Skip to content

Advertisement

  • Research
  • Open Access

The nonlocal inverse problem of the identification of the lowest coefficient and the right-hand side in a second-order parabolic equation with integral conditions

Boundary Value Problems20192019:11

https://doi.org/10.1186/s13661-019-1126-z

  • Received: 15 September 2018
  • Accepted: 10 January 2019
  • Published:

Abstract

In the present paper a nonlocal inverse boundary-value problem for a second-order parabolic equation is considered. This investigation introduces the identification of the lowest unknown coefficient and time-depend right-hand side in a second-order parabolic equation on overdetermination at the internal points. Sufficient conditions for the existence and uniqueness of the classical solution to inverse problem of a second-order parabolic equation are obtained for small time.

Keywords

  • Inverse problem
  • Parabolic equation
  • Overdetermination conditions
  • Classical solution

MSC

  • 35A02
  • 35A09
  • 35R30
  • 35K10

1 Introduction

In this article we study the unique solvability of the inverse problem of determining the triple of functions \(\{ u(x,t), a(t), b(t)\}\) satisfying the equation
$$ c(t)u_{t}(x,t) = u_{xx}(x,t) + a(t)u(x,t) + b(t)g(x,t) + f(x,t), $$
(1.1)
with the nonlocal initial condition
$$ u(x,0) + \delta u(x,T) + \int _{0}^{T} p(t)u(x,t)\,dt = \varphi (x)\quad (0 \le x \le 1), $$
(1.2)
periodic boundary condition
$$ u(0,t) = \beta u(1,t)\quad (0 \le t \le T), $$
(1.3)
nonlocal integral condition
$$ \int _{0}^{1} u(x,t)\,dx = 0\quad (0 \le t \le T), $$
(1.4)
and the overdetermination conditions
$$ u(x_{i},t) = h_{i}(t)\quad (i = 1,2; 0 \le t \le T). $$
(1.5)
Here \(D_{T}: = \{ (x,t): 0 < x < 1, 0 < t \le T\}\) is a rectangular domain, \(T > 0\), \(\beta , \delta \ge 0\), \(x_{i} \in (0,1)\) (\(i = 1,2\); \(x _{1} \ne x_{2}\)) are fixed numbers, \(0 < c(t)\), \(g(x,t)\), \(f(x,t)\), \(0 \le p(t)\), \(\varphi (x)\), \(h_{i}(t)\) (\(i = 1, 2\)) are given functions, and \(u(x,t)\), \(a(t)\), \(b(t)\) are the desired functions.

Recently, in theory of partial differential equations, investigations devoted to the inverse coefficient problems and problems with discontinuous coefficients took an important place. Inverse problems arise in situations, when the structure of the mathematical model of the studying process is known and it is necessary to set the problems of determining the parameters of the mathematical model itself. Such problems include the problems of determining the various coefficients of the equations, external influence, boundary conditions, initial conditions, and so on. Many important applied problems lead to inverse problems.

The theory of inverse problems, by virtue of its theoretical and applied importance, is one of the intensively developing sections of the contemporary theory of partial differential equations. It attracts the attention of many researchers, who are interested in both the theory itself and its applications.

Fundamentals of the theory and practice of research of inverse problems were established and developed in the works published by Tikhonov [1], Lavrent’ev [2], Isakov et al. [3].

A more detailed bibliography and a classification of recent works connected with the investigation of inverse problems for partial differential equations can be found in monographs and in articles [413].

Problems of the solvability of inverse problems for equations of parabolic type were considered in the papers by Ivanchov [14, 15], Kozhanov [16], Vasin [17], Pyatkov [18], Kabanikhin [19], Ismailov [2023], and many others. Regarding recent results on the theory of partial differential equation with discontinuous coefficients, we refer the reader to [2426]. But the statement of problems and the proof technique used in this study are different from representations in these papers.

Nonlocal boundary-value problems are usually called problems in which, by together specifying the values of the solution or its derivatives on a fixed part of the boundary, a relationship is established between these values and the values of the same functions on other internal or boundary manifolds. The theory of nonlocal boundary value problems is important as a section of the general theory of boundary value problems for partial differential equations, it is also important as a branch of the theory of inverse problems.

Definition 1.1

The triplet \(\{ u(x,t), a(t), b(t)\}\) is said to be a classical solution of problem (1.1)–(1.5), if the functions \(u(x,t)\), \(a(t)\), and \(b(t)\) satisfy the following conditions:
  1. 1.

    The function \(u(x,t)\) and its derivatives \(u_{t}(x,t)\), \(u_{x}(x,t)\), \(u_{xx}(x,t)\), are continuous in the domain \(D_{T} \);

     
  2. 2.

    The functions \(a(t)\) and \(b(t)\) are continuous on the interval \([0,T]\);

     
  3. 3.

    Eq. (1.1) and conditions (1.2)–(1.5) are satisfied in the classical (usual) sense.

     

Theorem 1.2

([27])

Suppose that the following conditions are satisfied: \(\delta \ge 0\), \(0 < c(t) \in C[0,T]\), \(0 \le p(t) \in C[0,T]\), \(f(x,t) \in C(D_{T})\), \(\varphi (x) \in C[0,1]\), \(\int _{0}^{1} f(x,t)\,dx = 0\) (\(0 \le t \le T\)), \(g(x,t) \in C(D_{T})\), \(\int _{0}^{1} g(x,t)\,dx = 0\) (\(0 \le t \le T\)), \(h_{i}(t) \in C^{1}[0,T]\), (\(i = 1, 2\)), \(h(t) \equiv h_{1}(t)g(x_{2},t) - h_{2}(t)g(x_{1},t) \ne 0\) (\(0 \le t \le T\)), and the compatibility conditions
$$\begin{aligned}& \int _{0}^{1} \varphi (x)\,dx = 0, \\& h_{i}(0) + \delta h_{i}(T) + \int _{0}^{T} p(t)h_{i}(t)\,dt = \varphi (x _{i})\quad (i = 1, 2). \end{aligned}$$
Then, the problem of finding a classical solution of (1.1)(1.5) is equivalent to the problem of determining functions \(u(x,t) \in C^{2, 1}(D_{T})\), \(a(t) \in C[0,T]\), and \(b(t) \in C[0,T]\), satisfying Eq. (1.1), conditions (1.2) and (1.3), and the conditions
$$\begin{aligned}& u_{x}(0,t) = u_{x}(1,t)\quad (0 \le t \le T), \end{aligned}$$
(1.6)
$$\begin{aligned}& c(t)h'_{i}(t) = u_{xx}(x_{i},t) + a(t)h_{i}(t) + b(t)g(x_{i},t) + f(x _{i},t) \end{aligned}$$
(1.7)
for \(i = 1,2\); \(0 \le t \le T\).

Notice that, in the case of \(\beta = 1\), the considered problem was investigated in [27]. In this article we assume that \(\beta \ne \pm 1\). In particular, for \(\beta = 0\), we have an Ionkin type boundary condition [28].

2 The auxiliary spectral problem

Now, in order to investigate problem (1.1)–(1.3) and (1.6), we cite some known facts. Consider the following spectral problem [28, 29]:
$$\begin{aligned}& X''(x) + \lambda X(x) = 0 \quad (0 \le x \le 1), \end{aligned}$$
(2.1)
$$\begin{aligned}& X(0) = \beta X(1),\qquad X'(0) = X'(1) \quad (\beta \ne \pm 1). \end{aligned}$$
(2.2)
Obviously, the boundary value problem (2.1)–(2.2) is not self-adjoint and the problem
$$\begin{aligned}& Y''(x) + \lambda Y(x) = 0 \quad (0 \le x \le 1), \end{aligned}$$
(2.3)
$$\begin{aligned}& Y(0) = Y(1), \qquad Y'(1) = \beta Y'(0) \end{aligned}$$
(2.4)
will be a conjugated problem.
We denote the system of eigen- and adjoint functions of problem (2.3)–(2.4) in the following [29] as:
$$\begin{aligned}& \begin{gathered} X_{0}(x) = ax + b,\qquad \ldots,\qquad X_{2k - 1}(x) = (ax + b) \cos \lambda _{k}x,\\ X _{2k}(x) = \sin \lambda _{k}x,\qquad \ldots, \end{gathered} \end{aligned}$$
(2.5)
$$\begin{aligned}& \begin{gathered} Y_{0}(x) = 2,\qquad \ldots, \qquad Y_{2k - 1}(x) = 4 \cos \lambda _{k}x,\\ Y_{2k}(x) = 4(1 - b - ax) \sin \lambda _{k}x,\qquad \ldots, \end{gathered} \end{aligned}$$
(2.6)
where
$$ \lambda _{k} = 2k\pi\quad (k = 0,1,2,\ldots),\qquad a = (1 - \beta )/(1 + \beta ) \ne 0,\qquad b = \beta /(1 + \beta ). $$
(2.7)
It is straightforward to verify that the biorthogonality conditions
$$ (X_{i},Y_{j}) = \int _{0}^{1} X_{i}(x)Y_{j}(x)\,dx = \delta _{ij} $$
are fulfilled.

Here, \(\delta _{ij}\) is Kronecker’s symbol.

