An Operator Method for Investigation of the Stability of Time-Dependent Source Identification Telegraph Type Differential Problems

Abstract

This article is devoted to the study of the stability of time-dependent source identification telegraph type differential problems with dependent coefficients. Time-dependent source identification problems (SIPs) for telegraph differential equations (TDEs) with constant coefficients can be solved by classical integral-transform methods. However, these classical methods can be used, basically, in cases where the differential equation has constant coefficients. We establish the basic theorem of the stability of the time-dependent SIPs for the second-order linear differential equation (DE) in a Hilbert space with a self-adjoint positive definite operator (SAPDO) and damping term. In practice, stability estimates for the solution of the three types of SIPs for one-dimensional and for multidimensional TDEs with dependent coefficients and classic and non-classic conditions are obtained.

1. Introduction

SIPs for partial differential equations (DEs) play a significant role in natural science, applied sciences, engineering, quantum mechanics, diffusion equations, and heat equations (see, e.g., [1,2,3,4]). The study of space-dependent and time-dependent source identification problems for partial DEs plays a significant role in engineering and applied sciences (e.g., communication, machines and buildings, mathematical physics, and chemical physics) and has been investigated by many authors [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. In [5], the singular boundary method was applied to solve the mixed problem.

$$\left\{\begin{array}{c}{W}_{\tau \tau }(\tau ,\eta )-{\left(a\left(\eta \right)W_{\eta }(\tau ,\eta )\right)}_{\eta }+\delta W(\tau ,\eta )\hfill \\ \\ =P\left(\tau \right)G\left(\eta \right)+F(\tau ,\eta ),\tau \in \left(0,\mathsf{\Lambda }\right),\eta \in \left(0,b\right),\hfill \\ W\left(0,\eta \right)=\varpi \left(\eta \right),w_{\tau }\left(0,\eta \right)=\chi \left(\eta \right),\eta \in \left[0,b\right],\hfill \\ W\left(\tau ,0\right)=W\left(\tau ,b\right)=0,\hfill \\ \\ {\int }_0^{b}W\left(\tau ,y\right)dy=\xi \left(\tau \right),\tau \in \left[0,\mathsf{\Lambda }\right],\hfill \end{array}\right.$$

(1)

for the two-dimensional TDE with Dirichlet boundary condition on $S_D$ and Neumann boundary condition on $S_N$. Here, $\chi$ denotes the (bounded and connected) computational domain, and $S_{D}N=\partial \chi ,S_D\bigcap S_N=\varphi .$ The finite difference method was applied to discretize the first and second order time derivatives. The differential equation was split into a system of partial differential equations, which was solved using the method of a particular solution in combination with the singular boundary method to obtain the homogeneous solution. Finally, three numerical examples were considered to demonstrate the proposed method. The well-posedness of time-dependent SIPs for the one-dimensional hyperbolic equation with classical and non-classical boundary conditions

$$\left\{\begin{array}{c}{W}_{\tau \tau }(\tau ,\eta )-{\left(a\left(\eta \right)W_{\eta }(\tau ,\eta )\right)}_{\eta }+\delta W(\tau ,\eta )\hfill \\ \\ =P\left(\tau \right)G\left(\eta \right)+F(\tau ,\eta ),\tau \in \left(0,\mathsf{\Lambda }\right),\eta \in \left(0,b\right),\hfill \\ W\left(0,\eta \right)=\varpi \left(\eta \right),w_{\tau }\left(0,\eta \right)=\chi \left(\eta \right),\eta \in \left[0,b\right],\hfill \\ W\left(\tau ,0\right)=W\left(\tau ,b\right),W_{\eta }\left(\tau ,0\right)=W_{\eta }\left(\tau ,b\right),\hfill \\ \\ {\int }_0^{b}W\left(\tau ,y\right)dy=\xi \left(\tau \right),\tau \in \left[0,\mathsf{\Lambda }\right]\hfill \end{array}\right.$$

(2)

was investigated in the papers [22,23,24]. Assume that all compatibility conditions for the given smooth data functions $F(\tau ,\eta ),a\left(\eta \right)$, $\varpi \left(\eta \right),\chi \left(\eta \right),\xi \left(\tau \right),G\left(\eta \right)$ are fulfilled, that $G\left(0\right)=G\left(b\right)=0$ for (1), that $G\left(0\right)=G\left(b\right),G^{\prime }\left(0\right)=G^{\prime }\left(b\right)$ and $a\left(\eta \right)\ge a_0>0$, $a\left(b\right)=a\left(0\right)$ for (2), and that ${\int }_0^{b}G\left(y\right)dy\ne 0$. The authors proved the stability estimates for solutions to these types of problems. The time-dependent source identification problem

$$\left\{\begin{array}{c}{W}_{\tau \tau }(\tau ,\eta )+\beta W_{\eta }(\tau ,\eta )-{\left(a\left(\eta \right)W_{\eta }(\tau ,\eta )\right)}_{\eta }-\gamma {\left(a(-\eta )W_{\eta }(\tau ,-\eta )\right)}_{\eta }+\delta W(\tau ,\eta )\hfill \\ \\ =P\left(\tau \right)G\left(\eta \right)+F(\tau ,\eta ),\tau \in \left(0,\mathsf{\Lambda }\right),\eta \in \left(-b,b\right),\hfill \\ W\left(0,\eta \right)=\varpi \left(\eta \right),w_{\tau }\left(0,\eta \right)=\chi \left(\eta \right),\eta \in \left[-b,b\right],\hfill \\ W_{\eta }\left(\tau ,-b\right)=W_{\eta }\left(\tau ,b\right)=0,{\int }_{-b}^{b}W\left(\tau ,y\right)dy=\xi \left(\tau \right),\tau \in \left[0,\mathsf{\Lambda }\right]\hfill \end{array}\right.$$

(3)

for a one dimensional telegraph equation with involution and Neumann conditions was studied in paper [25]. Assume that all compatibility conditions for the given smooth data functions $F(\tau ,\eta ),a\left(\eta \right)$, $\varpi \left(\eta \right),\chi \left(\eta \right),\xi \left(\tau \right),G\left(\eta \right)$ are fulfilled, that $G^{\prime }(-b)=G^{\prime }\left(b\right)=0$ and $a\ge a\left(\eta \right)=a(-\eta )\ge \delta >0,\beta >0,$$\delta -a\left|\gamma \right|\ge 0,$, and that ${\int }_{-b}^{b}G\left(y\right)dy\ne 0$. The authors have proved the stability estimates for solutions to this problem. The SIPs for hyperbolic-parabolic type DEs have been studied in papers by Ashyralyev [26], Ashyralyev, Ashyraliyev and Ashyralyyeva [27].

