ) + \frac{r_{2}} {2} + \frac{1} {2}\int \limits _{0}^{l}\varphi _{ k}^{2}(x)\,dx\right ). {}\\ \end{array}$$

The assumptions (A1)–(A2) imply that A _k, i are bounded. Therefore, the Cauchy problem has a unique solution c _k ^N ∈ C ¹(0, T) and all approximations u ^N(x, t) are defined. The next aim is to show that the sequence {u ^N(x, t)} converges to the solution to the problem (19.1), (19.2), (19.5), (19.6). To this aim we multiply each (19.21) by c _i ^N(t), sum it up from i = 0 to i = N and integrate the result with respect to t from 0 to t ₁ < T. Thus we obtain

$$\displaystyle\begin{array}{rcl} \frac{1} {2}\int \limits _{0}^{l}\left (u^{N}(x,t_{ 1})\right )^{2}\,dx +\int \limits _{ Q_{t_{1}}}\left (u_{x}^{N}\right )^{2}\,dx\,dt& =& \frac{1} {2}\int \limits _{0}^{l}\varphi ^{2}(x)\,dx +\int \limits _{ Q_{t_{1}}}c(u^{N})^{2}\,dx\,dt {}\\ & & +\,\int \limits _{0}^{t_{1} }K_{2}(l,t)(u^{N}(l,t))^{2}\,dt {}\\ & & -\,\int \limits _{0}^{t_{1} }K_{1}(l,t)u^{N}(0,t)u^{N}(l,t)\,dt {}\\ & & +\,\int \limits _{0}^{t_{1} }K_{1}(0,t)(u^{N}(0,t))^{2}\,dt {}\\ & & -\,\int \limits _{0}^{t_{1} }K_{2}(0,t)u^{N}(0,t)u^{N}(l,t)\,dt {}\\ & & +\,\int \limits _{Q_{t_{ 1}}}R_{1}(x,t)u^{N}(x,t)\,dx\,u^{N}(l,t)\,dt {}\\ & & -\,\int \limits _{Q_{t_{ 1}}}R_{2}(x,t)u^{N}(x,t)\,dx\,u^{N}(0,t)\,dt. {}\\ \end{array}$$

Therefore, from Lemmas 19.2 and 19.3 it follows that $\left \vert u^{N}\right \vert _{Q_{T}} \leq \mathrm{ Const}$. It implies that there exists a subsequence of {u ^N(x, t)} which converges weakly in L ₂(0, l) and uniformly with respect to t ∈ [0, T] to some function u(x, t) [18]. We shall prove that this function u(x, t) satisfies the integral identity (19.7) from the definition of a generalized solution to the problem (19.1), (19.2), (19.5), (19.6). To this end, we multiply each (19.21) by a smooth function d _i (t), d _i (T) = 0, sum it up from i = 1 to i = N, integrate with respect to t from 0 to T and denote $\varPhi ^{N^{{\prime}} } =\sum _{ i=1}^{N^{{\prime}} }d_{i}(t)\varphi _{i}(x)$. As a result we obtain

$$\displaystyle\begin{array}{rcl} & & \int \limits _{0}^{T}\left (-(u^{N},\,\varPhi _{ t}^{N^{{\prime}} }) + (u_{x}^{N},\,\varPhi _{ x}^{N^{{\prime}} }) - (cu^{N},\,\varPhi ^{N^{{\prime}} })\right ) \\ & & \quad =\int \limits _{ 0}^{l}\varphi (x)\varPhi ^{N^{{\prime}} }(x,0)\,dx +\int \limits _{ 0}^{T}\int \limits _{ 0}^{l}R_{ 1}(x,t)u^{N}(x,t)\,dx\,\varPhi ^{N^{{\prime}} }(0,t)\,dt \\ & & \qquad -\,\int \limits _{0}^{T}\int \limits _{ 0}^{l}R_{ 2}(x,t)u^{N}(x,t)\,dx\,\varPhi ^{N^{{\prime}} }(l,t)\,dt -\int \limits _{0}^{T}K_{ 2}(0,t)u^{N}(0,t)\varPhi ^{N^{{\prime}} }(l,t)\,dt \\ & & \qquad +\,\int \limits _{ 0}^{T}K_{ 2}(l,t)u^{N}(l,t)\varPhi ^{N^{{\prime}} }(l,t)\,dt -\int \limits _{0}^{T}K_{ 1}(l,t)u^{N}(l,t)\varPhi ^{N^{{\prime}} }(0,t)\,dt \\ & & \qquad -\,\int \limits _{0}^{T}K_{ 1}(0,t)u^{N}(l,t)\varPhi ^{N^{{\prime}} }(0,t)\,dt. {}\end{array}$$