Theorem 2.1

([30])

The system of functions (2.5) forms a Riesz basis in the space \(L_{2}(0,1)\), and the estimates
$$ r \bigl\Vert \psi (x) \bigr\Vert _{L_{2}(0,1)} \le \sum _{k = 0}^{\infty } \psi _{k}^{2} \le R \bigl\Vert \psi (x) \bigr\Vert _{L_{2}(0,1)} $$
(2.8)
are true for any function \(\psi (x) \in L_{2}(0,1)\), where
$$\begin{aligned}& \psi _{k} = \bigl(\psi (x),Y_{k}(x)\bigr) = \int _{0}^{1} \psi (x)Y_{k}(x)\,dx \quad (k = 0,1,\ldots), \\& r = \biggl\{ \frac{1}{3} \biggl( \biggl( a + \frac{3}{2}b \biggr)^{2} + \frac{3}{4}b^{2} \biggr) + \frac{1}{2} \bigl( 1 + \bigl\Vert (ax + b)^{2} \bigr\Vert _{C[0,1]} \bigr) \biggr\} ^{ - 1}, \\& R = 8 \bigl( 1 + \bigl\Vert ( 1 - b - ax )^{2} \bigr\Vert _{C [ 0,1 ]} \bigr). \end{aligned}$$
Under the assumptions
$$\begin{aligned}& \psi (x) \in C^{2i - 1}[0, 1], \qquad \psi ^{(2i)}(x) \in L_{2}(0,1), \\& \psi ^{(2s)}(0) = \beta \psi ^{(2s)}(1), \qquad \psi ^{(2s + 1)}(0) = \psi ^{(2s + 1)}(1) \quad (s = \overline{0,i - 1}; i \ge 1), \end{aligned}$$
we establish the validity of the estimates:
$$\begin{aligned}& \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{2i}\psi _{2k - 1}\bigr)^{2} \Biggr) ^{\frac{1}{2}} \le 2\sqrt{2} \bigl\Vert \psi ^{(2i)}(x) \bigr\Vert _{L _{2}(0,1)}, \end{aligned}$$
(2.9)
$$\begin{aligned}& \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{2i}\psi _{2k}\bigr)^{2} \Biggr) ^{\frac{1}{2}} \le 2\sqrt{2} \bigl\Vert \psi ^{(2i)}(x) (1 - b - ax) - 2ai \psi ^{(2i - 1)}(x) \bigr\Vert _{L_{2}(0,1)}. \end{aligned}$$
(2.10)
Further, under the assumptions
$$\begin{aligned}& \psi (x) \in C^{2i}[0,1], \qquad \psi ^{(2i + 1)}(x) \in L_{2}(0,1), \\& \psi ^{(2s)}(0) = \beta \psi ^{(2s)}(1), \qquad \psi ^{(2s - 1)}(0) = \psi ^{(2s - 1)}(1)\quad (i \ge 1; s = \overline{0,i}), \end{aligned}$$
we obtain
$$\begin{aligned}& \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{2i + 1}\psi _{2k - 1}\bigr)^{2} \Biggr)^{\frac{1}{2}} \le 2\sqrt{2} \bigl\Vert \psi ^{(2i + 1)}(x) \bigr\Vert _{L_{2}(0,1)}, \end{aligned}$$
(2.11)
$$\begin{aligned}& \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{2i + 1}\psi _{2k}\bigr)^{2} \Biggr) ^{\frac{1}{2}} \le 2\sqrt{2} \bigl\Vert \psi ^{(2i + 1)}(x) (1 - b - ax) - a(2i + 1)\psi ^{(2i)}(x) \bigr\Vert _{L_{2}(0,1)}. \end{aligned}$$
(2.12)
Now, denote by \(B_{2,T}^{\alpha } \) [30] the space consisting of functions of the form
$$ u(x,t) = \sum_{k = 0}^{\infty } u_{k}(t)X_{k}(x) $$
in domain \(D_{T}\), where the functions \(u_{k}(t)\) (\(k = 0,1,2,\ldots\)) are continuous on the interval \([0,T]\), and satisfy the condition
$$ J(u) \equiv \bigl\Vert u_{0}(t) \bigr\Vert _{C[0,T]} + \Biggl( \sum_{k = 1} ^{\infty } \bigl(\lambda _{k}^{\alpha } \bigl\Vert u_{2k - 1}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} + \Biggl( \sum _{k = 1}^{\infty } \bigl(\lambda _{k}^{\alpha } \bigl\Vert u_{2k}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr) ^{\frac{1}{2}} < + \infty $$
whenever \(\alpha \ge 0\). The norm in the space \(B_{2,T}^{\alpha } \) is
$$ \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{\alpha }} = J(u). $$
In particular, when \(\alpha = 3\), the function \(u(x,t)\), as an element of the space \(B_{2,T}^{3}\), has the following properties:
$$\begin{aligned}& u(x,t), u_{x}(x,t), u_{xx}(x,t) \in C(D_{T}),\qquad u_{xxx}(x,t) \in C\bigl([0,T]; L_{2}(0,1)\bigr); \\& u(0,t) = \beta u(1,t),\qquad u_{x}(0,t) = u_{x}(1,t),\qquad u_{xx}(0,t) = \beta u _{xx}(1,t)\quad (0 \le t \le T). \end{aligned}$$
We denote by \(E_{T}^{\alpha } \) the Banach space \(B_{2,T}^{\alpha } \times C[0,T] \times C[0,T]\) of vector functions \(z(x,t) = \{ u(x,t),a(t), b(t)\}\) with the norm
$$ \bigl\Vert z(x,t) \bigr\Vert _{E_{2,T}^{\alpha }} = \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{\alpha }} + \bigl\Vert a(t) \bigr\Vert _{C[0,T]} + \bigl\Vert b(t) \bigr\Vert _{C[0,T]}. $$

It is known that \(B_{2,T}^{\alpha } \) and \(E_{T}^{\alpha } \) are Banach spaces.

3 Existence and uniqueness of the solution of the inverse problem

Since system (2.5) forms a Riesz basis in \(L_{2}(0,1)\) and systems (2.5)–(2.6) form a system of biorthogonal functions in \(L_{2}(0,1)\), we’ll seek the first component \(u(x,t)\) of classical solution \(\{ u(x,t), a(t), b(t)\}\) of problem (1.1)–(1.3), (1.6), and (1.7) in the form
$$ u(x,t) = \sum_{k = 0}^{\infty } u_{k}(t)X_{k}(x), $$
(3.1)
where
$$ u_{k}(t) = \int _{0}^{1} u(x,t)Y_{k}(x)\,dx \quad (k = 0,1, \ldots). $$
(3.2)

Moreover, \(X_{k}(x)\) and \(Y_{k}(x)\) are defined by relations (2.5) and (2.6), respectively.