Diverse fields of engineering and science have made classical and non-classical problems for TDEs a primary focus of their study in [28,29,30,31,32,33,34,35,36,37,38,39,40,41]. The well-posedness of the Cauchy problem for abstract TDEs

$$\left\{\begin{array}{c}{W}_{\tau \tau }\left(\tau \right)+\beta W_{\tau }\left(\tau \right)+AW\left(\tau \right)=F\left(\tau \right),\tau \in (0,\mathsf{\Lambda }),\hfill \\ \\ W\left(0\right)=\varpi ,W_{\tau }\left(0\right)=\chi \hfill \end{array}\right.$$

in a Hilbert space H with the SAPDO A was investigated in [28]. The well-posedness of the space-dependent SIPs for telegraph and elliptic-telegraph DEs was studied by Ashyralyev and Cekic [42] and by Ashyralyev and Al-Hammouri [43], respectively. In particular, the well-posedness of the source identification problem for a TDE with unknown parameter P

$$\left\{\begin{array}{c}{W}_{\tau \tau }\left(\tau \right)+\beta W_{\tau }\left(\tau \right)+AW\left(\tau \right)=P+F\left(\tau \right),\tau \in (0,\mathsf{\Lambda }),\hfill \\ \\ W\left(0\right)=\varpi ,W_{\tau }\left(0\right)=\chi ,W\left(\mathsf{\Lambda }\right)u=\zeta \hfill \end{array}\right.$$

in a Hilbert space H with the SAPDO A was proved in [42]. The authors established stability estimates for the solution to this problem. In applications, three source identification problems for telegraph equations were investigated. In addition, in [43], the authors investigated the source identification problems for the elliptic-telegraph differential equation with unknown parameter P in a Hilbert space with a self-adjoint positive definite operator. In [11], Kozhanov and Safiullaeva studied the solvability of the inverse problems on finding a solution $W(\tau ,\eta )$ and an unknown coefficient c for a TDE

$$W_{\tau \tau }(\tau ,\eta )-\Delta W(\tau ,\eta )+cW(\tau ,\eta )=F(\tau ,\eta ).$$

Theorems on the existence of the regular solutions were established. A feature of the problems was the presence of new overdetermination conditions for the considered class of equations. In [44], Kozhanov and Safiullaeva studied the solvability of the parabolic and hyperbolic inverse problems of finding a solution together with an unknown right hand side when the general overdetermination condition was given. Several theorems of the unique existence of regular solutions were established. In [45], Kozhanov and Teleshova investigated nonlinear inverse coefficient problems for non stationary higher- order Des of the pseudo-hyperbolic type. In [32], the problem of determining a pair of functions $(U(\tau ,\eta ),(V_1\left(\eta \right),V_2\left(\eta \right),V_3\left(\eta \right))$

$$\left\{\begin{array}{c}{U}_{\tau \tau }(\tau ,\eta )-\Delta U(\tau ,\eta )+{\displaystyle \sum _{r=1}^3}{V}_r\left(\eta \right)U_{{\eta }_r}(\tau ,\eta )\hfill \\ +F(\tau ,\eta ),\tau \in (0,\mathsf{\Lambda }),\eta =({\eta }_1,{\eta }_2,{\eta }_3)\in \chi ,\hfill \\ U(0,\eta )=\varpi \left(\eta \right),U_{\tau }(0,\eta )=\chi \left(\eta \right),\eta \in \chi ,\hfill \\ U_{\mu }(\tau ,\eta )={\int }_{\chi }K(\eta ,y)U(\tau ,y)dy,\tau \in (0,\mathsf{\Lambda }),\eta \in S,\hfill \\ {\int }_{[0,\mathsf{\Lambda }]}W(\tau ,\eta )U(\tau ,\eta )d\tau =H\left(\eta \right),\tau \in (0,\mathsf{\Lambda }),\eta \in \chi \hfill \end{array}\right.$$

(4)

for the three-dimensional hyperbolic DE with the Neumann boundary condition on S. Here, $\chi$ denotes the bounded domain, and $S=\partial \chi .$$\mu$ is an outward normal to the boundary $S.$ The authors reduced this problem to the optimal control problem for the second-order hyperbolic DE with a nonlocal condition. An existence theorem for the optimal control problem was established, and the necessary optimality condition was derived. Moreover, the gradient of the considered functional was calculated. Finally, the stability of time-dependent SIPs for TDEs has been not well-investigated. Time-dependent SIPs for TDEs with constant coefficients can be solved by classical methods, such as the Fourier transform method, the Fourier series method, and the Laplace method. However, these classical methods can be used, basically, in case when the DE has constant coefficients. It is well known that the most useful method for solving source identification for problems partial DEs with dependent coefficients is the operator method. Our goal in the present paper is to investigate the stability of time-dependent SIPs for TDEs with dependent coefficients. We have developed an operator approach for treating the stability of time-dependent SIPs for TDEs with dependent coefficients. Namely, the basic theorem on the stability of time-dependent SIPs for second-order ODEs in a Hilbert space with the SAPDO and damping term established. This operator approach allows us to prove the stability of the wider class of TDEs with dependent coefficients. In practice, newstability estimates for the solutions of three types of time-dependent SIPs for TDEs with dependent coefficients are obtained. The study of this paper is organized as follows. Section 1 is the introduction. In Section 2, auxiliary statements are given. In Section 3, theorems on the well-posedness of SIPs for TDEs are established. In Section 4, stability estimates for the solution of the three types of SIPs for one dimensional and for multidimensional TDEs with classic and non-classic conditions are obtained. Finally, Section 4 presents our conclusion and our future plans.

2. The Auxiliary Statements

It is known that various time-dependent SIPs for TDEs can be reduced to the time-dependent SIPs for second-order ODEs,

$$\left\{\begin{array}{c}{W}_{\tau \tau }\left(\tau \right)+\beta W_{\tau }\left(\tau \right)+AW\left(\tau \right)=P\left(\tau \right)G+F\left(\tau \right),\tau \in (0,\mathsf{\Lambda }),\hfill \\ \\ W\left(0\right)=\varpi ,W_{\tau }\left(0\right)=\chi ,C\left[W\left(\tau \right)\right]=\xi \left(\tau \right),\tau \in [0,\mathsf{\Lambda }]\hfill \end{array}\right.$$

(5)

in a Hilbert space $\mathit{H}$ with the SAPDO A with dense domain $D\left(A\right)$ in $\mathit{H}$, $A\ge \delta I,\delta >0,\beta \ge 0$. Here, $C:\mathit{H}\to R$ is a given linear bounded functional, $\xi \left(\tau \right):[0,\mathsf{\Lambda }]\to R$ is a given smooth function and $G\in D\left(A\right),CG\ne 0$.