(19.24)

Since $\varPhi ^{N^{{\prime}} },\,\varPhi _{t}^{N^{{\prime}} },\,\varPhi _{x}^{N^{{\prime}} } \in L_{2}(Q_{T})$, the subsequence $\left \{u^{N_{m}}(x,t)\right \}$ converges weakly in L ₂(Q _T ), so it is possible to pass to the limit in (19.24) as $m \rightarrow \infty$ for any fixed $\varPhi ^{N^{{\prime}} }$. Thus, for any u(x, t) ∈ V ₂(Q _T ) the following identity holds

$$\displaystyle\begin{array}{rcl} & & \int \limits _{0}^{T}\left (-(u,\,\varPhi _{ t}^{N^{{\prime}} }) + (u_{x},\,\varPhi _{x}^{N^{{\prime}} }) - (cu,\,\varPhi ^{N^{{\prime}} })\right ) \\ & & \quad =\int \limits _{ 0}^{l}\varphi (x)\varPhi ^{N^{{\prime}} }(x,0)\,dx +\int \limits _{ 0}^{T}\int \limits _{ 0}^{l}R_{ 1}(x,t)u(x,t)\,dx\,\varPhi ^{N^{{\prime}} }(0,t)\,dt \\ & & \qquad -\int \limits _{0}^{T}\int \limits _{ 0}^{l}R_{ 2}(x,t)u(x,t)\,dx\,\varPhi ^{N^{{\prime}} }(l,t)\,dt -\int \limits _{0}^{T}K_{ 2}(0,t)u(0,t)\varPhi ^{N^{{\prime}} }(l,t)\,dt \\ & & \qquad +\int \limits _{ 0}^{T}K_{ 2}(l,t)u(l,t)\varPhi ^{N^{{\prime}} }(l,t)\,dt -\int \limits _{0}^{T}K_{ 1}(l,t)u(l,t)\varPhi ^{N^{{\prime}} }(0,t)\,dt \\ & & \qquad +\int \limits _{ 0}^{T}K_{ 1}(0,t)u(l,t)\varPhi ^{N^{{\prime}} }(0,t)\,dt. {}\end{array}$$

(19.25)

Denote $\varPhi =\bigcup _{ N^{{\prime}}=1}^{\infty }\varPhi ^{N^{{\prime}} }$. The set Φ is dense in W ₂ ¹(Q _T ) and hence, there exists a function Φ(x, t) ∈ W ₂ ¹(Q _T ) that is the limit of the sequence $\varPhi ^{N^{{\prime}} }$. Finally, we conclude that the relation (19.25) holds for all functions Φ(x, t) ∈ W ₂ ¹(Q _T ) and therefore, there exists the solution u(x, t) ∈ V ₂(Q _T ) to the problem (19.1), (19.2), (19.5), (19.6) in sense of Definition 19.1. Assume that there exist two different generalized solutions u ₁(x, t), u ₂(x, t) ∈ V ₂(Q _T ) to the problem (19.1), (19.2), (19.5), (19.6). Then