Then by applying the method of separation of variables, from (1.1) and (1.2) we have
$$\begin{aligned}& c(t)u'_{0}(t) = F_{0}(t; u, a, b)\quad (0 \le t \le T), \end{aligned}$$
(3.3)
$$\begin{aligned}& c(t)u'_{2k - 1}(t) + \lambda _{k}^{2}u_{2k - 1}(t) = F_{2k - 1}(t; u, a, b) \quad (k = 1,2 \ldots ; 0 \le t \le T), \end{aligned}$$
(3.4)
$$\begin{aligned}& c(t)u'_{2k}(t) + \lambda _{k}^{2}u_{2k}(t) = F_{2k}(t; u, a, b) - 2a \lambda _{k}u_{2k - 1}(t)\quad (k = 1,2 \ldots ; 0 \le t \le T), \end{aligned}$$
(3.5)
$$\begin{aligned}& u_{k}(0) + \delta u_{k}(T) + \int _{0}^{T} p(t)u_{k}(t)\,dt = \varphi _{k}\quad (k = 0,1,2, \ldots ), \end{aligned}$$
(3.6)
where
$$\begin{aligned}& \lambda _{k} = 2k\pi\quad (k = 1,2, \ldots ), \\& F_{k}(t; u, a, b) = f_{k}(t) + b(t)g_{k}(t) + a(t)u_{k}(t),\quad (k = 0,1,2, \ldots ), \\& f_{k}(t) = \int _{0}^{1} f(x,t)Y_{k}(x)\,dx, \qquad g_{k}(t) = \int _{0}^{1} g(x,t)Y_{k}(x)\,dx, \\& \varphi _{k} = \int _{0}^{1} \varphi (x)Y_{k}(x)\,dx \quad (k = 0,1, \ldots ). \end{aligned}$$
Solving problem (3.3)–(3.6), we obtain
$$\begin{aligned}& \begin{aligned}[b] u_{0}(t) &= (1 + \delta )^{ - 1} \biggl( \varphi _{0} - \int _{0}^{T} p(t)u _{0}(t)\,dt - \delta \int _{0}^{T} \frac{1}{c(t)}F_{10}(t; u, a,b)\,dt \biggr) \\ &\quad {}+ \int _{0}^{t} \frac{1}{c(\tau )}F_{0}( \tau ; u, a,b)\,d\tau ,\end{aligned} \end{aligned}$$
(3.7)
$$\begin{aligned}& \begin{aligned}[b] u_{2k - 1}(t) &= \frac{e^{ - \int _{0}^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{ \lambda _{k}^{2}}{c(s)}\,ds}} \biggl( \varphi _{2k - 1} - \int _{0}^{T} p(t)u _{2k - 1}(t)\,dt \biggr)\\ &\quad {} + \int _{0}^{t} \frac{1}{c(\tau )}F_{2k - 1}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau \\ &\quad {}- \frac{\delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}} \int _{0} ^{T} \frac{1}{c(\tau )}F_{2k - 1}( \tau ; u, a, b)e^{ - \int _{\tau } ^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau ,\quad (k = 1,2, \ldots ),\end{aligned} \end{aligned}$$
(3.8)
$$\begin{aligned}& \begin{aligned}[b] u_{2k}(t) &= \frac{e^{ - \int _{0}^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}} \biggl( \varphi _{2k} - \int _{0}^{T} p(t)u_{2k}(t)\,dt \biggr)\\ &\quad {} + \int _{0}^{t} \frac{1}{c( \tau )}F_{2k}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{\lambda _{k} ^{2}}{c(s)}\,ds} \,d\tau \\ &\quad {}- \frac{\delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}} \int _{0} ^{T} \frac{1}{c(\tau )}F_{2k}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{ \lambda _{k}^{2}}{c(s)}\,ds} \,d\tau + q_{k}(t,T), \end{aligned} \end{aligned}$$
(3.9)
where
$$\begin{aligned} q_{k}(t,T) &= - 2a\lambda _{k} \biggl( \varphi _{2k - 1} - \int _{0}^{T} p(t)u _{2k - 1}(t)\,dt \biggr) \\ &\quad {}\times \frac{\delta e^{ - \int _{0}^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{\lambda _{k} ^{2}}{c(s)}\,ds}} \biggl[ \int _{0}^{t} \frac{d\tau }{c(\tau )} - \frac{ \delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}} \int _{0}^{T} \frac{d \tau }{c(\tau )} \biggr] \\ &\quad {}- 2a\lambda _{k} \biggl[ \int _{0}^{t} \frac{1}{c(\tau )} \biggl( \int _{0}^{\tau } \frac{1}{c(\xi )}F_{2k - 1}( \xi ; u, a, b)e^{ - \int _{ \xi }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau \biggr)\,d\xi \\ &\quad {} - \frac{\delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{ \lambda _{k}^{2}}{c(s)}\,ds}} \int _{0}^{T} \frac{1}{c(\tau )} \biggl( \int _{0}^{\tau } \frac{1}{c(\xi )}F_{2k - 1}( \xi ; u, a, b)e^{ - \int _{ \xi }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\xi \biggr)\,d\tau \biggr] \\ &\quad {}+ 2a\lambda _{k}\frac{\delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{ \lambda _{k}^{2}}{c(s)}\,ds}} \int _{0}^{T} \frac{1}{c(\xi )}F_{2k - 1}( \xi ; u, a, b)e^{ - \int _{\xi }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d \xi \\ &\quad {}\times \biggl[ \int _{0}^{t} \frac{1}{c(\tau )}\,d\tau - \frac{\delta e ^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}} \int _{0}^{T} \frac{1}{c( \tau )}\,d\tau \biggr] \quad (k = 1,2, \ldots ). \end{aligned}$$
(3.10)
After substituting expressions \(u_{0}(t)\), \(u_{2k - 1}(t)\), and \(u_{2k}(t)\) (\(k = 1,2, \ldots \)), respectively described by (3.7), (3.8), and (3.9), into (3.1), we have
$$\begin{aligned} u(x,t) &= (1 + \delta )^{ - 1} \biggl( \varphi _{0} - \int _{0}^{T} p(t)u _{0}(t)\,dt - \delta \int _{0}^{T} \frac{1}{c(t)}F_{10}(t; u, a,b)\,dt \biggr) \\ &\quad {} + \int _{0}^{t} \frac{1}{c(\tau )}F_{0}( \tau ; u, a,b)\,d\tau \\ &\quad {}+ \sum_{k = 1}^{\infty } \biggl\{ \frac{e^{ - \int _{0}^{t} \frac{ \lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}} \biggl( \varphi _{2k - 1} - \int _{0}^{T} p(t)u_{2k - 1}(t)\,dt \biggr) \\ &\quad {}+ \int _{0}^{t} \frac{1}{c(\tau )}F_{2k - 1}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau \\ &\quad {}- \frac{e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}} \int _{0} ^{T} \frac{1}{c(\tau )}F_{2k - 1}( \tau ; u, a, b)e^{ - \int _{\tau } ^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau \biggr\} X_{2k - 1} \\ &\quad {} + \sum _{k = 1}^{\infty } \biggl\{ \frac{e^{ - \int _{0}^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{\lambda _{k} ^{2}}{c(s)}\,ds}} \biggl( \varphi _{2k} - \int _{0}^{T} p(t)u_{2k}(t)\,dt \biggr) \\ &\quad {}+ \int _{0}^{t} \frac{1}{c(\tau )}F_{2k}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau \\ &\quad {}- \frac{\delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}} \int _{0}^{T} \frac{1}{c( \tau )}F_{2k}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{\lambda _{k} ^{2}}{c(s)}\,ds} \,d\tau + q_{k}(t,T) \biggr\} X_{2k}(x). \end{aligned}$$
(3.11)
Using (1.7) gives
$$\begin{aligned}& \begin{aligned}[b] a(t) &= \bigl[h(t)\bigr]^{ - 1}\Biggl\{ \bigl(c(t)h'_{1}(t) - f(x_{1}, t)\bigr)g(x_{2}, t) - \bigl(c(t)h'_{2}(t) - f(x_{2}, t)\bigr)g(x_{1}, t) \\ &\quad {}+ \sum_{k = 1}^{\infty } \lambda _{k}^{2}u_{2k - 1}(t) \bigl(g(x_{2},t)X_{2k - 1}(x_{1}) - g(x_{1},t)X_{2k - 1}(x_{2})\bigr) \\ &\quad {} + \sum_{k = 1}^{\infty } \lambda _{k}^{2}u_{2k}(t) \bigl(g(x_{2},t)X _{2k}(x_{1}) - g(x_{1},t)X_{2k}(x_{2}) \bigr) \Biggr\} , \end{aligned} \end{aligned}$$
(3.12)
$$\begin{aligned}& \begin{aligned}[b] b(t) &= \bigl[h(t)\bigr]^{ - 1}\Biggl\{ h_{1}(t) \bigl(c(t)h'_{2}(t) - f(x_{2}, t)\bigr) - h_{2}(t) \bigl(c(t)h'_{1}(t) - f(x_{1}, t)\bigr) \\ &\quad {}+ \sum_{k = 1}^{\infty } \lambda _{k}^{2}u_{2k - 1}(t) \bigl(h_{1}(t)X_{2k - 1}(x_{2}) - h_{2}(t)X_{2k - 1}(x_{1})\bigr) \\ &\quad {}+ \sum_{k = 1}^{\infty } \lambda _{k}^{2}u_{2k}(t) \bigl(h_{1}(t)X _{2k}(x_{2}) - h_{2}(t)X_{2k}(x_{1}) \bigr) \Biggr\} , \end{aligned} \end{aligned}$$
(3.13)
where
$$ h(t) \equiv h_{1}(t)g(x_{2},t) - h_{2}(t)g(x_{1},t) \ne 0\quad (0 \le t \le T). $$
We substitute relations (3.8) and (3.9) into (3.12) and (3.13), respectively, to obtain
$$\begin{aligned}& a(t) = \bigl[h(t)\bigr]^{ - 1}\Biggl\{ \bigl(c(t)h'_{1}(t) - f(x_{1}, t)\bigr)g(x_{2}, t) - \bigl(c(t)h'_{2}(t) - f(x_{2}, t)\bigr)g(x_{1}, t) \\& \hphantom{a(t) =} {}+ \sum_{k = 1}^{\infty } \lambda _{k}^{2} \biggl[ \frac{e^{ - \int _{0} ^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{ \lambda _{k}^{2}}{c(s)}\,ds}} \biggl( \varphi _{2k - 1} - \int _{0}^{T} p(t)u _{2k - 1}(t)\,dt \biggr) \\& \hphantom{a(t) =}{}+ \int _{0}^{t} \frac{1}{c(\tau )}F_{2k - 1}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau \\& \hphantom{a(t) =} {}- \frac{\delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{ \lambda _{k}^{2}}{c(s)}\,ds}} \int _{0}^{T} \frac{1}{c(\tau )}F_{2k - 1}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau \biggr] \\& \hphantom{a(t) =}{}\times \bigl(g(x_{2},t)X_{2k - 1}(x_{1}) - g(x_{1},t)X_{2k - 1}(x _{2})\bigr) \\& \hphantom{a(t) =} {}+ \sum_{k = 1}^{\infty } \lambda _{k}^{2} \biggl[ \frac{e^{ - \int _{0} ^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{ \lambda _{k}^{2}}{c(s)}\,ds}} \biggl( \varphi _{2k} - \int _{0}^{T} p(t)u_{2k}(t)\,dt \biggr) \\& \hphantom{a(t) =}{}+ \int _{0}^{t} \frac{1}{c(\tau )}F_{2k}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau \\& \hphantom{a(t) =} {}- \frac{\delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{ \lambda _{k}^{2}}{c(s)}\,ds}} \int _{0}^{T} \frac{1}{c(\tau )}F_{2k}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau + q_{k}(t,T) \biggr] \\& \hphantom{a(t) =} {}\times \bigl(g(x_{2},t)X_{2k - 1}(x_{1}) - g(x_{1},t)X_{2k - 1}(x _{2})\bigr) \Biggr\} , \end{aligned}$$
(3.14)
$$\begin{aligned}& b(t) = \bigl[h(t)\bigr]^{ - 1}\Biggl\{ h_{1}(t) \bigl(c(t)h'_{2}(t) - f(x_{2}, t)\bigr) - h_{2}(t) \bigl(c(t)h'_{1}(t) - f(x_{1}, t)\bigr) \\& \hphantom{b(t) =} {}+ \sum_{k = 1}^{\infty } \lambda _{k}^{2} \biggl[ \frac{e^{ - \int _{0} ^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{ \lambda _{k}^{2}}{c(s)}\,ds}} \biggl( \varphi _{2k - 1} - \int _{0}^{T} p(t)u _{2k - 1}(t)\,dt \biggr) \\& \hphantom{b(t) =}{}+ \int _{0}^{t} \frac{1}{c(\tau )}F_{2k - 1}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau \\& \hphantom{b(t) =} {}- \frac{\delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{ \lambda _{k}^{2}}{c(s)}\,ds}} \int _{0}^{T} \frac{1}{c(\tau )}F_{2k - 1}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau \biggr] \\& \hphantom{b(t) =}\times{}\bigl(h_{1}(t)X_{2k - 1}(x_{2}) - h_{2}(t)X_{2k - 1}(x_{1})\bigr) \\& \hphantom{b(t) =} {}+ \sum_{k = 1}^{\infty } \lambda _{k}^{2} \biggl[ \frac{e^{ - \int _{0} ^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{ \lambda _{k}^{2}}{c(s)}\,ds}} \biggl( \varphi _{2k} - \int _{0}^{T} p(t)u_{2k}(t)\,dt \biggr) \\& \hphantom{b(t) =}{}+ \int _{0}^{t} \frac{1}{c(\tau )}F_{2k}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau \\& \hphantom{b(t) =} {}- \frac{\delta e^{ - \int _{0}^{T} \frac{\lambda _{k}^{2}}{c(s)}\,ds}}{1 + \delta e^{ - \int _{0}^{T} \frac{ \lambda _{k}^{2}}{c(s)}\,ds}} \int _{0}^{T} \frac{1}{c(\tau )}F_{2k}( \tau ; u, a, b)e^{ - \int _{\tau }^{t} \frac{\lambda _{k}^{2}}{c(s)}\,ds} \,d\tau + q_{k}(t,T) \biggr] \\& \hphantom{b(t) =}{}\times \bigl(h_{1}(t)X_{2k}(x_{2}) - h_{2}(t)X_{2k}(x _{2})\bigr)\Biggr\} . \end{aligned}$$
(3.15)

Thus, the problem of finding the solutions (1.1)–(1.3), (1.6), and (1.7) reduces to finding a solution of system (3.11), (3.14), and (3.15) with unknown functions \(u(x,t)\), \(a(t)\), and \(b(t)\), respectively.