By the solution of the time-dependent SIP (5) we refer to a pair $\left(W\right(\tau ),P(\tau \left)\right)$ with the following requirements:

• The element $W\left(\tau \right)$ belongs to $D\left(A\right)$ for all $\tau \in [0,\mathsf{\Lambda }]$, and the function $AW\left(\tau \right)$ is continuous on $[0,\mathsf{\Lambda }]$, $P\left(\tau \right)\in C[0,\mathsf{\Lambda }]$.
• $W\left(\tau \right)$ is twice continuously differentiable on $[0,\mathsf{\Lambda }]$.
• $\left(W\right(\tau ),P(\tau \left)\right)$ satisfies the DE and initial and additional conditions.

The solution of the time-dependent SIP (5) defined in this form will be referred to, in the sequel, as the solution of SIP (5) in the space $\tilde{C}\left(\mathit{H}\right)=C\left(\mathit{H}\right)\times C[0,\mathsf{\Lambda }]$. Here $C\left(\mathit{H}\right)=C(\mathit{H},[0,\mathsf{\Lambda }\left]\right)$ stands for the Banach space of all continuous $\mathit{H}$-valued functions $W\left(\tau \right)$ defined on $[0,\mathsf{\Lambda }]$, equipped with the norm

$${\parallel W\parallel }_{C\left(\mathit{H}\right)}=\underset{\tau \in [0,\mathsf{\Lambda }]}{max}{\parallel W\left(\tau \right)\parallel }_{\mathit{H}}.$$

(6)

Moreover, $C^1\left(\mathit{H}\right)=C^1([0,\mathsf{\Lambda }],\mathit{H})$ is the space of all continuously differentiable $\mathit{H}$-valued functions $W\left(\tau \right)$ defined on $[0,\mathsf{\Lambda }]$, equipped with the norm

$${\parallel W\parallel }_{C^1\left(\mathit{H}\right)}={\parallel W\parallel }_{C\left(\mathit{H}\right)}+\underset{\tau \in [0,\mathsf{\Lambda }]}{max}{\parallel W^{\prime }\left(\tau \right)\parallel }_{\mathit{H}}.$$

(7)

We first establish the basic theorem of the stability of SIP (5).

3. The Well-Posedness of SIP (5)

It is not difficult to prove that for $\frac{{\beta }^2}{4}<\delta$, $B=A-\frac{{\beta }^2}{4}I$ is the SAPDO in the space $\mathit{H}$. Through this paper, let $\left\{\mathrm{c}\right(\tau ),\tau \ge 0\}$ be a strongly continuous cosine and $\left\{\mathrm{s}\right(\tau ),\tau \ge 0\}$ be a sine operator-functions defined by the formulas

$$\mathrm{c}\left(\tau \right)W=\frac{e^{it{B}^{\frac{1}{2}}}+e^{-it{B}^{\frac{1}{2}}}}{2}W,$$

$$\mathrm{s}\left(\tau \right)W={\int }_0^{\tau }\mathrm{c}\left(y\right)Wdy.$$

Then, it follows that

$$\mathrm{s}\left(\tau \right)W=B^{-\frac{1}{2}}\frac{e^{it{B}^{\frac{1}{2}}}-e^{-it{B}^{\frac{1}{2}}}}{2i}W.$$

Let us formulate one lemma that will be needed to study of SIPs (38).

Lemma 1

([46]). For any $\tau \ge 0$, the following estimates hold:

$${\parallel \mathrm{s}\left(\tau \right)\parallel }_{\mathit{H}\to \mathit{H}}\le \tau ,{\parallel \mathrm{c}\left(\tau \right)\parallel }_{\mathit{H}\to \mathit{H}}\le 1,{∥B^{\frac{1}{2}}\mathrm{s}\left(t\right)∥}_{\mathit{H}\to \mathit{H}}\le 1.$$

(8)

In the present paper, $M(\delta ,G,\beta )$ denotes positive constants that may differ in time, and, thus, it is not a subject of precision.

Theorem 1.

Suppose $\beta \ge 0$ and $\varpi \in D\left(A\right),\chi \in D\left(A^{\frac{1}{2}}\right)$, $f\in C^1\left(\mathit{H}\right)$ and $\xi \in C^2[0,\mathsf{\Lambda }]$. Then the time-dependent SIP (5) has a unique solution $\left(u\right(\tau ),P(\tau \left)\right)$ in $C\left(\mathit{H}\right)\times C[0,\mathsf{\Lambda }]$.

Proof. 

Let $z\left(\tau \right)$ be the solution of the initial value problem (IVP),

$$\left\{\begin{array}{c}{z}_{\tau \tau }\left(\tau \right)+\beta z_{\tau }\left(\tau \right)+Az\left(\tau \right)=\nu \left(\tau \right)AG+F\left(\tau \right),\tau \in (0,\mathsf{\Lambda }),\hfill \\ \\ {\displaystyle z\left(0\right)=\varpi ,z_{\tau }\left(0\right)=\chi }\hfill \end{array}\right.$$

(9)

and let $\nu \left(\tau \right)$ be the function defined by the formula

$$\nu \left(\tau \right)={\int }_0^{\tau }(\tau -y)P\left(y\right)dy,0\le \tau \le \mathsf{\Lambda }.$$

(10)

Then,

$$W\left(\tau \right)=z\left(\tau \right)+\nu \left(\tau \right)G.$$

(11)

Using $C\left[W\right(\tau \left)\right]=\xi \left(\tau \right)$ and formula (11), we can obtain

$$\nu \left(\tau \right)=\frac{1}{CG}(\xi -C\left[z\left(\tau \right)\right]).$$

(12)

Since

$$P\left(\tau \right)={\nu }_{\tau \tau }\left(\tau \right),\tau \in (0,\mathsf{\Lambda }),\nu \left(0\right)=0,{\nu }^{\prime }\left(0\right)=0,$$

(13)

we obtain

$$P\left(\tau \right)=\frac{1}{CG}({\xi }_{\tau \tau }-C\left[z_{\tau \tau }\left(\tau \right)\right]),\tau \in (0,\mathsf{\Lambda }).$$

(14)

Thus, the next theorem completes the proof of Theorem 1. □

Theorem 2.

Suppose that the assumptions of Theorem 1 hold. Then, IVP (9) has a unique solution,$z\left(\tau \right)\in C\left(\mathit{H}\right)$.

Proof. 