$$\displaystyle{ u = u_{1} - u_{2} \in V _{2}(Q_{T}) }$$

satisfies the following identity

$$\displaystyle\begin{array}{rcl} & & \int \limits _{Q_{T}}\left (-u\eta _{t} + u_{x}\eta _{x} - cu\eta \right )\,dx\,dt \\ & & \quad =\int \limits _{Q_{T}}\left (R_{1}(x,t)\eta (l,t) - R_{2}(0,t)\eta (0,t)\right )u(x,t)\,dx\,dt \\ & & \quad \quad +\,\int \limits _{ 0}^{T}\left (K_{ 1}(0,t)\eta (0,t) - K_{2}(0,t)\eta (l,t)\right )u(0,t)\,dt \\ & & \quad \quad +\,\int \limits _{ 0}^{T}\left (K_{ 2}(l,t)\eta (l,t) - K_{1}(l,t)\eta (0,t)\right )u(l,t)\,dt.{}\end{array}$$

(19.26)

We take

$$\displaystyle{ \eta (x,t) = \left \{\begin{array}{ll} 0, &b \leq t \leq T,\\ \int \limits _{ b}^{t}u(x,\tau )\,d\tau,&0 \leq t \leq b, \end{array} \right. }$$

where b ∈ [0, T] is arbitrary. Note that

$$\displaystyle{ \eta (x,t) \in W_{2}^{1}(Q_{ T}),\eta (x,T) = 0 }$$

and since η _xt  = u _x , so η _xt  ∈ L ₂(Q _T ). We substitute η(x, t) into (19.26) and express u, u _x in terms of η. Then (19.26) becomes

$$\displaystyle\begin{array}{rcl} & & \int _{Q_{b}}\left (-\eta _{t}^{2} +\eta _{ tx}\eta _{x} - c\eta \eta _{t}\right )\,dx\,dt \\ & & \quad =\int _{Q_{b}}\left (R_{1}(x,t)\eta (l,t) - R_{2}(0,t)\eta (0,t)\right )\eta _{t}(x,t)\,dx\,dt \\ & & \quad \quad +\,\int _{ 0}^{b}\left (K_{ 1}(0,t)\eta (0,t) - K_{2}(0,t)\eta (l,t)\right )\eta _{t}(0,t)\,dt \\ & & \quad \quad +\,\int _{ 0}^{T}\left (K_{ 2}(l,t)\eta (l,t) - K_{1}(l,t)\eta (0,t)\right )\eta _{t}(l,t)\,dt.{}\end{array}$$

(19.27)

Integrating by parts in (19.27) we obtain

$$\displaystyle\begin{array}{rcl} & & \frac{1} {2}\int \limits _{0}^{l}\eta _{ x}^{2}(x,0)\,dx +\int \limits _{ Q_{b}}\eta _{t}^{2}\,dx\,dt \\ & & \quad = \frac{1} {2}\left (-\int \limits _{0}^{l}c(x,0)\eta ^{2}(x,0)\,dx +\int \limits _{ Q_{b}}c_{t}\eta ^{2}\,dt\right ) \\ & & \qquad +\,\int \limits _{Q_{b}}\left (-R_{1}(x,t)\eta (l,t) + R_{2}(0,t)\eta (0,t)\right )\eta _{t}(x,t)\,dx\,dt \\ & & \qquad +\, \frac{1} {2}K_{1}(0,0)\eta ^{2}(0,0) + \frac{1} {2}\int \limits _{0}^{b}(K_{ 1}(0,t))_{t}\eta ^{2}(0,t)\,dt \\ & & \qquad +\, \frac{1} {2}K_{2}(l,0)\eta ^{2}(l,0) + \frac{1} {2}\int \limits _{0}^{b}(K_{ 2}(l,t))_{t}\eta ^{2}(l,t)\,dt \\ & & \qquad -\, K_{2}(0,0)\eta (l,0)\eta (0,0) -\int \limits _{0}^{b}K_{ 2}(0,t)\eta _{t}(l,t)\eta (0,t)\,dt \\ & & \qquad +\,\int \limits _{ 0}^{b}K_{ 1}(l,t)\eta (l,t)_{t}\eta (0,t)\,dt -\int \limits _{0}^{b}(K_{ 2}(0,t))_{t}\eta (l,t)\eta (0,t)\,dt.{}\end{array}$$