Lemma 3.1

If \(\{ u(x,t), a(t), b(t)\}\) is a classical solution of problem (1.1)(1.3), (1.6), and (1.7), then the functions
$$\begin{aligned}& u_{0}(t) = \int _{0}^{1} u(x,t)Y_{0}(x)\,dx, \\& u_{2k - 1}(t) = \int _{0}^{1} u(x,t)Y_{2k - 1}(x)\,dx\quad (k = 1,2, \ldots), \\& u_{2k}(t) = \int _{0}^{1} u(x,t)Y_{2k}(x)\,dx\quad (k = 1,2, \ldots) \end{aligned}$$
satisfy system (3.7), (3.8), and (3.9) on the interval \([0,T]\).

Corollary 3.2

Suppose that system (3.11), (3.14) and (3.15) has a unique solution. Then the problem (1.1)(1.3), (1.6), and (1.7), cannot have more than one solution; in other words, if problem (1.1)(1.3), (1.6), and (1.7) has a solution, then it is unique.

Now, consider the operator
$$ \varPhi (u,a,b) = \bigl\{ \varPhi _{1}(u,a,b), \varPhi _{2}(u,a,b), \varPhi _{3}(u,a,b)\bigr\} $$
in the space \(E_{T}^{3}\), where
$$\begin{aligned}& \varPhi _{1}(u,a,b) = \tilde{u}(x,t) = \sum_{k = 0}^{\infty } \tilde{u} _{k}(t)X_{k}(x), \\& \varPhi _{2}(u,a,b) = \tilde{a}(t), \qquad \varPhi _{3}(u,a,b) = \tilde{b}(t), \end{aligned}$$
and the functions \(\tilde{u}_{0}(t)\), \(\tilde{u}_{2k - 1}(t)\), \(\tilde{u}_{2k}(t)\) (\(k = 1, 2,\ldots\)), \(\tilde{a}(t)\), and \(\tilde{b}(t)\) are equal to the right-hand sides of (3.7), (3.8), (3.9), (3.14), and (3.15), respectively.
Assume that the data for problem (1.1)–(1.3) and (1.7) satisfy the following conditions:
  1. (A)

    \(\varphi (x) \in C^{2}[0,1]\), \(\varphi '''(x) \in L_{2}(0,1)\), \(\varphi (0) = \beta \varphi (1)\), \(\varphi '(0) = \varphi '(1)\), \(\varphi ''(0) = \beta \varphi ''(1)\) (\(\beta \ne \pm 1\));

     
  2. (B)
    \(f(x,t) \in C_{x,t}^{2,0}(D_{T})\), \(f_{xxx}(x,t) \in L_{2}(D _{T})\), \(f(0,t) = \beta f(1,t)\), \(f_{x}(0,t) = f_{x}(1,t)\),
    $$ f_{xx}(0,t) = \beta f_{xx}(1,t)\quad (\beta \ne \pm 1), (0 \le t \le T); $$
     
  3. (C)
    \(g(x,t) \in C_{x,t}^{2,0}(D_{T})\), \(g_{xxx}(x,t) \in L_{2}(D _{T})\), \(g(0,t) = \beta g(1,t)\), \(g_{x}(0,t) = g_{x}(1,t)\),
    $$ g_{xx}(0,t) = \beta g_{xx}(1,t) \quad (\beta \ne \pm 1), (0 \le t \le T); $$
     
  4. (D)

    \(\delta \ge 0\), \(c(t) \in C[0,T]\), \(c(t) > 0\) (\(0 \le t \le T\)), \(0 \le p(t) \in C[0,T]\), \(h_{i}(t) \in C^{1}[0,T]\) (\(i = 1,2\)), \(h(t) \equiv h_{1}(t)g(x_{2},t) - h_{2}(t)g(x_{1},t) \ne 0\).