IVP (9) is equivalent to the following integral equation

$$\begin{array}{c}z\left(\tau \right)=e^{-\frac{\beta }{2}\tau }(\mathrm{c}\left(\tau \right)+\frac{\beta }{2}\mathrm{s}\left(\tau \right))z\left(0\right)+{\mathrm{e}}^{-\frac{\beta }{2}\tau }s\left(\tau \right)z_{\tau }\left(0\right)\hfill \\ \\ +{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}\mathrm{s}(\tau -y)\left\{-\frac{1}{CG}(\xi \left(y\right)-C\left[z\left(y\right)\right])AG+F\left(y\right)\right\}dy.\hfill \end{array}$$

(15)

Taking the second-order derivative, with respect to $\tau$, we obtain

$$\begin{array}{c}{z}_{\tau \tau }\left(\tau \right)=-e^{-\frac{\beta }{2}\tau }\left(B+\frac{{\beta }^2}{4}I\right)\left(-\frac{\beta }{2}\mathrm{s}\left(\tau \right)+\mathrm{c}\left(\tau \right)\right)\varpi +e^{-\frac{\beta }{2}\tau }\left(-\left(B-\frac{{\beta }^2}{4}I\right)\mathrm{s}\left(\tau \right)-\beta \mathrm{c}\left(\tau \right)\right)\chi \hfill \\ \\ +\frac{1}{PG}(\xi \left(\tau \right)-P\left[z\left(\tau \right)\right])AG+F\left(\tau \right)\hfill \\ \\ +{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}\left(-\left(B-\frac{{\beta }^2}{4}I\right)\mathrm{s}(\tau -y)-\beta \mathrm{c}(\tau -y)\right)\left\{\frac{1}{CG}(\xi \left(y\right)-C\left[z\left(y\right)\right])AG+F\left(y\right)\right\}dy\hfill \\ \\ =-e^{-\frac{\beta }{2}\tau }\left\{\left(B+\frac{{\beta }^2}{4}I\right)\left(-\frac{\beta }{2}\mathrm{s}\left(\tau \right)+\mathrm{c}\left(\tau \right)\right)\varpi +\left(-\left(B-\frac{{\beta }^2}{4}I\right)\mathrm{s}\left(\tau \right)-\beta \mathrm{c}\left(\tau \right)\right)\chi \right\}\hfill \\ \\ +{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}\left(\frac{{\beta }^2}{4}\mathrm{s}(\tau -y)-\beta \mathrm{c}(\tau -y)\right)\left\{\frac{1}{CG}(\xi \left(y\right)-C\left[W\left(y\right)\right])AG+F\left(y\right)\right\}dy\hfill \\ \\ +e^{-\frac{\beta }{2}\tau }\mathrm{c}\left(\tau \right)\left\{\frac{1}{CG}(\xi \left(0\right)-C\left[\varpi \right])AG+F\left(0\right)\right\}\hfill \\ \\ +{\int }_0^{\tau }\mathrm{c}(\tau -y)e^{-\frac{\beta }{2}(\tau -y)}(\frac{\beta }{2}\left\{F\left(y\right)+\frac{1}{CG}(W\left(y\right)-C\left[W\left(y\right)\right])AG\right\}\hfill \\ \\ +\frac{1}{CG}({\xi }_y\left(y\right)-C\left[W_y\left(y\right)\right])AG+F_y\left(y\right))dy.\hfill \end{array}$$

(16)

Applying formulas

$${\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}B\mathrm{s}(\tau -y)g\left(y\right)dy=F\left(\tau \right)-e^{-\frac{\beta }{2}}\mathrm{c}\left(\tau \right)g\left(0\right)$$

$$-{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}\mathrm{c}(\tau -y)\left(\frac{\beta }{2}g\left(y\right)+g_y\left(y\right)\right)dy,$$

$$z\left(t\right)=z\left(0\right)+z_{\tau }\left(0\right)\tau +{\int }_0^{\tau }(\tau -y)z_y\left(y\right)dy,z_y\left(y\right)=z_y\left(0\right)+{\int }_0^{\tau }{z}_{yy}\left(y\right)dy$$

and substitution $z_{\tau \tau }\left(\tau \right)=v\left(\tau \right)$, we obtain

$$\begin{array}{c}v\left(\tau \right)=-e^{-\frac{\beta }{2}\tau }\left\{\left(B+\frac{{\beta }^2}{4}I\right)\left(-\frac{\beta }{2}\mathrm{s}\left(\tau \right)+\mathrm{c}\left(\tau \right)\right)\varpi +\left(-\left(B-\frac{{\beta }^2}{4}I\right)\mathrm{s}\left(\tau \right)-\beta \mathrm{c}\left(\tau \right)\right)\chi \right\}\hfill \\ \\ +{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}\left(\frac{{\beta }^2}{4}\mathrm{s}(\tau -y)-\beta \mathrm{c}(\tau -y)\right)\hfill \\ \\ \times \left\{\frac{1}{CG}(\xi \left(y\right)-C[\varpi +y\chi +{\int }_0^y(y-s)v\left(s\right)ds])AG+F\left(y\right)\right\}dy\hfill \\ \\ +e^{-\frac{\beta }{2}\tau }\mathrm{c}\left(\tau \right)\left\{\frac{1}{CG}(\xi \left(0\right)-C\left[\varpi \right])AG+F\left(0\right)\right\}+{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}\mathrm{c}(\tau -y)\hfill \\ \\ \times (\frac{\beta }{2}\left\{\frac{1}{CG}(W\left(y\right)-C[\varpi +y\chi +{\int }_0^y(y-r)v\left(r\right)dr])AG+F\left(r\right)\right\}\hfill \\ \\ +\frac{1}{CG}({\xi }_y\left(y\right)-C[\chi +{\int }_0^{y}v\left(r\right)dr])AG+F_y\left(y\right))dy.\hfill \end{array}$$

(17)

Note that (17) is a linear Volterra equation of the second kind with respect to $\tau$ for the $z_{\tau \tau }\left(\tau \right)=v\left(\tau \right)$ in $C\left(\mathit{H}\right)$. Thus, the proof of Theorem 2 is based on the fixed-point theorem. We use the following recursive formula:

$$\begin{array}{c}{v}_0\left(\tau \right)=-e^{-\frac{\beta }{2}\tau }\left(B+\frac{\beta 2}{4}I\right)\left(-\frac{\beta }{2}\mathrm{s}\left(\tau \right)+\mathrm{c}\left(\tau \right)\right)\varpi \hfill \\ \\ -e^{-\frac{\beta }{2}\tau }\left(-\left(B-\frac{{\beta }^2}{4}I\right)\mathrm{s}\left(\tau \right)-\beta \mathrm{c}\left(\tau \right)\right)\chi \},\hfill \\ \\ v_j\left(\tau \right)=-e^{-\frac{\beta }{2}\tau }\left(B+\frac{{\beta }^2}{4}I\right)\left(-\frac{\beta }{2}\mathrm{s}\left(\tau \right)+\mathrm{c}\left(\tau \right)\right)\varpi \hfill \\ \\ -e^{-\frac{\beta }{2}\tau }\left(-\left(B-\frac{{\beta }^2}{4}I\right)\mathrm{s}\left(\tau \right)-\beta \mathrm{c}\left(\tau \right)\right)\chi \}\hfill \\ \\ +{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}\left(\frac{{\beta }^2}{4}\mathrm{s}(\tau -y)-\beta \mathrm{c}(\tau -y)\right)\hfill \\ \\ \times \left\{\frac{1}{CG}(\xi \left(y\right)-C[\varpi +y\chi +{\int }_0^y(y-r)v_{j-1}\left(r\right)dr])AG+F\left(y\right)\right\}dy\hfill \\ \\ +e^{-\frac{\beta }{2}\tau }\mathrm{c}\left(\tau \right)\left\{\frac{1}{CG}(\xi \left(0\right)-C\left[\varpi \right])AG+F\left(0\right)\right\}+{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}\mathrm{c}(\tau -y)\hfill \\ \\ \times [\frac{\beta }{2}\left\{\frac{1}{CG}(W\left(y\right)-C[\varpi +y\chi +{\int }_0^y(y-r)v_{j-1}\left(r\right)dr])AG+F\left(y\right)\right\}\hfill \\ \\ +\frac{1}{CG}({\xi }_y\left(y\right)-C[\chi +{\int }_0^{y}v\left(r\right)dr])AG+F_y\left(y\right))dy,j\ge 1\hfill \end{array}$$

