1 Introduction

Let \(\Omega \subset {{\mathbb {R}}}^d,\,d\ge 1,\) be a bounded domain with smooth boundary \(\partial \Omega \) and \(\tau >0\) be fixed. For \(F:[0,\tau ]\times L^2(\Omega )\rightarrow L^2(\Omega )\) and \(g\in L^2(\Omega )\) we consider the problem of recovering \(u:[0,\tau ]\rightarrow L^2(\Omega )\) that satisfies the following final value problem (FVP) with a non-local term

$$\begin{aligned} \begin{array}{ll} u_t(x,t)-\Delta u(x,t)=F(t,u(x,t))+\int _t^\tau u(x,s)\,ds,&{}\quad \text {in}\quad \Omega \times (0,\tau ),\\ u(x,t)=0,&{}\quad \text {on}\quad \partial \Omega \times (0,\tau ),\\ u(x,\tau )=g(x),&{}\quad \text {in}\quad \Omega . \end{array} \end{aligned}$$
(1)

This type of PDE appears in various context of engineering such as modelling the heat conduction for materials with memory [17], analysis of space-time dependent nuclear dynamics [18], epidemic phenomena in biology [13]. For further details about applications of some parabolic equations with non-local term, the interested reader may refer to [11, 13] and the references therein. Additionally, it is worth to note that these type of PDE with non-local terms have attracted a lot of attention in control theory in the recent past (see for e.g., [6, 9, 22]).

As mentioned earlier, in this paper we are interested in recovering \(u:[0,\tau ]\rightarrow L^2(\Omega )\) from the knowledge of the final value \(g\in L^2(\Omega )\) that satisfies (1) for a general source function \(F:[0,\tau ]\times L^2(\Omega )\rightarrow L^2(\Omega ).\) Throughout, we assume that the possibly non linear source function F satisfies the following hypothesis:

(F1):

There exists \(\kappa >0\) such that \(\left\| F(s,\psi _1)-F(s,\psi _2)\right\| \le \kappa \left\| \psi _1-\psi _2\right\| \) for all \(s\in [0,\tau ]\) and \(\psi _1,\psi _2\in L^2(\Omega ).\)

(F2):

There exists \(\phi \in L^2(\Omega )\) such that \(s\mapsto F(s,\phi )\in L^1(0,\tau ;L^2(\Omega )).\)

Note that the assumptions (F1) and (F2) are quite standard. Indeed, (F1) requires that F is Lipschitz with respect to the second variable, and the condition (F2) is assumed just to ensure some integrability of F (see Lemma 3.3 (i)).

Since we are interested in the backward problem, we assume that \(u\in C([0,\tau ];L^2(\Omega ))\) is the solution of (1) corresponding to the exact final value \(g\in L^2(\Omega ).\) (In the next section we shall provide an example (see Example 2.1) of a FVP of the form (1) for which we obtain explicitly the analytical representation of the solution of the corresponding FVP.) An important question one would like to ask is whether the solution is stable with respect to the perturbation in the data. We shall show that solving the FVP (1) is an ill-posed problem (see Example 2.2) in the sense that a small perturbation in the final value g may lead to a large deviation in the solution u(t) for each \(0\le t<\tau .\) Thus, the considered FVP possesses the ill-posedness behaviour similar to that of the classical backward heat conduction equation [8].

In real world scenario, it is obvious that the final value g will be available from some measurements, and hence noise in the data is inevitable. Thus, we assume that for \(\delta >0,\) \(g^\delta \in L^2(\Omega )\) is the noisy data corresponding to the exact final value g satisfying

$$\begin{aligned} \left\| g^\delta -g\right\| _{L^2(\Omega )}\le \delta . \end{aligned}$$
(2)

Thus, the problem at hand is to recover \(u(t),\,0\le t<\tau ,\) where u is the unique solution of the FVP (1) for the exact final value \(g\in L^2(\Omega ),\) from the knowledge of \(g^\delta \) satisfying (2). Since the considered FVP is ill-posed, we have to employ some regularization method (see for e.g. [4, 15]) in order to obtain some stable approximations. In this regard, we shall obtain regularized solution by solving some non linear integral equation derived using a truncated version of the Fourier expansion of the sought solution. The level at which the Fourier expansion is truncated plays the role of regularization parameter.

The literature on the regularization aspect for the classical backward heat conduction problem is quite rich and several regularization methods (see for example, quasi reversibility method [10, 21]; quasi boundary value method [7, 14]; truncated regularization method [3, 16, 25]) were studied extensively over the last few decades. In comparison to that, there seems to be a very limited work devoted to the study of regularization aspect for the ill-posed FVP with non-local terms. To the best of our knowledge, a recent work of Luan and Khanh [13] is perhaps the first work, where the authors investigated the FVP/backward problem of identification of \(v(\cdot ,t)\) for \(0\le t<\tau \) from the knowledge of the final value \(h\in L^2(\Omega ),\) where v is the solution of

$$\begin{aligned} \begin{array}{ll} v_t(x,t)-\Delta v(x,t)=\beta \int _0^t v(x,s)\,ds,&{}\quad \text {in}\quad \Omega \times (0,\tau ),\\ v(x,t)=0,&{}\quad \text {on}\quad \partial \Omega \times (0,\tau ),\\ v(x,\tau )=h(x),&{}\quad \text {in}\quad \Omega , \end{array} \end{aligned}$$
(3)

for a fixed \(\beta >0.\) They have shown that for \(0<t<\tau ,\) the backward problem is well-posed in the sense that it has a unique solution and the solution depends continuously on the final value h, whereas for \(t=0,\) the backward problem is ill-posed (see [13, Theorem 3.2]). That is, the problem of identification of \(v_0:=v(\cdot ,0)\) from the knowledge of the final value h is ill-posed. In order to obtain stable approximations they have employed the truncated spectral regularization method and obtained error estimates of Hölder type for different type of parameter choice strategies.