(19.28)

Under the assumptions (A1)–(A2) from (19.28) we obtain the following estimate

$$\displaystyle{ \int \limits _{0}^{l}\eta _{ x}^{2}(x,0)\,dx +\int \limits _{ Q_{b}}\eta _{t}^{2}\,dx\,dt \leq (c_{ 1}b + P_{1}b^{2})\int \limits _{ Q_{b}}\eta _{t}^{2}\,dx\,dt + P_{ 2}\int \limits _{Q_{b}}\eta _{x}^{2}\,dx\,dt, }$$

where

$$\displaystyle{ P_{1} = 2 + c_{2} + \frac{k_{3} + 2k_{4}} {l},\,P_{2} = 2(2l^{2} + k_{ 3}l + 3k_{4}l). }$$

Since b > 0 is an arbitrary, so let b be such that $1 - c_{1}b - Cb^{2} > 0$ and in particular,

$$\displaystyle{ 1 - c_{1}b - Cb^{2} \geq \frac{1} {2}. }$$

That is,

$$\displaystyle{ b \in [0,\nu ]),\,\nu = \frac{-c_{1} + \sqrt{c_{1 }^{2 } + 2C}} {2C}. }$$

Then for all b ∈ [0, ν]

$$\displaystyle{ \int \limits _{0}^{l}\eta _{ x}^{2}(x,0)\,dx + \frac{1} {2}\int \limits _{Q_{b}}\eta _{t}^{2}\,dx\,dt \leq P_{ 2}\int \limits _{Q_{b}}\eta _{x}^{2}\,dx\,dt. }$$

(19.29)

Define the function $u(x,t) =\int \limits _{ 0}^{t}u(x,\tau )\,d\tau$. Then $\eta (x,t) = y(x,t) - y(x,b)$ for t ∈ [0, b] and (19.29) can be represented as

$$\displaystyle{ \int \limits _{0}^{l}y_{ x}^{2}(x,b)\,dx + \frac{1} {2}\int \limits _{Q_{b}}y_{t}^{2}\,dx\,dt \leq P_{ 2}\int \limits _{Q_{b}}(y(x,t) - y(x,b))_{x}^{2}\,dx\,dt, }$$

which implies that

$$\displaystyle\begin{array}{rcl} \int \limits _{0}^{l}y_{ x}^{2}(x,b)\,dx& \leq & P_{ 2}\int \limits _{Q_{b}}(y(x,t) - y(x,b))_{x}^{2}\,dx\,dt {}\\ & \leq & 2P_{2}\int \limits _{Q_{b}}y_{x}^{2}(x,t)\,dx\,dt + 2P_{ 2}b\int _{0}^{l}y_{ x}^{2}(x,b)\,dx. {}\\ \end{array}$$

In particular, for $b \leq \dfrac{1} {4P_{2}}$ we obtain

$$\displaystyle{ \int \limits _{0}^{l}y_{ x}^{2}(x,b)\,dx \leq 4P_{ 2}\int \limits _{Q_{b}}y_{x}^{2}(x,t)\,dx\,dt. }$$

(19.30)

The estimate (19.30) is valid for all b ∈ [0, b ₁], where $b_{1} =\min \left \{ \dfrac{1} {4P_{2}},\,\nu \right \}$. From (19.30), Gronwall’s lemma and the condition y _x (x, 0) = 0 we obtain that y _x ²(x, b) = 0 for all b ∈ [0, b ₁]. And hence, η _x (x, t) = 0, t ∈ [0, b ₁]. Then from (19.30) it follows that $\eta _{t}(x,t) = u(x,t) = 0,\,t \in [0,b_{1}]$. Likewise, we repeat the above arguments and obtain u(x, t) = 0 for all t ∈ [b ₁, 2b ₁] and so on. Finally, we conclude that u(x, t) = 0 in Q _T that in turn implies uniqueness of the solution to the problem (19.1), (19.2), (19.5), (19.6).