(18)

for the solution of IVP (9). Theorem 2 is established.

Now, we state the main results as follows. □

Theorem 3.

Suppose that the assumptions of Theorem 1 hold. Then the solution of SIP (5) obeys the stability inequality

$$\begin{array}{c}\parallel W_{\tau \tau }{\parallel }_{C\left(\mathit{H}\right)}+{\parallel AW\parallel }_{C\left(\mathit{H}\right)}+{\parallel P\parallel }_{C[0,\mathsf{\Lambda }]}\hfill \\ \\ \le M(\delta ,G,\beta )\left[{\parallel A\varpi \parallel }_{\mathit{H}}+\parallel A^{\frac{1}{2}}{\chi \parallel }_{\mathit{H}}+{\parallel F\parallel }_{C^1\left(\mathit{H}\right)}+{\parallel {\xi }_{\tau \tau }\parallel }_{C[0,\mathsf{\Lambda }]}\right].\hfill \end{array}$$

(19)

Proof. 

Using Formula (14), estimates (8), and $CG\ne 0$, we obtain the estimate

$$\left|P\left(\tau \right)\right|\le M_1(\delta ,G,\beta )\left[\parallel {\xi }_{\tau \tau }{\parallel }_{C[0,\mathsf{\Lambda }]}+{\parallel W_{\tau \tau }\parallel }_{\mathit{H}}\right]$$

(20)

for all $\tau \in [0,\mathsf{\Lambda }]$ and

$${\parallel P\parallel }_{C[0,\mathsf{\Lambda }]}\le M_1(\delta ,G,\beta )\left[\parallel {\xi }_{\tau \tau }{\parallel }_{C[0,\mathsf{\Lambda }]}+{\parallel W_{\tau \tau }\parallel }_{\mathit{H}}\right].$$

(21)

Now, using Formulas (11) and (13), we can obtain

$$u_{\tau \tau }\left(\tau \right)=W_{\tau \tau }\left(\tau \right)+P\left(\tau \right)G,0<\tau \le \mathsf{\Lambda }.$$

Using the triangle inequality, the formula gives us

$$\parallel W_{\tau \tau }{\parallel }_{C\left(\mathit{H}\right)}\le \parallel z_{\tau \tau }{\parallel }_{C\left(\mathit{H}\right)}+{\parallel P\parallel }_{C[0,\mathsf{\Lambda }]}{\parallel G\parallel }_{\mathit{H}}.$$

(22)

From that, the proof of estimate (19) is based on Equation (9), estimates (20) and (22), and on the following result on the stability estimate. Theorem 3 is established. □

Theorem 4.

Assume that the assumptions of Theorem 1 hold. Then the solution of IVP (9) obeys the stability inequality

$$\parallel z_{\tau \tau }{\parallel }_{C\left(\mathit{H}\right)}\le M(\delta ,G,\beta )\left[{\parallel A\varpi \parallel }_{\mathit{H}}+\parallel A^{\frac{1}{2}}{\chi \parallel }_{\mathit{H}}+{\parallel F\parallel }_{C^1\left(\mathit{H}\right)}+{\parallel \xi \parallel }_{C^2[0,\mathsf{\Lambda }]}\right].$$

(23)

Proof. 

Applying Formula (16), we can obtain

$$\begin{array}{c}{z}_{\tau \tau }\left(\tau \right)=-e^{-\frac{\beta }{2}\tau }\left\{\left(B+\frac{{\beta }^2}{4}I\right)\left(-\frac{\beta }{2}\mathrm{s}\left(\tau \right)+\mathrm{c}\left(\tau \right)\right)\varpi +\left(-\left(B-\frac{{\beta }^2}{4}I\right)\mathrm{s}\left(\tau \right)-\beta \mathrm{c}\left(\tau \right)\right)\chi \right\}\hfill \\ \\ +{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}\left(\frac{{\beta }^2}{4}\mathrm{s}(\tau -y)-\beta \mathrm{c}(\tau -y)\right)\hfill \\ \\ \times \left\{\frac{1}{CG}(\xi \left(y\right)-C[\varpi +y\chi +{\int }_0^y(y-r)v\left(r\right)dr])AG+F\left(y\right)\right\})dy\hfill \\ \\ +e^{-\frac{\beta }{2}\tau }\mathrm{c}\left(\tau \right)\left\{\frac{1}{CG}(\xi \left(0\right)-C\left[\varpi \right])AG+F\left(0\right)\right\}\hfill \\ \\ +{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}\mathrm{c}(\tau -y)\frac{\beta }{2}\left\{\frac{1}{CG}\left(W\left(y\right)-C[\varpi +y\chi +{\int }_0^y(y-r)v\left(r\right)dr]\right)AG+F\left(y\right)\right\}dy\hfill \\ \\ +{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}\mathrm{c}(\tau -y)\left\{\frac{1}{CG}({\xi }_y\left(y\right)-C[\chi +{\int }_0^{y}v\left(r\right)dr])AG+F_y\left(y\right)\right\}dy.\hfill \end{array}$$

(24)