It is clear from (1) and (3) that the non-local term associated with these equations are different. Also, it is worth noting that with the type of non-local term as considered in this paper, the ill-posedness behaviour of the backward problem changes completely, as discussed earlier. Thus, in view of this, studying the problem associated with (1) is indeed an interesting problem.

We now point out the important contributions of this paper.

  • To the best of our knowledge, the FVP (1) with a general source term that allows some non-linearity and involving a different version of the non-local term is studied for the first time from the regularization point of view.

  • We establish that the considered FVP is ill-posed for every \(0\le t<\tau \) and thus obtain stable approximations by employing a regularization method using the truncated Fourier expansion.

  • Under different Gevrey smoothness assumption, we obtain the corresponding error estimates. One of the key tool in deriving such rates is a version of Grönwalls’ inequality for iterated integrals, which perhaps is proposed and analysed for the first time in this paper.

The rest of the paper is organised as follows: In Sect. 2, we recall some well known results associated to the Dirichlet eigenvalue problem for a Laplace operator. Using the eigenvalues of the Dirichlet eigenvalue problem, we obtain an integral representation of the exact solution of the FVP (1), which plays the key role in defining the stable approximations. We also recall the definition of Gevrey spaces, that will play the role of source sets. In Sect. 3, we state and prove (wherever required) various necessary results that plays the essential role in deriving error estimates and defining the stable approximations. Moreover, in this section we provide a new version of Grönwalls’ type inequality for iterated integrals. Section 4 contains the main results of this paper. In this section we shall deal with regularization and obtain various error estimates. Moreover, we shall provide a parameter choice strategy and obtain the corresponding estimates.

2 Preliminaries

It is well known that the Dirichlet eigenvalue problem

$$\begin{aligned} \begin{array}{ll} -\Delta v=\lambda v,&{}\quad \text {in}\quad \Omega ,\\ v=0,&{}\quad \text {on}\quad \partial \Omega , \end{array} \end{aligned}$$
(4)

admits a sequence of eigenvalues \(0<\lambda _1\le \lambda _2\le \cdots \rightarrow \infty \) and the corresponding eigenfunctions \(\varphi _n\in H^1_0(\Omega )\cap H^2(\Omega )\) are such that \(\left\{ \varphi _n:\,n\in {{\mathbb {N}}}\right\} \) forms an orthonormal basis of \(L^2(\Omega )\) (see for e.g., [5]). It is also known that (see for e.g., [2]) there exist \(e_1,e_2>0\) such that

$$\begin{aligned} e_1 n^{2/d}\le \lambda _n\le e_2n^{2/d},\quad n\in {{\mathbb {N}}}. \end{aligned}$$
(5)

We first give an example of a FVP that is of the form (1) and show that it has a unique solution.

Example 2.1

Let \(d=1\) and \(\Omega =(0,1).\) Let \(\lambda _j\) be the eigenvalues for the Dirichlet problem (4) and \(\left\{ \varphi _j:\,j\in {{\mathbb {N}}}\right\} \) be the corresponding eigenfunctions. Then it is known that \(\lambda _j=j^2\pi ^2,\,j\in {{\mathbb {N}}}.\) Now consider the FVP

$$\begin{aligned} \begin{array}{ll} v_t(x,t)-v_{xx}(x,t)=v(x,t)+\int _t^\tau v(x,s)\,ds,&{}\quad \text {in}\quad \Omega \times (0,\tau ),\\ v(x,t)=0,&{}\quad \text {on}\quad \partial \Omega \times (0,\tau ),\\ v(x,\tau )=\varphi _n(x),&{}\quad \text {in}\quad \Omega . \end{array} \end{aligned}$$
(6)

Let

$$\begin{aligned} \alpha _j&=\frac{1}{2}\left( -\left( \lambda _j-1\right) +\sqrt{\left( \lambda _j-1\right) ^2-4}\right) \quad \text {and}\\ \beta _j&=\frac{1}{2}\left( -\left( \lambda _j-1\right) -\sqrt{\left( \lambda _j-1\right) ^2-4}\right) . \end{aligned}$$

Then it can be verified easily that

$$\begin{aligned} v(x,t)=\frac{\alpha _ne^{-\alpha _n(\tau -t)}-\beta _ne^{-\beta _n(\tau -t)}}{\alpha _n-\beta _n}\varphi _n(x). \end{aligned}$$

is the solution of (6).

We now give an example to show that the considered FVP (1) is ill-posed in the sense that a small perturbation in the final value g may lead to a large deviation in the solution u of the FVP (1).

Example 2.2

Let \(\Omega ,\lambda _j,\varphi _j\,\alpha _j\,\beta _j\) be as considered in Example 2.1. Consider the FVP of recovering the solution \(v(\cdot ,t)\) for \(0\le t<\tau \) such that

$$\begin{aligned} \begin{array}{ll} v_t-v_{xx}=v+\int _t^\tau v(s)\,ds,&{}\quad \text {in}\quad \Omega \times (0,\tau ),\\ v=0,&{}\quad \text {on}\quad \partial \Omega \times (0,\tau ),\\ v(\cdot ,\tau )=h^n,&{}\quad \text {in}\quad \Omega , \end{array} \end{aligned}$$
(7)

where \(h^n=\frac{1}{\left| \beta _n\right| }\varphi _n.\) It can be verified easily that for every \(n\in {{\mathbb {N}}},\)

$$\begin{aligned} v^n(x,t)=\frac{\alpha _ne^{-\alpha _n(\tau -t)}-\beta _ne^{-\beta _n(\tau -t)}}{\left( \alpha _n-\beta _n\right) \left| \beta _n\right| }\varphi _n(x) \end{aligned}$$

is the solution of (7). Recall that \(\lambda _n=n^2\pi ^2,\,\,\left| \alpha _n\right| \le \left| \beta _n\right| ,\,\,\text {for all}\,\,n\in {{\mathbb {N}}}\) and

$$\begin{aligned} \left| \beta _n\right|&=\frac{\left( \lambda _n-1\right) +\sqrt{\left( \lambda _n-1\right) ^2-4}}{2}\\&=\frac{n^2\pi ^2-1+\sqrt{\left( n^2\pi ^2-1\right) ^2-4}}{2}\rightarrow \infty \,\,\text {as}\,\, n\rightarrow \infty . \end{aligned}$$