     
From (3.10), it is easy to see that
$$\begin{aligned}& \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \bigl\Vert q_{k}(t,T) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} \\& \quad \le 4\sqrt{3} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}(1 + \delta ) \Biggl[ \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \Vert \varphi _{2k - 1} \Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} \\& \qquad {}+ T \bigl\Vert p(t) \bigr\Vert _{C[0,T]} \Biggl( \sum _{k = 1}^{ \infty } \bigl(\lambda _{k}^{3} \bigl\Vert u_{2k - 1}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} \Biggr] \\& \qquad {}+ 6\sqrt{2} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} ^{2}\sqrt{ \bigl\Vert c(t) \bigr\Vert _{C[0,T]}} T(1 + \delta )^{2} \Biggl[ \sqrt{T} \Biggl( \int _{0}^{T} \sum_{k = 1}^{\infty } \bigl(\lambda _{k} ^{3} \bigl\vert f_{2k - 1}(\tau ) \bigr\vert \bigr)^{2} \,d\tau \Biggr)^{ \frac{1}{2}} \\& \qquad {}+ T \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \Biggl( \sum _{k = 1}^{ \infty } \bigl(\lambda _{k}^{3} \bigl\Vert u_{2k - 1}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}}\\& \qquad {} + \sqrt{T} \bigl\Vert b(t) \bigr\Vert _{C[0,T]} \Biggl( \int _{0}^{T} \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \bigl\vert g _{2k - 1}(\tau ) \bigr\vert \bigr)^{2} \,d\tau \Biggr)^{\frac{1}{2}} \Biggr]. \end{aligned}$$
Hence, taking into account inequalities (2.9)–(2.12), we have
$$\begin{aligned}& \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \bigl\Vert q_{k}(t,T) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} \\& \quad \le 4\sqrt{3} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}(1 + \delta )\Biggl[2\sqrt{2} \bigl\Vert \varphi '''(x) \bigr\Vert _{L_{2}(0,1)} \\& \qquad {}+ T \bigl\Vert p(t) \bigr\Vert _{C[0,T]} \Biggl( \sum _{k = 1}^{ \infty } \bigl(\lambda _{k}^{3} \bigl\Vert u_{2k - 1}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} \Biggr] \\& \qquad {}+ 6\sqrt{2} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} ^{2}\sqrt{ \bigl\Vert c(t) \bigr\Vert _{C[0,T]}} T(1 + \delta )^{2}\Biggl[2 \sqrt{2T} \bigl\Vert f_{xxx}(x,t) \bigr\Vert _{L_{2}(D_{T})} \\& \qquad {} + T \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \Biggl( \sum _{k = 1}^{ \infty } \bigl(\lambda _{k}^{3} \bigl\Vert u_{2k - 1}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} \\& \qquad {}+ 2\sqrt{2T} \bigl\Vert b(t) \bigr\Vert _{C[0,T]} \bigl\Vert g_{xxx}(x,t) \bigr\Vert _{L_{2}(D_{T})} \Biggr]. \end{aligned}$$
(3.16)
Now from (3.7)–(3.9), respectively, we have
$$\begin{aligned}& \begin{aligned}[b] &\bigl\Vert \tilde{u}_{0}(t) \bigr\Vert _{C[0,T]} \\ &\quad \le (1 + \delta )^{ - 1}\bigl( \vert \varphi _{0} \vert + T \bigl\Vert p(t) \bigr\Vert _{C[0,T]} \bigl\Vert u_{0}(t) \bigr\Vert _{C[0,T]}\bigr) \\ &\qquad {}+ \bigl(1 + \delta (1 + \delta )^{ - 1}\bigr) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \biggl[ \sqrt{T} \biggl( \int _{0}^{T} \bigl\vert f_{0}(t) \bigr\vert ^{2}\,dt \biggr)^{\frac{1}{2}} \\ &\qquad {}+ T \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \bigl\Vert u_{0}(t) \bigr\Vert _{C[0,T]}+ \sqrt{T} \bigl\Vert b(t) \bigr\Vert _{C[0,T]} \biggl( \int _{0}^{T} \bigl\vert g_{0}(t) \bigr\vert ^{2}\,dt \biggr) ^{\frac{1}{2}} \biggr],\end{aligned} \end{aligned}$$
(3.17)
$$\begin{aligned}& \begin{aligned}[b] &\Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \bigl\Vert \tilde{u}_{2k - 1}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} \\ &\quad \le \sqrt{5} \Biggl[ \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \Vert \varphi _{2k - 1} \Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} + T \bigl\Vert p(t) \bigr\Vert _{C[0,T]} \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k} ^{3} \bigl\Vert u_{2k - 1}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{ \frac{1}{2}} \Biggr] \\ &\qquad {}+ \sqrt{5} (1 + \delta )) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \Biggl[ \sqrt{T} \Biggl( \int _{0}^{T} \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\vert f_{2k - 1}(\tau ) \bigr\vert \bigr)^{2} \,d\tau \Biggr) ^{\frac{1}{2}} \\ &\qquad {}+ T \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \Biggl( \sum _{k = 1} ^{\infty } \bigl(\lambda _{k}^{3} \bigl\Vert u_{2k - 1}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} \\ &\qquad {}+ T \bigl\Vert b(t) \bigr\Vert _{C[0,T]} \Biggl( \int _{0}^{T} \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \bigl\Vert g_{2k - 1}(\tau ) \bigr\Vert _{C[0,T]}\bigr)^{2} \,d\tau \Biggr)^{\frac{1}{2}} \Biggr], \end{aligned} \end{aligned}$$
(3.18)
$$\begin{aligned}& \begin{aligned}[b] &\Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \bigl\Vert \tilde{u}_{2k}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}}\\ &\quad \le \sqrt{6} \Biggl[ \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k} ^{3} \Vert \varphi _{2k} \Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}}+ T \bigl\Vert p(t) \bigr\Vert _{C[0,T]} \Biggl( \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\Vert u_{2k}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr) ^{\frac{1}{2}} \Biggr] \\ &\qquad {}+ \sqrt{6} (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \Biggl[ \sqrt{T} \Biggl( \int _{0}^{T} \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\vert f_{2k}(\tau ) \bigr\vert \bigr)^{2} \,d\tau \Biggr)^{ \frac{1}{2}}\\ &\qquad {} + T \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \Biggl( \sum_{k = 1} ^{\infty } \bigl(\lambda _{k}^{3} \bigl\Vert u_{2k}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} \\ &\qquad {}+ \sqrt{T} \bigl\Vert b(t) \bigr\Vert _{C[0,T]} \Biggl( \int _{0}^{T} \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \bigl\vert g_{2k}( \tau ) \bigr\vert \bigr)^{2} \,d\tau \Biggr)^{\frac{1}{2}} \Biggr]\\ &\qquad {} + \sqrt{6} \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \bigl\Vert q _{k}(t,T) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}}. \end{aligned} \end{aligned}$$
(3.19)
Taking into consideration (3.16), from (3.17)–(3.19), by virtue of (2.9)–(2.12), we have the following estimates:
$$\begin{aligned}& \begin{aligned}[b] \bigl\Vert \tilde{u}_{0}(t) \bigr\Vert _{C[0,T]} &\le A_{1}(T) + B_{1}(T) \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T} ^{3}}\\ &\quad {} + C_{1}(T) \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{3}} + D_{1}(T) \bigl\Vert b(t) \bigr\Vert _{C[0,T]},\end{aligned} \end{aligned}$$
(3.20)
$$\begin{aligned}& \begin{aligned}[b] \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \bigl\Vert \tilde{u}_{2k - 1}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} &\le A_{2}(T) + B_{2}(T) \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \bigl\Vert u(x,t) \bigr\Vert _{B _{2,T}^{3}} \\ &\quad {}+ C_{2}(T) \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{3}} + D_{2}(T) \bigl\Vert b(t) \bigr\Vert _{C[0,T]},\end{aligned} \end{aligned}$$
(3.21)
$$\begin{aligned}& \begin{aligned}[b] \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \bigl\Vert \tilde{u}_{2k}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} &\le A_{3}(T) + B_{3}(T) \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{3}} \\ &\quad {}+ C_{3}(T) \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{3}} + D_{3}(T) \bigl\Vert b(t) \bigr\Vert _{C[0,T]}, \end{aligned} \end{aligned}$$
(3.22)
where
$$\begin{aligned}& A_{1}(T) = 2(1 + \delta )^{ - 1} \bigl\Vert \varphi (x) \bigr\Vert _{L_{2}(0,1)} + 2\sqrt{T} \bigl(1 + \delta (1 + \delta )^{ - 1} \bigr) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \bigl\Vert f(x,t) \bigr\Vert _{L_{2}(D _{T})}, \\& B_{1}(T) = \bigl(1 + \delta (1 + \delta )^{ - 1}\bigr)T \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}, \\& C_{1}(T) = (1 + \delta )^{ - 1}T \bigl\Vert p(t) \bigr\Vert _{C[0,T]}, \\& D_{1}(T) = \bigl(1 + \delta (1 + \delta )^{ - 1}\bigr) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}\sqrt{T} \bigl\Vert b(t) \bigr\Vert _{C[0,T]} \bigl\Vert g(x,t) \bigr\Vert _{L_{2}(D_{T})}, \\& A_{2}(T) = 2\sqrt{10} \bigl\Vert \varphi '''(x) \bigr\Vert _{L_{2}(0,1)} + 2\sqrt{10T} (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \bigl\Vert f_{xxx}(x,t) \bigr\Vert _{L_{2}(D_{T})}, \\& B_{2}(T) = \sqrt{6} T(1 + \delta )) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}, \\& C_{2}(T) = \sqrt{6} T \bigl\Vert p(t) \bigr\Vert _{C[0,T]}, \\& D_{2}(T) = 4\sqrt{3T} (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \bigl\Vert g_{xxx}(x,t) \bigr\Vert _{L_{2}(D_{T})}, \\& \begin{aligned} A_{3}(T) &= 4\sqrt{3} \bigl\Vert \varphi '''(x) (1 - b - ax) - 3a\varphi ''(x) \bigr\Vert _{L_{2}(0,1)} \\ &\quad {}+ 4\sqrt{3T} (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \bigl\Vert f_{xxx}(x,t) (1 - b - ax) - 3af_{xx}(x,t) \bigr\Vert _{L_{2}(D_{T})} \\ &\quad {}+ 12\sqrt{2} \vert a \vert (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \bigl\Vert \varphi '''(x) \bigr\Vert _{L _{2}(0,1)} \\ &\quad {}+ 12\sqrt{3} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}^{2}\sqrt{ \bigl\Vert c(t) \bigr\Vert _{C[0,T]}} T\sqrt{T} (1 + \delta )^{2} \bigl\Vert f_{xxx}(x,t) \bigr\Vert _{L_{2}(D_{T})}, \end{aligned} \\& B_{3}(T) = \sqrt{6} T(1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} + 6\sqrt{2} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}^{2}\sqrt{ \bigl\Vert c(t) \bigr\Vert _{C[0,T]}} T^{2}(1 + \delta )^{2}, \\& C_{3}(T) = \sqrt{3} T \bigl\Vert p(t) \bigr\Vert _{C[0,T]} \biggl[ \sqrt{2} + 4 \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}(1 + \delta ) \biggr], \\& \begin{aligned} D_{3}(T) &= 6\sqrt{2T} (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}\biggl( \bigl\Vert g_{xxx}(x,t) (1 - b - ax) - 3ag_{xx}(x,t) \bigr\Vert _{L_{2}(D_{T})} \\ &\quad {}+ \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}\sqrt{ \bigl\Vert c(t) \bigr\Vert _{C[0,T]}} T(1 + \delta ) \bigl\Vert g_{xxx}(x,t) \bigr\Vert _{L_{2}(D_{T})} \biggr). \end{aligned} \end{aligned}$$
We conclude from (3.20)–(3.22) that
$$ \begin{aligned}[b] \bigl\Vert \tilde{u}(x,t) \bigr\Vert _{B_{2,T}^{3}} &\le A_{4}(T) + B_{4}(T) \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T} ^{3}} \\ &\quad {}+ C_{4}(T) \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{3}} + D_{4}(T) \bigl\Vert b(t) \bigr\Vert _{C[0,T]}, \end{aligned} $$
(3.23)
where
$$\begin{aligned}& A_{4}(T) = A_{1}(T) + A_{2}(T) + A_{3}(T), B_{4}(T) = B_{1}(T) + B _{2}(T) + B_{3}(T), \\& C_{4}(T) = C_{1}(T) + C_{2}(T) + C_{3}(T), D_{4}(T) = D_{1}(T) + D _{2}(T) + D_{3}(T). \end{aligned}$$
Now from (3.14) and (3.15) we have