Next, by the triangle inequality and estimates (8) and $CG\ne 0$, we obtain

$$\begin{array}{c}\parallel z_{\tau \tau }{\parallel }_{\mathit{H}}\le e^{-\frac{\beta }{2}\tau }\left(\frac{\beta }{2}+1\right){\parallel A\varpi \parallel }_{\mathit{H}}\hfill \\ +e^{-\frac{\beta }{2}\tau }\left(\parallel \left(B-\frac{{\beta }^2}{4}I\right)A^{-1}{\parallel }_{\mathit{H}\to \mathit{H}}\parallel A^{\frac{1}{2}}{\mathrm{s}\left(\tau \right)\parallel }_{\mathit{H}o\mathit{H}}+\beta {\parallel \mathrm{c}\left(\tau \right)\parallel }_{\mathit{H}\to \mathit{H}}\right){\parallel A^{\frac{1}{2}}\chi \parallel }_{\mathit{H}}\hfill \\ +{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}\left(\frac{{\beta }^2}{4}{\parallel \mathrm{s}(\tau -y)\parallel }_{\mathit{H}\to \mathit{H}}+\beta {\parallel \mathrm{c}(\tau -y)\parallel }_{\mathit{H}\to \mathit{H}}\right)\hfill \\ \\ \times \left\{\frac{1}{\left|CG\right|}\left(\left|\xi \left(y\right)\right|+\left|C[\varpi +z\chi +{\int }_0^y(y-r)v_{j-1}\left(r\right)dr]\right|\right){\parallel AG\parallel }_{\mathit{H}}+{\parallel F\left(y\right)\parallel }_{\mathit{H}}\right\}dy\hfill \\ \\ +{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}{\parallel \mathrm{c}(\tau -y)\parallel }_{\mathit{H}\to \mathit{H}}\frac{\beta }{2}\hfill \\ \\ \times \left\{\frac{1}{\left|CG\right|}\left(\left|\xi \left(y\right)\right|+\left|C[\varpi +y\chi +{\int }_0^y(y-r)v_{j-1}\left(r\right)dr]\right|\right){\parallel AG\parallel }_{\mathit{H}}+{\parallel F\left(y\right)\parallel }_{\mathit{H}}\right\}dy\hfill \\ \\ +{\int }_0^{\tau }{e}^{-\frac{\beta }{2}(\tau -y)}{(\beta \parallel \mathrm{s}(\tau -y)\parallel }_{\mathit{H}\to \mathit{H}}+{2\parallel \mathrm{c}(\tau -y)\parallel }_{\mathit{H}\to \mathit{H}}\left)\right\{\frac{1}{\left|CG\right|}(\left|{\xi }_y\left(y\right)\right|\hfill \\ \\ +\left|C[\chi +{\int }_0^y{v}_{j-1}\left(r\right)dr]\right|{)\parallel AG\parallel }_{\mathit{H}}+{\parallel F_y\left(y\right)\parallel }_{\mathit{H}}\}dy\hfill \\ \\ \le M_3(\delta ,G,\beta )\left[{\parallel A\varpi \parallel }_{\mathit{H}}+{\parallel \xi \parallel }_{C^2[0,\mathsf{\Lambda }]}+{\parallel F\parallel }_{C^1\left(\mathit{H}\right)}\right]+M_4(\delta ,G,\beta ){\int }_0^{\tau }{\parallel z_{yy}\left(y\right)\parallel }_{\mathit{H}}dy\hfill \end{array}$$

(25)

for all $\tau ,\tau \in [0,\mathsf{\Lambda }]$. In the next step, applying the integral inequality, we establish that

$$\parallel z_{\tau \tau }{\left(\tau \right)\parallel }_{\mathit{H}}\le M(\delta ,G,\beta )\left[{\parallel A\varpi \parallel }_{\mathit{H}}+\parallel A^{\frac{1}{2}}{\chi \parallel }_{\mathit{H}}+\parallel {\xi }_{\tau \tau }{\parallel }_{C[0,T]}+{\parallel F_{\tau }\parallel }_{C\left(\mathit{H}\right)}\right]e^{M_4(\delta ,G,\beta )\tau }$$

(26)

is satisfied for the solution of IVP (9) for every $\tau ,\tau \in [0,\mathsf{\Lambda }]$. From estimate (26), it follows estimate (23). Theorem 4 is proved. □

4. Applications

In the present section, we present the applications of the stability results of the main Theorem 3 to the three types of time-dependent SIPs for TDEs with local and nonlocal conditions.

First, we study the time-dependent SIP

$$\left\{\begin{array}{c}{W}_{\tau \tau }(\tau ,\eta )+\beta W_{\tau }(\tau ,\eta )-{\left(a\left(\eta \right)W_{\eta }(\tau ,\eta )\right)}_{\eta }+\delta W(\tau ,\eta )\hfill \\ \\ =P\left(\tau \right)G\left(\eta \right)+F(\tau ,\eta ),\tau \in \left(0,\mathsf{\Lambda }\right),\eta \in \left(0,b\right),\hfill \\ \\ W\left(0,\eta \right)=\varpi \left(\eta \right),w_{\tau }\left(0,\eta \right)=\chi \left(\eta \right),\eta \in \left[0,b\right],\hfill \\ \\ W\left(\tau ,0\right)=W\left(\tau ,b\right),W_{\eta }\left(\tau ,0\right)=W_{\eta }\left(\tau ,b\right),\hfill \\ \\ {\int }_0^{b}W\left(\tau ,y\right)dy=\xi \left(\tau \right),\tau \in \left[0,\mathsf{\Lambda }\right]\hfill \end{array}\right.$$

(27)

for the one-dimensional TDE with nonlocal conditions. Assume that all compatibility conditions for the given smooth data functions $F(\tau ,\eta ),a\left(\eta \right)$, $\varpi \left(\eta \right),\chi \left(\eta \right),\xi \left(\tau \right),G\left(\eta \right)$ are fulfilled, that $G\left(0\right)=G\left(b\right),G^{\prime }\left(0\right)=G^{\prime }\left(b\right)$, and that ${\int }_0^{b}G\left(y\right)dy\ne 0$, $a\left(\eta \right)\ge a_0>0$, $a\left(b\right)=a\left(0\right)$. Then the problem (27) has a unique solution $\left(W\right(\tau ,\eta ),P(\tau \left)\right)$. Problem (27) can be written as the time-dependent SIP (5) in a Hilbert space $\mathit{H}=L_2[0,l]$, with SAPDO $A=A^{\eta }$ defined by the formula

$$A^{\eta }z\left(\eta \right)=-{\left(a\left(\eta \right)z_{\eta }\left(\eta \right)\right)}_{\eta }+\delta z\left(\eta \right)$$

(28)

with the domain $D\left(A^{\eta }\right)=\{z\in W_2^2[0,l]:z\left(0\right)=z\left(b\right),z_{\eta }\left(0\right)=z_{\eta }\left(b\right)\}$. Thus, Theorem 3 enables us to obtain the result on the stability of the solution of problem (27).

Theorem 5.