Now, we observe that

$$\begin{aligned} \left\| h^n\right\| _{L^2(\Omega )}=\frac{1}{\left| \beta _n\right| }\rightarrow 0\quad \text {as}\,\,n\rightarrow \infty . \end{aligned}$$

Moreover, for every \(0\le t<\tau ,\) we have

$$\begin{aligned} \left\| v^n(t)\right\| _{L^2(\Omega )}&=\left| \frac{\alpha _ne^{-\alpha _n(\tau -t)}-\beta _ne^{-\beta _n(\tau -t)}}{\left( \alpha _n-\beta _n\right) \left| \beta _n\right| }\right| \\&=\frac{\left| \beta _n\right| e^{\left| \beta _n\right| (\tau -t)}-\left| \alpha _n\right| e^{\left| \alpha _n\right| (\tau -t)}}{\left( \left| \beta _n\right| -\left| \alpha _n\right| \right) \left| \beta _n\right| }\\&\ge \frac{\left| \beta _n\right| e^{\left| \beta _n\right| (\tau -t)}-\left| \alpha _n\right| e^{\left| \beta _n\right| (\tau -t)}}{\left( \left| \beta _n\right| -\left| \alpha _n\right| \right) \left| \beta _n\right| }=\frac{e^{\left| \beta _n\right| (\tau -t)}}{\left| \beta _n\right| }, \end{aligned}$$

and hence,

$$\begin{aligned} \left\| v^n(t)\right\| _{L^2(\Omega )}\rightarrow \infty \,\,\text {as}\,\,n\rightarrow \infty ,\,\,0\le t<\tau . \end{aligned}$$

This shows that a small perturbation in the final value may lead to a large deviation in the corresponding solution of the FVP for every \(0\le t<\tau .\) Thus, the problem of solving the FVP (7) is ill-posed.

Before proceeding further, let us fix some notations that we shall follow in the rest of the paper.

  • We shall denote the inner-product on \(L^2(\Omega )\) by \(\langle \cdot ,\cdot \rangle \) and the norm by \(\left\| \cdot \right\| .\)

  • For a Banach space \({\mathcal {X}},\) \({\mathcal {B}}({\mathcal {X}})\) shall denote the Banach space of all bounded linear operators from \({\mathcal {X}}\) to \({\mathcal {X}}.\) The norm on \({\mathcal {B}}\left( {\mathcal {X}}\right) \) will be denoted by \(\left\| \cdot \right\| _{{\mathcal {X}}\rightarrow {\mathcal {X}}}.\)

  • For a Banach space \({\mathcal {X}},\) \(C([0,\tau ];{\mathcal {X}})\) denotes the Banach space of all continuous functions \(v:[0,\tau ]\rightarrow {\mathcal {X}}\) with the norm \(\displaystyle \left\| v\right\| _{C([0,\tau ];{\mathcal {X}})}:=\max _{0\le t\le \tau } \left\| v(t)\right\| _{\mathcal {X}}.\)

  • \(L^1(0,\tau ;L^2(\Omega ))\) denotes the Banach space of all measurable functions \(v:[0,\tau ]\rightarrow L^2(\Omega )\) such that \(\int _0^\tau \left\| v(t)\right\| \,dt<\infty \) and the norm is given by \(\left\| v\right\| _{L^1(0,\tau ;L^2(\Omega ))}:=\int _0^\tau \left\| v(t)\right\| \,dt.\)

  • \(L^\infty (0,\tau ;L^2(\Omega ))\) denotes the Banach space of all measurable functions \(v:[0,\tau ]\rightarrow L^2(\Omega )\) such that there exists \(M>0\) satisfying \(\left\| v(t)\right\| \le M\) for almost all \(t\in [0,\tau ],\) and the norm is given by \(\left\| v\right\| _{L^\infty (0,\tau ;L^2(\Omega ))}:=\inf \left\{ M:\, \left\| v(t)\right\| \le M\,\,\text {for almost all}\,\,t\in [0,\tau ]\right\} .\)

  • For \(w\in L^1(0,\tau ;L^2(\Omega ))\) we shall use the standard notation w(t) to denote the function \(w(\cdot ,t).\) Moreover, we may denote \(\langle w(t),\varphi _j\rangle \) by \(w_j(t)\) and \(\langle F(t,w((t)),\varphi _j\rangle \) by \(F_j(t,w(t)).\)

  • For \(f\in L^2(\Omega )\) we shall denote \(\langle f,\varphi _j\rangle \) by \(f_j\).

Recall that \(u\in C([0,\tau ];L^2(\Omega ))\) is the solution of (1) for the exact final value \(g\in L^2(\Omega )\). Thus, taking the \(L^2(\Omega )\) inner product on both sides of the governing equation of (1) with \(\varphi _j\), we have for all \(j\in {{\mathbb {N}}},\)

$$\begin{aligned} \begin{array}{ll} \frac{d}{dt}\left( u_j(t)\right) +\lambda _ju_j(t)=F_j(t,u(t))+\int _t^\tau u_j(s)\,ds,&{}\quad t\in (0,\tau )\\ u_j(\tau )=g_j. \end{array} \end{aligned}$$
(8)

Now, it can be verified easily that

$$\begin{aligned} u_j(t)=e^{\lambda _j(\tau -t)}g_j-\int _t^\tau e^{\lambda _j(s-t)}F_j(s,u(s))\,ds-\int _t^\tau e^{\lambda _j(s-t)}\int _s^\tau u_j(\xi )\,d\xi \,ds. \end{aligned}$$
(9)

Thus, the solution u to the FVP (1) is

$$\begin{aligned} u(t)=\sum _{j=1}^\infty u_j(t)\varphi _j \end{aligned}$$

where \(u_j\) is as given in (9).