$$\begin{aligned}& \bigl\Vert \tilde{a}(t) \bigr\Vert _{C[0,T]} \\& \quad \le \bigl\Vert \bigl[h(t)\bigr]^{ - 1} \bigr\Vert _{C[0,T]}\Biggl\{ \bigl\Vert \bigl(c(t)h'_{1}(t) - f(x_{1}, t) \bigr)g(x_{2}, t) - \bigl(c(t)h'_{2}(t) - f(x_{2}, t)\bigr)g(x_{1}, t)) \bigr\Vert _{C[0,T]} \\& \qquad {}+ \Biggl( \sum_{k = 1}^{\infty } \lambda _{k}^{ - 2} \Biggr)^{ \frac{1}{2}} \Biggl[ \Biggl( \sum _{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \Vert \varphi _{2k - 1} \Vert _{C[0,T]} \bigr)^{2} \Biggr) ^{\frac{1}{2}} \\& \qquad {} + T \bigl\Vert p(t) \bigr\Vert _{C[0,T]} \Biggl( \sum_{k = 1} ^{\infty } \bigl( \lambda _{k}^{3} \bigl\Vert u_{2k - 1}(t) \bigr\Vert _{C[0,T]} \bigr)^{2} \Biggr)^{\frac{1}{2}} \\& \qquad {}+ (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \Biggl[ \sqrt{T} \Biggl( \int _{0}^{T} \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\vert f_{2k - 1}(\tau ) \bigr\vert \bigr) ^{2}\,d\tau \Biggr)^{\frac{1}{2}} \\& \qquad {}+ T \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \Biggl( \sum _{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\Vert u_{2k - 1}(t) \bigr\Vert _{C[0, T]} \bigr)^{2} \Biggr)^{\frac{1}{2}} \\& \qquad {}+ \sqrt{T} \bigl\Vert b(t) \bigr\Vert _{C[0,T]} \Biggl( \int _{0} ^{T} \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\vert g_{2k - 1}( \tau ) \bigr\vert \bigr)^{2}\,d\tau \Biggr)^{\frac{1}{2}} \Biggr] \\& \qquad {}+ \Biggl( \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \Vert \varphi _{2k} \Vert _{C[0,T]} \bigr)^{2} \Biggr)^{\frac{1}{2}}+ T \bigl\Vert p(t) \bigr\Vert _{C[0,T]} \Biggl( \sum _{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\Vert u_{2k}(t) \bigr\Vert _{C[0,T]} \bigr) ^{2} \Biggr)^{\frac{1}{2}} \\& \qquad {}+ (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \Biggl[ \sqrt{T} \Biggl( \int _{0}^{T} \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\vert f_{2k}(\tau ) \bigr\vert \bigr)^{2}\,d \tau \Biggr)^{\frac{1}{2}} \\& \qquad {}+ T \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \Biggl( \sum _{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\Vert u_{2k}(t) \bigr\Vert _{C[0, T]} \bigr)^{2} \Biggr)^{\frac{1}{2}} \\& \qquad {}+ \sqrt{T} \bigl\Vert b(t) \bigr\Vert _{C[0,T]} \Biggl( \int _{0}^{T} \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\vert g _{2k}(\tau ) \bigr\vert \bigr)^{2}\,d\tau \Biggr)^{\frac{1}{2}} \Biggr] \\& \qquad {} + \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \bigl\Vert q_{k}(t,T) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} \Biggr] \\& \qquad {}\times \bigl\Vert \bigl\vert g(x_{2},t) \bigr\vert + \bigl\vert g(x_{1},t) \bigr\vert \bigr\Vert _{C[0,T]} \Vert ax + b \Vert _{C[0,1]}\Biggr\} , \end{aligned}$$
(3.24)
$$\begin{aligned}& \bigl\Vert \tilde{b}(t) \bigr\Vert _{C[0,T]} \\& \quad \le \bigl\Vert \bigl[h(t)\bigr]^{ - 1} \bigr\Vert _{C[0,T]}\Biggl\{ \bigl\Vert h_{1}(t) \bigl(c(t)h'_{2}(t) - f(x_{2}, t)\bigr) - h_{2}(t) \bigl(c(t)h'_{1}(t) - f(x_{1}, t)\bigr) \bigr\Vert _{C[0,T]} \\& \qquad {}+ \Biggl( \sum_{k = 1}^{\infty } \lambda _{k}^{ - 2} \Biggr)^{ \frac{1}{2}} \Biggl( \sum _{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \Vert \varphi _{2k - 1} \Vert _{C[0,T]} \bigr)^{2} \Biggr) ^{\frac{1}{2}} \\& \qquad {}\times T \bigl\Vert p(t) \bigr\Vert _{C[0,T]} \Biggl( \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\Vert u_{2k - 1}(t) \bigr\Vert _{C[0,T]} \bigr)^{2} \Biggr)^{\frac{1}{2}} \\& \qquad {}+ (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \Biggl[ \sqrt{T} \Biggl( \int _{0}^{T} \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\vert f_{2k - 1}(\tau ) \bigr\vert \bigr)^{2}\,d\tau \Biggr) ^{\frac{1}{2}} \\& \qquad {}+ T \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \Biggl( \sum_{k = 1} ^{\infty } \bigl( \lambda _{k}^{3} \bigl\Vert u_{2k - 1}(t) \bigr\Vert _{C[0, T]} \bigr)^{2} \Biggr)^{\frac{1}{2}} \\& \qquad {}+ T \bigl\Vert b(t) \bigr\Vert _{C[0,T]} \Biggl( \int _{0}^{T} \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\vert g_{2k - 1}( \tau ) \bigr\vert \bigr)^{2}\,d\tau \Biggr)^{\frac{1}{2}} \Biggr] \\& \qquad {}+ \Biggl( \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \Vert \varphi _{2k} \Vert _{C[0,T]} \bigr)^{2} \Biggr)^{\frac{1}{2}}+ T \bigl\Vert p(t) \bigr\Vert _{C[0,T]} \Biggl( \sum _{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\Vert u_{2k}(t) \bigr\Vert _{C[0,T]} \bigr) ^{2} \Biggr)^{\frac{1}{2}} \\& \qquad {}+ (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \Biggl[ \sqrt{T} \Biggl( \int _{0}^{T} \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\vert f_{2k}(\tau ) \bigr\vert \bigr)^{2}\,d \tau \Biggr)^{\frac{1}{2}} \\& \qquad {}+ T \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \Biggl( \sum _{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\Vert u_{2k}(t) \bigr\Vert _{C[0, T]} \bigr)^{2} \Biggr)^{\frac{1}{2}} \\& \qquad {}+ T \bigl\Vert b(t) \bigr\Vert _{C[0,T]} \Biggl( \int _{0}^{T} \sum_{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\vert g_{2k}(\tau ) \bigr\vert \bigr)^{2}\,d\tau \Biggr)^{\frac{1}{2}} \Biggr] \\& \qquad {}+ \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k}^{3} \bigl\Vert q_{k}(t,T) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} \bigl\Vert \bigl\vert h_{1}(t) \bigr\vert + \bigl\vert h_{2}(t) \bigr\vert \bigr\Vert _{C[0,T]} \Vert ax + b \Vert _{C[0,1]}\Biggr\} . \end{aligned}$$
(3.25)
Using (3.16), by virtue of (2.9)–(2.12), from the relations (3.24) and (3.25), we obtain
$$\begin{aligned}& \begin{aligned}[b] \bigl\Vert \tilde{a}(t) \bigr\Vert _{C[0,T]} &\le A_{5}(T) + B_{5}(T) \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{3}} \\ &\quad {}+ C _{5}(T) \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{3}} + D_{5}(T) \bigl\Vert b(t) \bigr\Vert _{C[0,T]},\end{aligned} \end{aligned}$$
(3.26)
$$\begin{aligned}& \begin{aligned}[b] \bigl\Vert \tilde{b}(t) \bigr\Vert _{C[0,T]} &\le A_{6}(T) + B_{6}(T) \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{3}} \\ &\quad {}+ C _{6}(T) \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{3}} + D_{6}(T) \bigl\Vert b(t) \bigr\Vert _{C[0,T]}, \end{aligned} \end{aligned}$$
(3.27)
where
$$\begin{aligned}& A_{5}(T) = \bigl\Vert \bigl[h(t)\bigr]^{ - 1} \bigr\Vert _{C[0,T]}\Biggl\{ \bigl\Vert h_{1}(t) \bigl(c(t)h'_{2}(t) - f(x_{2}, t)\bigr) - h_{2}(t) \bigl(c(t)h'_{1}(t) - f(x _{1}, t)\bigr) \bigr\Vert _{C[0,T]} \\& \hphantom{A_{5}(T) =}{}+ \Biggl( \sum_{k = 1}^{\infty } \lambda _{k}^{ - 2} \Biggr)^{ \frac{1}{2}} \biggl[ 2\sqrt{2} \biggl[ \biggl( 1 + 4\sqrt{3} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}(1 + \delta ) \biggr) \bigl\Vert \varphi '''(x) \bigr\Vert _{L_{2}(0,1)} \\& \hphantom{A_{5}(T) =}{}+ \bigl\Vert \varphi '''(x) (1 - b - ax) - 3a\varphi ''(x) \bigr\Vert _{L_{2}(0,1)}\biggr] + 2 \sqrt{2T} (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \\& \hphantom{A_{5}(T) =}{}\times \biggl[ \biggl( 1 + 6\sqrt{2} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \sqrt{ \bigl\Vert c(t) \bigr\Vert _{C[0,T]}} T(1 + \delta ) \biggr) \bigl\Vert f_{xxx}(x,t) \bigr\Vert _{L_{2}(D_{T})} \\& \hphantom{A_{5}(T) =}{}+ \bigl\Vert f_{xxx}(x,t) (1 - b - ax) - 3af_{xx}(x,t) \bigr\Vert _{L_{2}(D _{T})}\biggr]\biggr]\\& \hphantom{A_{5}(T) =}{}\times \bigl\Vert \bigl\vert g(x_{2},t) \bigr\vert + \bigl\vert g(x_{1},t) \bigr\vert \bigr\Vert _{C[0,T]} \Vert ax + b \Vert _{C[0,1]}\Biggr\} , \\& B_{5}(T) = 2T \bigl\Vert \bigl[h(t)\bigr]^{ - 1} \bigr\Vert _{C[0,T]} \Biggl( \sum_{k = 1}^{\infty } \lambda _{k}^{ - 2} \Biggr)^{\frac{1}{2}}(1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}\\& \hphantom{B_{5}(T) =}{}\times \biggl( 1 + 3 \sqrt{2} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \sqrt{ \bigl\Vert c(t) \bigr\Vert _{C[0,T]}} T(1 + \delta ) \biggr) \\& \hphantom{B_{5}(T) =}{}\times \bigl\Vert \bigl\vert g(x_{2},t) \bigr\vert + \bigl\vert g(x_{1},t) \bigr\vert \bigr\Vert _{C[0,T]} \Vert ax + b \Vert _{C[0,1]}, \\& C_{5}(T) = 2T \bigl\Vert \bigl[h(t)\bigr]^{ - 1} \bigr\Vert _{C[0,T]} \Biggl( \sum_{k = 1}^{\infty } \lambda _{k}^{ - 2} \Biggr)^{\frac{1}{2}} \bigl\Vert p(t) \bigr\Vert _{C[0,T]} \biggl( 1 + 2\sqrt{3} (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \biggr) \\& \hphantom{C_{5}(T) =}{}\times \bigl\Vert \bigl\vert g(x_{2},t) \bigr\vert + \bigl\vert g(x_{1},t) \bigr\vert \bigr\Vert _{C[0,T]} \Vert ax + b \Vert _{C[0,1]}, \\& D_{5}(T) = 2\sqrt{2T} (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \bigl\Vert \bigl[h(t)\bigr]^{ - 1} \bigr\Vert _{C[0,T]} \Biggl( \sum_{k = 1} ^{\infty } \lambda _{k}^{ - 2} \Biggr)^{\frac{1}{2}} \\& \hphantom{D_{5}(T) =}{}\times \biggl[ \biggl( 1 + 6\sqrt{2} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \sqrt{ \bigl\Vert c(t) \bigr\Vert _{C[0,T]}} T(1 + \delta ) \biggr) \bigl\Vert g_{xxx}(x,t) \bigr\Vert _{L_{2}(D_{T})} \\& \hphantom{D_{5}(T) =}{}+ \bigl\Vert g_{xxx}(x,t) (1 - b - ax) - 3ag_{xx}(x,t) \bigr\Vert _{L_{2}(D _{T})}\biggr]\\& \hphantom{D_{5}(T) =}{}\times \bigl\Vert \bigl\vert g(x_{2},t) \bigr\vert + \bigl\vert g(x_{1},t) \bigr\vert \bigr\Vert _{C[0,T]} \Vert ax + b \Vert _{C[0,1]}, \\& A_{6}(T) = \bigl\Vert \bigl[h(t)\bigr]^{ - 1} \bigr\Vert _{C[0,T]}\\& \hphantom{A_{6}(T) =}{}\times\Biggl\{ \bigl\Vert \bigl(c(t)h'_{1}(t) - f(x_{1}, t)\bigr)g(x_{2}, t) - \bigl(c(t)h'_{2}(t) - f(x_{2}, t)\bigr)g(x _{1}, t) \bigr\Vert _{C[0,T]} \\& \hphantom{A_{6}(T) =}{}+ \Biggl( \sum_{k = 1}^{\infty } \lambda _{k}^{ - 2} \Biggr)^{ \frac{1}{2}} \biggl[ 2\sqrt{2} \biggl[ \biggl( 1 + 4\sqrt{3} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}(1 + \delta ) \biggr) \bigl\Vert \varphi '''(x) \bigr\Vert _{L_{2}(0,1)} \\& \hphantom{A_{6}(T) =}{}+ \bigl\Vert \varphi '''(x) (1 - b - ax) - 3a\varphi ''(x) \bigr\Vert _{L_{2}(0,1)}\biggr] + 2 \sqrt{2T} (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \\& \hphantom{A_{6}(T) =}{}\times \biggl[ \biggl( 1 + 6\sqrt{2} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \sqrt{ \bigl\Vert c(t) \bigr\Vert _{C[0,T]}} T(1 + \delta ) \biggr) \bigl\Vert f_{xxx}(x,t) \bigr\Vert _{L_{2}(D_{T})} \\& \hphantom{A_{6}(T) =}{}+ \bigl\Vert f_{xxx}(x,t) (1 - b - ax) - 3af_{xx}(x,t) \bigr\Vert _{L_{2}(D _{T})}\biggr]\biggr]\\& \hphantom{A_{6}(T) =}{}\times \bigl\Vert \bigl\vert h_{1}(t) \bigr\vert + \bigl\vert h_{2}(t) \bigr\vert \bigr\Vert _{C[0,T]} \Vert ax + b \Vert _{C[0,1]}\Biggr\} , \\& B_{6}(T) = 2T \bigl\Vert \bigl[h(t)\bigr]^{ - 1} \bigr\Vert _{C[0,T]} \Biggl( \sum_{k = 1}^{\infty } \lambda _{k}^{ - 2} \Biggr)^{\frac{1}{2}}(1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \\& \hphantom{B_{6}(T) =}{}\times \biggl( 1 + 3\sqrt{2} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \sqrt{ \bigl\Vert c(t) \bigr\Vert _{C[0,T]}} T(1 + \delta ) \biggr)\\& \hphantom{B_{6}(T) =}{}\times\bigl\Vert \bigl\vert h_{1}(t) \bigr\vert + \bigl\vert h_{2}(t) \bigr\vert \bigr\Vert _{C[0,T]} \Vert ax + b \Vert _{C[0,1]}, \\& C_{6}(T) = 2T \bigl\Vert \bigl[h(t)\bigr]^{ - 1} \bigr\Vert _{C[0,T]} \Biggl( \sum_{k = 1}^{\infty } \lambda _{k}^{ - 2} \Biggr)^{\frac{1}{2}} \bigl\Vert p(t) \bigr\Vert _{C[0,T]} \biggl( 1 + 2\sqrt{3} (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \biggr) \\& \hphantom{C_{6}(T) =}{}\times \bigl\Vert \bigl\vert h_{1}(t) \bigr\vert + \bigl\vert h_{2}(t) \bigr\vert \bigr\Vert _{C[0,T]} \Vert ax + b \Vert _{C[0,1]}, \\& D_{6}(T) = 2\sqrt{2T} (1 + \delta ) \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \bigl\Vert \bigl[h(t)\bigr]^{ - 1} \bigr\Vert _{C[0,T]} \Biggl( \sum_{k = 1} ^{\infty } \lambda _{k}^{ - 2} \Biggr)^{\frac{1}{2}} \\& \hphantom{D_{6}(T) =}{}\times \biggl[ \biggl( 1 + 6\sqrt{2} \vert a \vert \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]} \sqrt{ \bigl\Vert c(t) \bigr\Vert _{C[0,T]}} T(1 + \delta ) \biggr) \bigl\Vert g_{xxx}(x,t) \bigr\Vert _{L_{2}(D_{T})} \\& \hphantom{D_{6}(T) =}{}+ \bigl\Vert g_{xxx}(x,t) (1 - b - ax) - 3ag_{xx}(x,t) \bigr\Vert _{L_{2}(D _{T})}\biggr]\\& \hphantom{D_{6}(T) =}{}\times \bigl\Vert \bigl\vert h_{1}(t) \bigr\vert + \bigl\vert h_{2}(t) \bigr\vert \bigr\Vert _{C[0,T]} \Vert ax + b \Vert _{C[0,1]}. \end{aligned}$$
From inequalities (3.23), (3.26), and (3.27), we deduce that
$$\begin{aligned}& \bigl\Vert \tilde{u}(x,t) \bigr\Vert _{B_{2,T}^{3}} + \bigl\Vert \tilde{a}(t) \bigr\Vert _{C[0,T]} + \bigl\Vert \tilde{b}(t) \bigr\Vert _{C[0,T]} \\& \quad \le A(T) + B(T) \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \bigl\Vert u(x,t) \bigr\Vert _{B _{2,T}^{3}} \\& \qquad {}+ C(T) \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{3}} + D(T) \bigl\Vert b(t) \bigr\Vert _{C[0,T]}, \end{aligned}$$
(3.28)
where
$$\begin{aligned}& A(T) = A_{4}(T) + A_{5}(T) + A_{6}(T),\qquad B(T) = B_{4}(T) + B_{5}(T) + B _{6}(T), \\& C(T) = C_{4}(T) + C_{5}(T) + C_{6}(T),\qquad D(T) = D_{4}(T) + D_{5}(T) + D _{6}(T). \end{aligned}$$