Suppose that $\delta >\frac{{\beta }^2}{4},\beta \ge 0$, $\varpi \in W_2^2[0,b],\chi \in W_2^1[0,b]$, $F\in C^1\left(L_2[0,b]\right)$ and $\xi \in C^2[0,\mathsf{\Lambda }]$. Then, the SIP (27) has a unique solution $W\in C\left(L_2[0,b]\right)=C([0,\mathsf{\Lambda }],L_2[0,b]),p\in C[0,\mathsf{\Lambda }]$, and the following stability estimates hold:

$$\begin{array}{c}{\parallel W\parallel }_{C^2\left(L_2[0,b]\right)}+{\parallel W\parallel }_{C\left(W_2^2[0,b]\right)}+{\parallel P\parallel }_{C[0,\mathsf{\Lambda }]}\\ \\ \le M_1(G,\beta )\left[{\parallel \varpi \parallel }_{W_2^2[0,b]}+{\parallel \chi \parallel }_{W_2^1[0,b]}+{\parallel F\parallel }_{C^1\left(L_2[0,b]\right)}+{\parallel \xi \parallel }_{C^2[0,\mathsf{\Lambda }])}\right].\end{array}$$

Here, the Sobolev space $W_2^r[0,b]$ for $r=1,2$ is the set of all functions $z\left(\eta \right)$ defined on $[0,b]$, such that $z\left(\eta \right)$ and the r-th order derivative function $z^{\left(r\right)}\left(\eta \right)$ are all locally integrable in $L_2[0,b]$, equipped by the norm,

$${\parallel z\parallel }_{W_2^r[0,b]}={\left({\int }_0^b{\left|z\left(y\right)dy\right|}^2\right)}^{\frac{1}{2}}+\sum _{j=1}^r{\left({\int }_0^b{\left|z^r\left(y\right)dy\right|}^2\right)}^{\frac{1}{2}}.$$

Proof. 

Our proof of Theorem 5 is based on three results: the first one is the stability result of Theorem 3; the second one is the SAPDO $A=A^{\eta }$, defined by Formula (28); the third one is the boundedness in $L_2[0,b]$ of a linear functional C defined by the formula

$$Cz(\tau ,.)={\int }_0^{l}z(\tau ,y)dy,\tau \in [0,\mathsf{\Lambda }].$$

Second, let $\chi \subset R^n$ be a bounded open domain with a smooth boundary S, $\overline{\chi }=\chi \cup S$. In $[0,\mathsf{\Lambda }]\times \chi$, we study the multidimensional time-dependent SIP

$$\left\{\begin{array}{c}{W}_{\tau \tau }(\tau ,\eta )+\beta W_t(\tau ,\eta )-{\displaystyle \sum _{r=1}^n}{\left(a\left(\eta \right)W_{\eta }(\tau ,\eta )\right)}_{\eta }+\delta W(\tau ,\eta )\hfill \\ \\ =P\left(\tau \right)G\left(\eta \right)+F(\tau ,\eta ),\tau \in (0,\mathsf{\Lambda }),\eta =({\eta }_1,\cdots ,{\eta }_n)\in \chi ,\hfill \\ \\ W(0,\eta )=\varpi \left(\eta \right),W_{\tau }(0,\eta )=\chi \left(\eta \right),\eta \in \overline{\chi },\hfill \\ \\ {\displaystyle W(\tau ,\eta )=0,\eta \in S,{\int }_{\chi }W(\tau ,y)d{y}_1\cdots d{y}_n=\xi \left(\tau \right),\tau \in [0,\mathsf{\Lambda }],}\hfill \end{array}\right.$$

(29)

for the TDE with a Dirichlet boundary condition. Assume that all compatibility conditions for the given smooth data functions $F(\tau ,\eta ),a_r\left(\eta \right)$, $\varpi \left(\eta \right),\chi \left(\eta \right)$, $\xi \left(\tau \right)$ are fulfilled and

$${\int }_{\chi }G\left(y\right)d{y}_1\cdots d{y}_n\ne 0.$$

(30)

$a_r\left(\eta \right)\ge a>0$. Then the problem (29) has a unique solution $\left(W\right(\tau ,x),P(\tau \left)\right)$. Problem (29) can be written as the time-dependent SIP (5) in a Hilbert space $H=L_2\left(\overline{\chi }\right)$, with SAPDO $A=A^x$ defined by the formula

$$\begin{array}{c}{A}^{\eta }z\left(\eta \right)=-{\displaystyle \sum _{r=1}^n}{\left(a_r\left(\eta \right)z_{{\eta }_r}\right)}_{{\eta }_r}+\delta z\left(\eta \right)\hfill \end{array}$$

(31)

with domain

$$D\left(A^{\eta }\right)=\{z\left(\eta \right):z\left(\eta \right),{\left(a_r\left(\eta \right)z_{{\eta }_r}\right)}_{{\eta }_r}\in L_2\left(\chi \right),1\le r\le n,z\left(\eta \right)=0,\eta \in S\}.$$

Therefore, Theorem 3 allows us to obtain the next result on the stability of problem (29). □

Theorem 6.

Suppose that $\delta >\frac{{\beta }^2}{4},\beta \ge 0$, $\varpi \in W_2^2\left(\chi \right)$, $\chi \in W_2^1\left(\chi \right)$, $F\in C^1\left(L_2\left(\overline{\chi }\right)\right))$ and $\xi \in C^2[0,\mathsf{\Lambda }]$. Then, the time-dependent SIP (29) has a unique solution $W\in C\left(L_2\left(\overline{\chi }\right)\right)=C([0,\mathsf{\Lambda }],L_2\left(\overline{\chi }\right)),P\in C[0,\mathsf{\Lambda }]$, for which the following stability estimates hold

$$\begin{array}{c}{\parallel W\parallel }_{C^2\left(L_2\overline{\chi }\right))}+{\parallel W\parallel }_{C\left(W_2^2\left(\chi \right)\right)}+{\parallel P\parallel }_{C[0,\mathsf{\Lambda }]}\\ \\ \le M_1(\delta ,G,\beta )\left[{\parallel \varpi \parallel }_{W_2^2\left(\chi \right)}+{\parallel \chi \parallel }_{W_2^1\left(\chi \right)}+{\parallel F\parallel }_{C^1\left(L_2\left(\overline{\chi }\right)\right)}+{\parallel \xi \parallel }_{C^2[0,\mathsf{\Lambda }]}\right].\end{array}$$

(32)

Proof. 

Our proof of Theorem 6 is based on the stability estimates of Theorem 3, on the SAPDO $A=A^{\eta }$ given by the Formula (31), on the boundedness in $L_2\left(\overline{\chi }\right)$ of a linear functional C defined by the formula

$$CW(\tau ,.)={\int }_{\chi }W(\tau ,y)d{y}_1\cdots d{y}_n,\tau \in [0,\mathsf{\Lambda }]$$

(33)

and on the next theorem on the well-posedness of the elliptic problem in $L_2\left(\overline{\chi }\right)$. □

Theorem 7.

For the solution of the Dirichlet boundary value problem [46]

$$\left\{\begin{array}{c}{A}^{\eta }W\left(\eta \right)=\chi \left(\eta \right),\eta \in \chi ,\hfill \\ \\ W\left(\eta \right)=0,\eta \in S,\hfill \end{array}\right.$$

the coercive inequality

$$\sum _{r=1}^n\parallel W_{{\eta }_r{\eta }_r}{\parallel }_{L_2\left(\chi \right)}\le M{\parallel \chi \parallel }_{L_2\left(\chi \right)}$$

(34)

holds.