Moreover, for \(\psi \in L^2(\Omega )\) if we define \({\mathcal {S}}(t)\psi :=\sum _{j=1}^\infty e^{\lambda _jt}\langle \psi ,\varphi _j\rangle \varphi _j\), then we have

$$\begin{aligned} u(t)={\mathcal {S}}(\tau -t)g-\int _t^\tau {\mathcal {S}}(s-t)F(s,u(s))\,ds-\int _t^\tau {\mathcal {S}}(s-t)\int _s^\tau u(\xi )\,d\xi \,ds. \end{aligned}$$
(10)

The above integral equation form of the solution will be helpful in our analysis in the upcoming sections. In fact, we obtain the approximations for the exact solution u by replacing \({\mathcal {S}}\) with some finite rank operators from \(L^2(\Omega )\) to \(L^2(\Omega ).\)

We end this section by recalling the definition of Gevrey space (see for e.g., [1, 24]) which shall play the role of source sets in deriving error estimates.

For \(p,q\ge 0,\) the Gevrey space \({\mathbb {G}}_{p,q}\) is defined as

$$\begin{aligned} {\mathbb {G}}_{p,q}:=\left\{ \psi \in L^2(\Omega ):\sum _{j=1}^\infty \lambda _j^{2p}e^{2q\lambda _j}\left| \langle \psi ,\varphi _j\rangle \right| ^2<\infty \right\} . \end{aligned}$$

It is well known that \({\mathbb {G}}_{p,q}\) is a Hilbert space with respect to the inner product

$$\begin{aligned} \langle \chi ,\psi \rangle _{{\mathbb {G}}_{p,q}}:=\sum _{j=1}^\infty \lambda _j^{2p}e^{2q\lambda _j}\langle \chi ,\varphi _j\rangle \langle \varphi _j,\psi \rangle ,\quad \chi ,\psi \in {\mathbb {G}}_{p,q}, \end{aligned}$$

so that the norm is given by

$$\begin{aligned} \left\| \psi \right\| _{{\mathbb {G}}_{p,q}}^2=\sum _{j=1}^\infty \lambda _j^{2p}e^{2q\lambda _j}\left| \langle \psi ,\varphi _j\rangle \right| ^2. \end{aligned}$$

3 Preparatory Results

We begin by proving a version of Grönwalls’ type inequality for iterated integrals that will play the crucial role in obtaining stability estimates in later part of the paper. This version, as mentioned in the introduction, is perhaps proposed for the first time. We should mention that this is motivated from a different version of Grönwalls’ inequality for iterated integrals established in [19, Theorem 1.4.1]. For various other aspects of Grönwalls’ inequality, the interested reader may refer to [12] as well.

Lemma 3.1

Let \(U:[0,\tau ]\rightarrow {{\mathbb {R}}}\) be a non-negative continuous function and \(c_0,c_1>0\) be fixed. Suppose that

$$\begin{aligned} U(t)\le c_0+c_1\int _t^\tau \left( U(s)+\int _s^\tau U(\xi )\,d\xi \right) ds,\quad \,0\le t\le \tau . \end{aligned}$$

Then

$$\begin{aligned} U(t)\le c_0e^{(1+c_1)(\tau -t)},\quad 0\le t\le \tau . \end{aligned}$$

Proof

Let

$$\begin{aligned} X(t):=c_0+c_1\int _t^\tau \left( U(s)+\int _s^\tau U(\xi )\,d\xi \right) ds,\quad 0\le t\le \tau . \end{aligned}$$

Then it follows that \(U(t)\le X(t)\) for all \(t\in [0,\tau ]\) and \(X(\tau )=c_0.\) Thus,

$$\begin{aligned} X'(t)=-c_1U(t)-c_1\int _t^\tau U(s)\,ds\ge -c_1\left( X(t)+\int _t^\tau X(s)\,ds\right) . \end{aligned}$$

Let

$$\begin{aligned} V(t):=X(t)+\int _t^\tau X(s)\,ds,\quad t\in [0,\tau ]. \end{aligned}$$

Clearly, \(V(\tau )=X(\tau )=c_0,\,0\le X(t)\le V(t)\) and \(X'(t)\ge -c_1V(t).\) Thus,

$$\begin{aligned} V'(t)=X'(t)-X(t)\ge -c_1V(t)-V(t)=-(1+c_1)V(t). \end{aligned}$$

That is,

$$\begin{aligned} V'(t)+(1+c_1)V(t)\ge 0, \quad t\in (0,\tau ),\,\,V(\tau )=c_0. \end{aligned}$$

Thus, it follows that

$$\begin{aligned} e^{(1+c_1)\tau }V(\tau )-e^{(1+c_1)t}V(t)\ge 0 \end{aligned}$$

and hence

$$\begin{aligned} V(t)\le c_0e^{(1+c_1)(\tau -t)}. \end{aligned}$$

Now using the relation \(U(t)\le X(t)\le V(t),\) we have

$$\begin{aligned} U(t)\le c_0e^{(1+c_1)(\tau -t)}. \end{aligned}$$

This completes the proof. \(\square \)

Remark 3.1

In Lemma 3.1 if we replace \(c_0\) and \(c_1\) by two non-negative continuous functions, say \(\alpha (t)\) and \(\beta (t),\) respectively, then also one can infer similar conclusion in terms of integrals of \(\alpha (t)\) and \(\beta (t).\) Since those generalizations are not needed for our purpose, we are not including that.