Theorem 3.3

If conditions (A)(D) and the condition
$$ \bigl(B(T) \bigl(A(T) + 2\bigr) + C(T) + D(T)\bigr) \bigl(A(T) + 2\bigr) < 1 $$
(3.29)
hold, then problem (1.1)(1.3), (1.6), and (1.7) has a unique solution in the ball \(K = K_{R}( \Vert z \Vert _{E_{T}^{3}} \le R \le A(T) + 2)\) of the space \(E_{T}^{3}\).

Remark 3.4

Inequality (3.29) is satisfied for sufficiently small values of T.

Proof

In the space \(E_{T}^{3}\), we consider the equation
$$ z = \varPhi z, $$
(3.30)
where \(z = \{ u, a, b\}\), and the components \(\varPhi _{i}(u,a, b)\) (\(i = 1,2,3\)) of operator \(\varPhi (u,a, b)\) are defined by the right-hand side of Eqs. (3.11), (3.14), and (3.15).
Consider the operator \(\varPhi (u,a, b)\) in the ball \(K = K_{R}\) of the space \(E_{T}^{3}\). Similarly, with the aid of (3.28), we obtain that for any \(z_{1},z_{2},z_{3} \in K_{R}\) the following inequalities hold:
$$\begin{aligned}& \Vert \varPhi z \Vert _{E_{T}^{3}} \\& \quad \le A(T) + B(T) \bigl\Vert a(t) \bigr\Vert _{C[0,T]} \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{3}} + C(T) \bigl\Vert u(x,t) \bigr\Vert _{B_{2,T}^{3}} + D(T) \bigl\Vert b(t) \bigr\Vert _{C[0,T]} \\& \quad \le A(T) + B(T) \bigl(A(T) + 2\bigr)^{2} + C(T) \bigl(A(T) + 2\bigr) + D(T) \bigl(A(T) + 2\bigr) < A(T) + 2, \end{aligned}$$
(3.31)
$$\begin{aligned}& \Vert \varPhi z_{1} - \varPhi z_{2} \Vert _{E_{T}^{3}} \le B(T)R\bigl( \bigl\Vert a_{1}(t) - a_{2}(t) \bigr\Vert _{C[0,T]} + \bigl\Vert u_{1}(x,t) - u_{2}(x,t) \bigr\Vert _{B_{2,T}^{3}}\bigr) \\& \hphantom{\Vert \varPhi z_{1} - \varPhi z_{2} \Vert _{E_{T}^{3}} \le}{}+ C(T) \bigl\Vert u_{1}(x,t) - u_{2}(x,t) \bigr\Vert _{B_{2,T}^{3}} + D(T) \bigl\Vert b_{1}(t) - b_{2}(t) \bigr\Vert _{C[0,T]}. \end{aligned}$$
(3.32)

Then by (3.29), from (3.31) and (3.32) it is clear that operator Φ on the set \(K = K_{R}\) satisfies the conditions of the contraction mapping principle. Therefore, operator Φ has a unique fixed point \(z = \{ u, a, b\}\), in the ball \(K = K_{R}\), which is a solution of Eq. (3.30), i.e., in the ball \(K = K_{R}\) it is the unique solution of the system (3.11), (3.14), and (3.15). Then the function \(u(x,t)\), as an element of space \(B_{2,T}^{3}\), is continuous and has continuous derivatives \(u_{x}(x,t)\) and \(u_{xx}(x,t)\) in \(D_{T}\).