Third, in $[0,T]\times \chi$, we investigate the multidimensional time-dependent SIP,

$$\left\{\begin{array}{c}{W}_{\tau \tau }(\tau ,\eta )+\beta W_{\tau }(\tau ,\eta )-{\displaystyle \sum _{r=1}^n}{\left(a\left(\eta \right)W_{\eta }(\tau ,\eta )\right)}_{\eta }+\delta W(\tau ,\eta )\hfill \\ \\ =P\left(\tau \right)G\left(\eta \right)+F(\tau ,\eta ),\tau \in (0,\mathsf{\Lambda }),\eta =({\eta }_1,\cdots ,{\eta }_n)\in \chi ,\hfill \\ \\ W(0,\eta )=\varpi \left(\eta \right),W_{\tau }(0,\eta )=\chi \left(\eta \right),\eta \in \overline{\chi },\hfill \\ \\ W_{\mu }(\tau ,\eta )=0,\eta \in S,{\int }_{\overline{\chi }}u\left(\tau ,y\right)d{y}_1\cdots d{y}_n=\xi \left(\tau \right),\tau \in [0,\mathsf{\Lambda }],\hfill \end{array}\right.$$

(35)

for the TDE with Neumann boundary condition. Here, $\mu$ is the normal vector to $\chi$. Assume that all compatibility conditions for the given smooth data functions $F(\tau ,\eta ),a_r\left(\eta \right)$, $\varpi \left(\eta \right),\chi \left(\eta \right)$,$\xi \left(\tau \right)$,$G\left(\eta \right)$, are fulfilled, that $G_{\eta }\left(\eta \right)=0$, $\eta \in S$, and that

$${\int }_{\chi }G\left(y\right)d{y}_1\cdots d{y}_n\ne 0.$$

(36)

$a_r\left(\eta \right)\ge a>0$. Then, problem (35) has a unique solution $\left(W\right(\tau ,\eta ),P(\tau \left)\right)$. Problem (35) can be written as the time-dependent SIP (5) in a Hilbert space $\mathit{H}=L_2\left(\overline{\chi }\right)$, with SAPDO $A=A^{\eta }$ defined by the formula

$$A^{\eta }z\left(\eta \right)=-\sum _{r=1}^n{\left(a\left(\eta \right)z_{\eta }\left(\eta \right)\right)}_{\eta }+\delta z\left(\eta \right)$$

(37)

with domain

$$D\left(A^{\eta }\right)=\{z\left(\eta \right):z\left(\eta \right),{\left(a_r\left(\eta \right)z_{{\eta }_r}\right)}_{{\eta }_r}\in L_2\left(\chi \right),1\le r\le n,v_{\mu }\left(\eta \right)=0,\eta \in S\}.$$

Thus, Theorem 3 helps us to obtain the next result on the stability of problem (35).

Theorem 8.

Suppose that the assumptions of Theorem 3 hold. Then, the solutions of the time-dependent SIP (35) satisfy the stability estimates (32).

Proof. 

The proof of Theorem 8 is based on the stability results of Theorem 3, on the boundedness in $L_2\left(\overline{\chi }\right)$ of a linear functional B defined by the formula (33), on the SAPDO A in $L_2\left(\overline{\chi }\right)$ defined by the formula (37), and on the theorem on well-posedness of the elliptic problem in $L_2\left(\overline{\chi }\right)$. □

Theorem 9.

For the solution of the Neumann boundary value problem [46]

$$\left\{\begin{array}{c}{A}^{\eta }W\left(\eta \right)=\chi \left(\eta \right),\eta \in \chi ,\hfill \\ \\ W_{\mu }\left(\eta \right)=0,\eta \in S\hfill \end{array}\right.$$

the coercive inequality (34) holds.

5. Conclusions and Our Future Plans

• In this article, we prove the main theorem of the stability of time-dependant SIPs for the second-order linear ordinary DE in a Hilbert space with SAPDO. In practice, the stability estimates for the solution of three types of time-dependent SIPs for the TDEs are given.
• We are interested in study of the stability of a high order of accuracy difference schemes uniformly with respect to the timestep size of approximate solutions of time-dependant SIPs for TDEs, in which stability is established without any assumptions in respect of the grid steps $\tau$ and $h.$ Note that absolute stable difference schemes of a high order of accuracy for the initial value problem for the hyperbolic partial differential equations were presented and investigated in paper [46]. Applying the methods of this paper and paper [46], the absolutely stable difference schemes for the numerical solutions of the time-dependent SIPs for TDEs could be investigated. Naturally, the stability of these difference schemes can be proved.
• Investigating the uniform two-step difference schemes and asymptotic formulas for the solution of the time-dependent SIPs for the TDEs

  $$\left\{\begin{array}{c}{\epsilon }^2{W}_{\tau \tau }\left(\tau \right)+\beta W_{\tau }\left(\tau \right)+AW\left(\tau \right)=P\left(\tau \right)G+F\left(\tau \right),\tau \in (0,\mathsf{\Lambda }),\hfill \\ \\ W\left(0\right)=\varpi ,W_{\tau }\left(0\right)=\chi ,C\left[W\left(\tau \right)\right]=\xi \left(\tau \right),\tau \in [0,\mathsf{\Lambda }]\hfill \end{array}\right.$$

  (38)

  in a Hilbert space $\mathit{H}$with the SAPDO A with dense domain $D\left(A\right)$ in $\mathit{H}$ and with the $\epsilon \in \left(0,\infty \right)$ parameter multiplying the highest order derivative term. Here, $A\ge \delta I,\delta >0,\beta \ge 0$ and $C:\mathit{H}\to R$ is a given linear bounded functional, $\xi \left(\tau \right):[0,\mathsf{\Lambda }]\to R$ is a given smooth function and $G\in D\left(A\right),CG\ne 0$. Note that the uniform difference schemes and asymptotic formulas for the solution of perturbation problem has been investigated in earlier paper [47].
• Study the time-dependent SIP for the stochastic TDE

  $$\begin{array}{c}d\stackrel{·}{V}\left(\tau \right)+\beta dV\left(\tau \right)+Au\left(\tau \right)d\tau =\left(P\left(\tau \right)G+F\left(\tau \right)\right)d{W}_{\tau },\tau \in \left(0,\mathsf{\Lambda }\right),\hfill \\ \\ V\left(0\right)=\phi ,\stackrel{·}{V}\left(0\right)=\omega ,G\left[V\left(\tau \right)\right]=\psi \left(\tau \right),\tau \in \left[0,\mathsf{\Lambda }\right]\hfill \end{array}$$

  in a Hilbert space H with the SAPDE $A.$ Here $W_{\tau }$ is a standard Wiener process given on the probability space $(\mathsf{\Omega },F,P)$.