Recall that \(g\in L^2(\Omega )\) is the exact final value and \(u\in C([0,\tau ];L^2(\Omega ))\) is the corresponding unique solution of the FVP (1). We have seen in Example 2.2 that the problem of recovering u(t) for (1) from the knowledge of final value g is an ill-posed problem for all \(0\le t<\tau .\) That is, a small perturbation in the final value may lead to a large deviation in the corresponding solution of the FVP. Since the accessible final value will be typically a measured data, noise in the final value is inevitable. Thus, it is reasonable to obtain some approximations for the exact solution from the knowledge of the measured final value/noisy data. Because of the ill-posedness of the problem, we have to employ a regularization method in order to obtain stable approximations. In this regard, we consider a Fourier truncation method for the regularization purpose. More specifically, we shall replace \({\mathcal {S}}\) in (10) by some finite rank operator described below, and then try to solve some non linear integral equation. The obtained solution will serve as the regularized approximations. Keeping this in mind, in this section we shall define those finite rank operators and study various important properties associated with it.

For \(N\in {{\mathbb {N}}}\) and \(t\in [0,\tau ],\) we define \({\mathcal {S}}_N(t):L^2(\Omega )\rightarrow L^2(\Omega )\) by

$$\begin{aligned} {\mathcal {S}}_N(t)\psi :=\sum _{j=1}^N e^{\lambda _jt}\langle \psi ,\varphi _j\rangle \varphi _j. \end{aligned}$$
(11)

From the definition of \({\mathcal {S}}_N\) it is easy to verify the following result.

Lemma 3.2

For \(N\in {{\mathbb {N}}}\) and \(t\in [0,\tau ],\) let \({\mathcal {S}}_N(t)\) be as defined in (11). Then the following holds:

  1. (i)

    \({\mathcal {S}}_N(t)\) is a bounded linear operator with \(\left\| S_N(t)\right\| _{L^2\rightarrow L^2}\le e^{t\lambda _N}\) for every \(t\in [0,\tau ].\)

  2. (ii)

    \(t\mapsto {\mathcal {S}}_N(t)\in C([0,\tau ];{\mathcal {B}}(L^2(\Omega )))\)

For \(v\in C([0,\tau ];L^2(\Omega )),\) we define

$$\begin{aligned} {\mathfrak {T}}v={\mathcal {S}}_N(\tau -t)g-\int _t^\tau {\mathcal {S}}_N(s-t)F(s,v(s))\,ds-\int _t^\tau {\mathcal {S}}_N(s-t)\int _s^\tau v(\xi )\,d\xi \,ds. \end{aligned}$$
(12)

Lemma 3.3

Let \(F:[0,\tau ]\times L^2(\Omega )\rightarrow L^2(\Omega )\) satisfies (F1) and (F2). Let \(v\in L^1(0,\tau ;L^2(\Omega )).\) Then the following holds:

  1. (i)

    \(t\mapsto F(t,v(t))\in L^1(0,\tau ;L^2(\Omega )).\)

  2. (ii)

    \(t\mapsto \int _t^\tau {\mathcal {S}}_N(s-t)\int _s^\tau v(\xi )\,d\xi \,ds\in C([0,\tau ];L^2(\Omega )).\)

  3. (iii)

    \(t\mapsto \int _t^\tau {\mathcal {S}}_N(s-t)F(s,v(s))\,ds\,\in C([0,\tau ];L^2(\Omega )).\)

Proof

Let \(\kappa >0,\phi \in L^2(\Omega )\) be as in (F1) and (F2), respectively.

(i) The proof follows by observing that

$$\begin{aligned} \int _0^\tau \left\| F(s,v(s))\right\| \,ds&\le \int _0^\tau \left( \left\| F(s,\phi )\right\| +\left\| F(s,\phi )-F(s,v(s))\right\| \right) ds\\&\le \int _0^\tau \left\| F(s,\phi )\right\| \,ds+\kappa \int _0^\tau \left\| \phi -v(s)\right\| \,ds. \end{aligned}$$

(ii) Let \(\Psi (t):=\int _t^\tau {\mathcal {S}}_N(s-t)\int _s^\tau v(\xi )\,d\xi \,ds,\,0\le t\le \tau .\) Then for each \(t\in [0,\tau ]\) it follows that \(\Psi (t)\in L^2(\Omega ).\) Indeed, by Lemma 3.2 (i), we have

$$\begin{aligned} \left\| \Psi (t)\right\| \le \int _t^\tau \left\| {\mathcal {S}}_N(s-t)\int _s^\tau v(\xi )\,d\xi \right\| \,ds\le \int _t^\tau e^{\lambda _N(s-t)}\int _s^\tau \left\| v(\xi )\right\| \,d\xi \,ds. \end{aligned}$$

We now show that \(t\mapsto \Psi (t),\,0\le t\le \tau ,\) is continuous.

Let \(t_0\in [0,\tau ].\) Then for \(t\le t_0,\) we have

$$\begin{aligned} \Psi (t)-\Psi (t_0)&=\int _t^\tau {\mathcal {S}}_N(s-t)\int _s^\tau v(\xi )\,d\xi \,ds-\int _{t_0}^\tau {\mathcal {S}}_N(s-t_0)\int _s^\tau v(\xi )\,d\xi \,ds\\&=\int _t^\tau \left( {\mathcal {S}}_N(s-t)-{\mathcal {S}}_N(s-t_0)\right) \int _s^\tau v(\xi )\,d\xi \,ds\\&\quad +\int _t^\tau {\mathcal {S}}_N(s-t_0)\int _s^\tau v(\xi )\,d\xi \,ds-\int _{t_0}^\tau {\mathcal {S}}_N(s-t_0)\int _s^\tau v(\xi )\,d\xi \,ds\\&= \int _t^\tau \left( {\mathcal {S}}_N(s-t)-{\mathcal {S}}_N(s-t_0)\right) \int _s^\tau v(\xi )\,d\xi \,ds\\&\quad +\int _t^{t_0}{\mathcal {S}}_N(s-t_0)\int _s^\tau v(\xi )\,d\xi \,ds. \end{aligned}$$