Next, from (3.4) and (3.5), it follows that \(u'_{k}(t)\) (\(k = 1,2, \ldots \)) are continuous on \([0,T]\), and consequently we have
$$\begin{aligned}& \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k} \bigl\Vert u'_{2k - 1}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} \\& \quad \le \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}\sqrt{2} \Biggl[ \Biggl( \sum _{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\Vert u_{2k - 1}(t) \bigr\Vert _{C[0, T]} \bigr) ^{2} \Biggr)^{\frac{1}{2}} \\& \qquad {}+ 2\sqrt{2} \bigl\Vert \bigl\Vert f_{x}(x,t) + a(t)u_{x}(x,t) + b(t)g_{x}(x,t) \bigr\Vert _{C[0,T]} \bigr\Vert _{L_{2}(0,1)} \Biggr] \\& \quad < + \infty , \\& \Biggl( \sum_{k = 1}^{\infty } \bigl(\lambda _{k} \bigl\Vert u'_{2k}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr)^{\frac{1}{2}} \\& \quad \le \biggl\Vert \frac{1}{c(t)} \biggr\Vert _{C[0,T]}\sqrt{3} \Biggl[ \Biggl( \sum _{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\Vert u_{2k}(t) \bigr\Vert _{C[0, T]} \bigr)^{2} \Biggr) ^{\frac{1}{2}} \\& \qquad {}+ 2\sqrt{2} \bigl\Vert \bigl\Vert \bigl(f_{x}(x,t) + a(t)u_{x}(x,t) + b(t)g _{x}(x,t)\bigr) (1 - b - ax) \\& \qquad {} + a\bigl(f(x,t) + a(t)u(x,t) + b(t)g(x,t)\bigr) \bigr\Vert _{C[0,T]} \bigr\Vert _{L_{2}(0,1)} \\& \qquad {}+ 2a \Biggl( \sum _{k = 1}^{\infty } \bigl( \lambda _{k}^{3} \bigl\Vert u_{2k - 1}(t) \bigr\Vert _{C[0,T]}\bigr)^{2} \Biggr) ^{\frac{1}{2}} \Biggr] \\& \quad < + \infty . \end{aligned}$$

Hence we conclude that the function \(u_{t}(t,x)\) is continuous in domain \(D_{T}\).

Further, it is possible to verify that Eq. (1.1) and conditions (1.2), (1.3), (1.6), and (1.7) are satisfied in the usual sense. Consequently, \(\{ u(x,t),a(t), b(t)\}\) is a solution of (1.1)–(1.3), (1.6), and (1.7), and by Corollary 3.2 it is unique in the ball \(K = K_{R}\). The proof is complete. □

From Theorems 1.2 and 3.3, the following assertion follows directly.

Theorem 3.5

Suppose that all assumptions of Theorem 3.3 and the compatibility conditions
$$\begin{aligned}& \int _{0}^{1} \varphi (x)\,dx = 0, \\& h_{i}(0) + \delta h_{i}(T) + \int _{0}^{T} p(t)h_{i}(t)\,dt = \varphi (x _{i})\quad (i = 1, 2) \end{aligned}$$
hold. If
$$\int _{0}^{1} f(x,t)\,dx = 0, \qquad \int _{0}^{1} g(x,t)\,dx = 0\quad (0 \le t \le T), $$
the, problem (1.1)(1.5) has a unique classical solution in the ball \(K = K_{R}\).

Declarations

Acknowledgements

The author sincerely thanks the editors and anonymous referees for the careful reading of the original manuscript and valuable comments, which have improved the quality of my work.

Availability of data and materials

Not applicable.

Funding

Not applicable.

Authors’ contributions

The author read and approved the final manuscript.

Competing interests

The author declares that he has no competing interests regarding the publication of this paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Computational Mathematics, Baku State University, Baku, Azerbaijan

References

  1. Tikhonov, A.N.: On stability of inverse problems. Dokl. Akad. Nauk SSSR 39(5), 195–198 (1943) (in Russian) MathSciNetGoogle Scholar
  2. Lavrent’ev, M.M., Romanov, V.G., Shishatskii, S.T.: Ill-Posed Problems of Mathematical Physics and Analysis. Moscow (1980) Google Scholar
  3. Isakov, V.M.: Inverse parabolic problems with the final overdetermination. Commun. Pure Appl. Math. 44(2), 185–209 (1991) MathSciNetView ArticleGoogle Scholar
  4. Bertsch, M., Smarrazzo, F., Tesei, A.: On a class of forward–backward parabolic equations: existence of solutions. Nonlinear Anal. 177(part A), 46–87 (2018) MathSciNetView ArticleGoogle Scholar
  5. Farroni, F., Moscariello, G.: A nonlinear parabolic equation with drift term. Nonlinear Anal. 177(part B), 397–412 (2018) MathSciNetView ArticleGoogle Scholar
  6. Fragnelli, G., Mugnai, D.: Carleman estimates for singular parabolic equations with interior degeneracy and non-smooth coefficients. Adv. Nonlinear Anal. 6(1), 61–84 (2017) MathSciNetMATHGoogle Scholar
  7. Giacomoni, J., Radulescu, V., Warnault, G.: Quasilinear parabolic problem with variable exponent: qualitative analysis and stabilization. Commun. Contemp. Math. 20, 8, 38 pp. (2018) MathSciNetView ArticleGoogle Scholar
  8. Grabowski, P.: Small-gain theorem for a class of abstract parabolic systems. Opusc. Math. 38(5), 651–680 (2018) MathSciNetView ArticleGoogle Scholar
  9. Belov, Yu.Ya., Lyubanova, A.Sh., Polyntseva, S.V., Sorokin, R.V., Frolenkov, I.V.: Inverse Problems of Mathematical Physics. Siberian Federal University, Krasnoyarsk (2008) Google Scholar
  10. Cannon, J.R., Lin, Y.P., Wang, S.M.: Determination of a control parameter in a parabolic partial-differential equation. J. Aust. Math. Soc. Ser. B, Appl. Math 33(2), 149–163 (1991) MathSciNetView ArticleGoogle Scholar
  11. Denisov, A.M.: Elements of the Theory of Inverse Problems. Inverse and Ill-Posed Problems Series. VSP, Utrecht (1999) Google Scholar
  12. Kamynin, V.L.: The inverse problem of the simultaneous determination of the right-hand side and the lowest coefficient in parabolic equation with many space variables. Math. Notes 97(3–4), 349–361 (2015) MathSciNetView ArticleGoogle Scholar
  13. Ramm, A.G.: Inverse Problems. Springer, Berlin (2005) MATHGoogle Scholar
  14. Ivanchov, M.I.: Inverse Problem for Equations of Parabolic Type. Mathematical Studies, vol. 10. VNTL Publishers, Lviv (2003) Monograph Series MATHGoogle Scholar
  15. Ivanchov, M.I., Pabyrivska, N.V.: On determination of two time-dependent coefficients in a parabolic equation. Sib. Math. J. 43(2), 323–329 (2002) MathSciNetView ArticleGoogle Scholar
  16. Kozhanov, A.I.: Parabolic equations with unknown time-dependent coefficients. Comput. Math. Math. Phys. 57(6), 961–972 (2017) MathSciNetView ArticleGoogle Scholar
  17. Vasin, I.A., Kamynin, V.L.: On the asymptotic behavior of solutions of inverse problems for parabolic equations. Sib. Math. J. 38(4), 647–662 (1997) MathSciNetView ArticleGoogle Scholar
  18. Pyatkov, S.G., Samkov, M.L.: Solvability of some inverse problems for the nonstationary heat-and-mass-transfer system. J. Math. Anal. Appl. 446(2), 1449–1465 (2017) MathSciNetView ArticleGoogle Scholar
  19. Kabanikhin, S.I.: Inverse and Ill-Posed Problems. Theory and Applications. de Gruyter, Berlin (2012) MATHGoogle Scholar
  20. Kerimov, N.B., Ismailov, M.I.: Direct and inverse problems for the heat equation with a dynamic-type boundary condition. IMA J. Appl. Math. 80(5), 1519–1533 (2015) MathSciNetView ArticleGoogle Scholar
  21. Sadybekov, M., Oralsyn, G., Ismailov, M.: Determination of a time-dependent heat source under not strengthened regular boundary and integral overdetermination conditions. Filomat 32(3), 809–814 (2018) MathSciNetView ArticleGoogle Scholar
  22. Hazanee, A., Lesnic, D., Ismailov, M.I., Kerimov, N.B.: Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions. Appl. Math. Comput. 346, 800–815 (2019) MathSciNetGoogle Scholar
  23. Kanca, F., Ismailov, M.I.: Inverse problem of finding the time-dependent coefficient of heat equation from integral overdetermination condition data. Inverse Probl. Sci. Eng. 20(4), 463–476 (2012) MathSciNetView ArticleGoogle Scholar
  24. Ragusa, M.A., Scapellato, A.: Mixed Morrey spaces and their applications to partial differential equations. Nonlinear Anal., Theory Methods Appl. 151, 51–65 (2017) MathSciNetView ArticleGoogle Scholar
  25. Scapellato, A.: New perspectives in the theory of some function spaces and their applications. In: International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2017), AIP Conference Proceedings, vol. 1978, UNSP 140002-1 (2018) Google Scholar
  26. Scapellato, A.: Homogeneous Herz spaces with variable exponents and regularity results. Electron. J. Qual. Theory Differ. Equ. 2018, 82, 1–11 (2018) MathSciNetView ArticleGoogle Scholar
  27. Azizbayov, E.I., Mehraliyev, Y.T.: Solvability of nonlocal inverse boundary-value problem for a second-order parabolic equation with integral conditions. Electron. J. Differ. Equ. 2017(125), 1 (2017) MathSciNetMATHGoogle Scholar
  28. Ionkin, N.I.: Solution of a boundary-value problem in heat conduction with a nonclassical boundary condition. Differ. Equ. 13, 204–211 (1977) MATHGoogle Scholar
  29. Kasumov, T.B., Mirzoyev, V.S.: On one generalization of Ionkin’s example. In: The Abstracts of Scientific Conference Devoted to 100th Anniversary of the Academician A.I. Huseynov, pp. 87–88 (2007) Google Scholar
  30. Khudaverdiyev, K.I., Azizbekov, E.I.: Investigation of classical solution of a one-dimensional not self-adjoint mixed problem for a class of semi-linear pseudohyperbolic equations of fourth order. News Baku State Univ., Ser. Phys.-Math. Sci. 2, 114–121 (2002) MATHGoogle Scholar

Copyright

© The Author(s) 2019

Advertisement