Therefore,

$$\begin{aligned} \left\| \Psi (t)-\Psi (t_0)\right\|&\le \int _t^\tau \left\| {\mathcal {S}}_N(s-t)-{\mathcal {S}}_N(s-t_0)\right\| _{L^2\rightarrow L^2}\int _s^\tau \left\| v(\xi )\right\| \,d\xi \,ds\\&\quad +\int _t^{t_0} \left\| {\mathcal {S}}_N(s-t_0)\right\| _{L^2\rightarrow L^2}\int _s^\tau \left\| v(\xi )\right\| \,d\xi \,ds\\&\le \left( \int _0^\tau \left\| {\mathcal {S}}_N(s-t)-{\mathcal {S}}_N(s-t_0)\right\| _{L^2\rightarrow L^2}\,ds\right) \left\| v\right\| _{L^1(0,\tau ;L^2(\Omega ))}\\&\quad +\left( \int _t^{t_0}\left\| {\mathcal {S}}_N(s-t_0)\right\| _{L^2\rightarrow L^2}\,ds\right) \left\| v\right\| _{L^1(0,\tau ;L^2(\Omega ))}. \end{aligned}$$

For \(t_0\le t,\) similarly we obtain

$$\begin{aligned} \left\| \Psi (t)-\Psi (t_0)\right\|&\le \left( \int _0^\tau \left\| {\mathcal {S}}_N(s-t)-{\mathcal {S}}_N(s-t_0)\right\| _{L^2\rightarrow L^2}\,ds\right) \left\| v\right\| _{L^1(0,\tau ;L^2(\Omega ))}\\&\quad +\left( \int _t^{t_0}\left\| {\mathcal {S}}_N(s-t_0)\right\| _{L^2\rightarrow L^2}\,ds\right) \left\| v\right\| _{L^1(0,\tau ;L^2(\Omega ))}. \end{aligned}$$

Now, from Lemma 3.2 (ii) it follows that \(\left\| {\mathcal {S}}_N(s-t)-{\mathcal {S}}_N(s-t_0)\right\| \rightarrow 0\) as \(t\rightarrow t_0,\) therefore by dominated convergence theorem it follows that

$$\begin{aligned} \int _0^\tau \left\| {\mathcal {S}}_N(s-t)-{\mathcal {S}}_N(s-t_0)\right\| \,ds\rightarrow 0 \end{aligned}$$

as \(t\rightarrow t_0.\) Also, \(\left| \int _t^{t_0}\left\| {\mathcal {S}}_N(s-t_0)\right\| \,ds\right| \rightarrow 0\) as \(t\rightarrow t_0.\) Thus, it follows that for each \(t_0\in [0,\tau ],\) \(\left\| \Psi (t)-\Psi (t_0)\right\| \rightarrow 0\) as \(t\rightarrow t_0.\)

(iii) Let \(\Phi (t):=\int _t^\tau {\mathcal {S}}_N(s-t)F(s,v(s))\,ds,\,\,0\le t\le \tau .\) By Lemma 3.2 (i), it follows that

$$\begin{aligned} \left\| \Phi (t)\right\| \le \int _t^\tau e^{\lambda _N(s-t)}\left\| F(s,v(s))\right\| \,ds. \end{aligned}$$

Thus, by Lemma 3.3 (i) it follows that \(\Phi (t)\in L^2(\Omega )\) for all \(t\in [0,\tau ].\) We now show that \(t\mapsto \Phi (t)\) is continuous. Let \(t_0\in [0,\tau ].\) Then similar to the proof of Lemma 3.3 (ii), we obtain

$$\begin{aligned} \left\| \Phi (t)-\Phi (t_0)\right\|&\le \int _0^\tau \left\| {\mathcal {S}}_N(s-t)-{\mathcal {S}}_N(s-t_0)\right\| _{L^2\rightarrow L^2}\left\| F(s,v(s))\right\| \,ds\\&\quad +\left| \int _t^{t_0}\left\| {\mathcal {S}}_N(s-t_0)\right\| _{L^2\rightarrow L^2}\left\| F(s,v(s))\right\| \,ds\right| . \end{aligned}$$

Thus, we conclude that \(\left\| \Phi (t)-\Phi (t_0)\right\| \rightarrow 0\) as \(t\rightarrow t_0.\) \(\square \)

Theorem 3.4

Let \(F:[0,\tau ]\times L^2(\Omega )\rightarrow L^2(\Omega )\) satisfies (F1) and (F2), and \({\mathfrak {T}}\) be as in (12). If \(v\in C([0,\tau ];L^2(\Omega )),\) then \({\mathfrak {T}}v\in C([0,\tau ];L^2(\Omega )).\)

Proof

By Lemma 3.2 (ii) we observe that \(t\mapsto {\mathcal {S}}_N(\tau -t)g\in C([0,\tau ];L^2(\Omega )).\) Now the proof follows by Lemma 3.3. \(\square \)

Theorem 3.5

Let F and \({\mathfrak {T}}:C([0,\tau ];L^2(\Omega ))\rightarrow C([0,\tau ];L^2(\Omega ))\) be as in Theorem 3.4. Then there exists a unique \(v_0\in C([0,\tau ];L^2(\Omega ))\) such that \({\mathfrak {T}}v_0=v_0.\)

Proof

For \(0\le t\le \tau ,\) we have

$$\begin{aligned} \left\| {\mathfrak {T}}v_1(t)-{\mathfrak {T}}v_2(t)\right\|&\le \int _t^\tau \left\| {\mathcal {S}}_N(s-t)\left( F(s,v_1(s))-F(s,v_2(s))\right) \right\| \,ds\\&\quad +\int _t^\tau \left\| {\mathcal {S}}_N(s-t)\left( \int _s^\tau \left( v_1(\xi )-v_2(\xi )\right) \,d\xi \right) \right\| \,ds\\&\le \int _t^\tau e^{\lambda _N(s-t)}\big (\left\| F(s,v_1(s))-F(s,v_2(s))\right\| \\&\quad +\int _s^\tau \left\| v_1(\xi )-v_2(\xi )\right\| \,d\xi \big )\,ds\\&\le \kappa \int _t^\tau e^{\lambda _N(s-t)}\left\| v_1(s)-v_2(s)\right\| \,ds\\&\quad + \left\| v_1-v_2\right\| _{C([0,\tau ];L^2(\Omega ))}\int _t^\tau e^{\lambda _N(s-t)}\int _s^\tau \,d\xi \,ds\\&\quad \le e^{\lambda _N\tau }\left\| v_1-v_2\right\| _{C([0,\tau ];L^2(\Omega ))}\left( \kappa \left( \tau -t\right) +\frac{\left( \tau -t\right) ^2}{2}\right) \\&\le e^{\lambda _N\tau }\left\| v_1-v_2\right\| _{C([0,\tau ];L^2(\Omega ))}\left( \kappa \left( \tau -t\right) +\tau \left( \tau -t\right) \right) \end{aligned}$$

and hence

$$\begin{aligned} \left\| {\mathfrak {T}}v_1(t)-{\mathfrak {T}}v_2(t)\right\| \le \kappa _0e^{\lambda _N\tau }\left( 1+\tau \right) \left( \tau -t\right) \left\| v_1-v_2\right\| _{C([0,\tau ];L^2(\Omega ))}, \end{aligned}$$

where \(\kappa _0=\max \{\kappa ,1\}.\) Using this, we obtain

$$\begin{aligned} \left\| {\mathfrak {T}}^2v_1(t)-{\mathfrak {T}}^2v_2(t)\right\|&\le \int _t^\tau \left\| {\mathcal {S}}_N(s-t)\left( F(s,{\mathfrak {T}}v_1(s))-F(s,{\mathfrak {T}}v_2(s))\right) \right\| \,ds\\&\quad +\int _t^\tau \left\| {\mathcal {S}}_N(s-t)\left( \int _s^\tau \left( {\mathfrak {T}}v_1(\xi )-{\mathfrak {T}}v_2(\xi )\right) \,d\xi \right) \right\| \,ds\\&\le \int _t^\tau e^{\lambda _N(s-t)}\big (\left\| F(s,{\mathfrak {T}}v_1(s))-F(s,{\mathfrak {T}}v_2(s))\right\| \\&\quad +\int _s^\tau \left\| {\mathfrak {T}}v_1(\xi )-{\mathfrak {T}}v_2(\xi )\right\| \,d\xi \big )\,ds\\&\le \kappa \int _t^\tau e^{\lambda _N(s-t)}\left\| {\mathfrak {T}}v_1(s)-{\mathfrak {T}}v_2(s)\right\| \,ds\\&\quad +\int _t^\tau e^{\lambda _N(s-t)}\int _s^\tau \left\| {\mathfrak {T}}v_1(\xi )-{\mathfrak {T}}v_2(\xi )\right\| \,d\xi \,ds\\&\le \kappa _0e^{\lambda _N\tau }\big (\int _t^\tau \left\| {\mathfrak {T}}v_1(s)-{\mathfrak {T}}v_2(s)\right\| \,ds\\&\quad +\tau \int _t^\tau \left\| {\mathfrak {T}}v_1(\xi )-{\mathfrak {T}}v_2(\xi )\right\| \,d\xi \big )\\&=\kappa _0e^{\lambda _N\tau }\left( 1+\tau \right) \int _t^\tau \left\| {\mathfrak {T}}v_1(s)-{\mathfrak {T}}v_2(s)\right\| \,ds\\&\le \left( \kappa _0e^{\lambda _N\tau }\left( 1+\tau \right) \right) ^2\left\| v_1-v_2\right\| _{C([0,\tau ];L^2(\Omega ))}\int _t^\tau \left( \tau -s\right) \,ds\\&= \left( \kappa _0e^{\lambda _N\tau }\left( 1+\tau \right) \right) ^2\frac{\left( \tau -t\right) ^2}{2!}\left\| v_1-v_2\right\| _{C([0,\tau ];L^2(\Omega ))}. \end{aligned}$$

Thus, inductively, for all \(m\in {{\mathbb {N}}}\) we have for \(0\le t\le \tau ,\)

$$\begin{aligned} \left\| {\mathfrak {T}}^mv_1(t)-{\mathfrak {T}}^mv_2(t)\right\| \le \frac{\left( \kappa _0e^{\lambda _N\tau }\left( 1+\tau \right) \left( \tau -t\right) \right) ^m}{m!}\left\| v_1-v_2\right\| _{C([0,\tau ];L^2(\Omega ))}, \end{aligned}$$

and hence

$$\begin{aligned} \left\| {\mathfrak {T}}^mv_1-{\mathfrak {T}}^mv_2\right\| _{C([0,\tau ];L^2(\Omega ))}\le \frac{\left( \kappa _0e^{\lambda _N\tau }\left( 1+\tau \right) \tau \right) ^m}{m!}\left\| v_1-v_2\right\| _{C([0,\tau ];L^2(\Omega ))}. \end{aligned}$$

Since \(\frac{\left( \kappa _0e^{\lambda _N\tau }\left( 1+\tau \right) \tau \right) ^m}{m!}\rightarrow 0\) as \(m\rightarrow \infty ,\) there exists \(m_0\in {{\mathbb {N}}}\) such that \({\mathfrak {T}}^{m_0}\) is a contraction. Therefore, by Banach fixed point theorem, there exists a unique \(v_0\in C([0,\tau ];L^2(\Omega ))\) such that \({\mathfrak {T}}^{m_0}v_0=v_0.\) This shows that \({\mathfrak {T}}\left( {\mathfrak {T}}^{m_0}v_0\right) ={\mathfrak {T}}v_0,\) that is, \({\mathfrak {T}}v_0\) is a fixed point of \({\mathfrak {T}}^{m_0}.\) Therefore, by uniqueness, we have \({\mathfrak {T}}v_0=v_0.\) Clearly, this \(v_0\) is the unique fixed point of \({\mathfrak {T}}.\) \(\square \)

Remark 3.2

Note that the unique \(v_0\in C([0,\tau ];L^2(\Omega ))\) as obtained in Theorem 3.5 depends on N. Thus, in order to visualize that dependence, from now onwards we shall denote that \(v_0\) by \(u^N.\) That is, \(u^N\in C([0,\tau ];L^2(\Omega ))\) is the unique element such that \({\mathfrak {T}}u^N=u^N.\)

4 Regularization and Error Analysis

Let \(u^N\in C([0,\tau ];L^2(\Omega ))\) be as described in Remark 3.2. In this section, we shall see that \(u^N\) are indeed regularized approximations. Moreover, we shall obtain estimates for the quantity \(\left\| u(t)-u^N(t)\right\| \) by assuming that u belongs to some Gevrey spaces, where we recall that u is the exact solution of (1). Note that, in order to obtain some estimate of \(\left\| u(t)-u^N(t)\right\| \) in terms of the regularization parameter N so that the estimate tends to zero as \(N\rightarrow \infty ,\) it is necessary to assume some priori condition on the solution u otherwise no such estimate can be derived (see for e.g., [20]). Finally, we provide a parameter choice strategy, that is a procedure to choose the regularization parameter N, and obtain the corresponding error estimates.

Theorem 4.1

Let \(F:[0,\tau ]\times L^2(\Omega )\rightarrow L^2(\Omega )\) satisfies (F1) and (F2). Let \(p,q>0,\) and \(\kappa _0=\max \{\kappa ,1\},\) where \(\kappa \) is as in (F1). For \(N\in {{\mathbb {N}}},\) let \(u^N\in C([0,\tau ];L^2(\Omega ))\) be as mentioned in Remark 3.2. Then the following hold:

  1. (i)

    If \(u\in L^\infty (0,\tau ;{\mathbb {G}}_{p,\tau })\) then

    $$\begin{aligned} \left\| u(t)-u^N(t)\right\| \le e^{\left( 1+\kappa _0\right) \left( \tau -t\right) }\left\| u\right\| _{L^\infty (0,\tau ;{\mathbb {G}}_{p,\tau })}\lambda _N^{-p}e^{-\lambda _Nt},\quad 0\le t\le \tau . \end{aligned}$$
  2. (ii)

    If \(u\in L^\infty (0,\tau ;{\mathbb {G}}_{0,q+\tau })\) then

    $$\begin{aligned} \left\| u(t)-u^N(t)\right\| \le e^{\left( 1+\kappa _0\right) \left( \tau -t\right) }\left\| u\right\| _{L^\infty (0,\tau ;{\mathbb {G}}_{0,q+\tau })}e^{-(q+t)\lambda _N},\quad 0\le t\le \tau . \end{aligned}$$

Proof

Recall that the exact solution u of (1) satisfies

$$\begin{aligned} u(t)={\mathcal {S}}(\tau -t)g-\int _t^\tau {\mathcal {S}}(s-t)F(s,u(s))\,ds-\int _t^\tau {\mathcal {S}}(s-t)\int _s^\tau u(\xi )\,d\xi \,ds \end{aligned}$$

and hence

$$\begin{aligned} \sum _{j=1}^N\langle u(t),\varphi _j\rangle \varphi _j&={\mathcal {S}}_N(\tau -t)g-\int _t^\tau {\mathcal {S}}_N(s-t)F(s,u(s))\,ds\\&\quad -\int _t^\tau {\mathcal {S}}_N(s-t)\int _s^\tau u(\xi )\,d\xi \,ds. \end{aligned}$$

Thus,

$$\begin{aligned} \left\| u(t)-u^N(t)\right\|&\le \left\| u(t)-\sum _{j=1}^N u_j(t)\varphi _j\right\| +\left\| \sum _{j=1}^N u_j(t)\varphi _j-u^N(t)\right\| \\&\le \sqrt{\sum _{j=N+1}^\infty \left| u_j(t)\right| ^2}\\&\quad +\int _t^\tau \left\| {\mathcal {S}}_N(s-t)\left( F(s,u^N(s))-F(s,u(s))\right) \right\| \,ds\\&\quad +\int _t^\tau \left\| {\mathcal {S}}_N(s-t)\left( \int _s^\tau \left( u^N(\xi )-u(\xi )\right) d\xi \right) \right\| \,ds. \end{aligned}$$

(i) Let \(u\in L^\infty (0,\tau ;{\mathbb {G}}_{p,\tau }).\) Then we have

$$\begin{aligned} \left\| u(t)-u^N(t)\right\|&\le \sqrt{\sum _{j=N+1}^\infty \lambda _j^{2p}e^{2t\lambda _j}\lambda _j^{-2p}e^{-2t\lambda _j}\left| u_j(t)\right| ^2}\\&\quad +\kappa \int _t^\tau e^{\lambda _N(s-t)}\left\| u^N(s)-u(s)\right\| \,ds\\&\quad + \int _t^\tau e^{\lambda _N(s-t)}\int _s^\tau \left\| u^N(\xi )-u(\xi )\right\| \,d\xi \,ds\\&\le \lambda _N^{-p}e^{-t\lambda _N}\left\| u(t)\right\| _{{\mathbb {G}}_{p,\tau }} +\kappa _0\left( \int _t^\tau e^{\lambda _N(s-t)}\left( \left\| u^N(s)-u(s)\right\| \right. \right. \\&\left. \left. \quad +\int _s^\tau \left\| u^N(\xi )-u(\xi )\right\| \,d\xi \right) \,ds\right) . \end{aligned}$$

Thus,

$$\begin{aligned} e^{t\lambda _N}\left\| u(t)-u^N(t)\right\|&\le \lambda _N^{-p}\left\| u\right\| _{L^\infty (0,\tau ;{\mathbb {G}}_{p,\tau })}+\kappa _0\left( \int _t^\tau e^{s\lambda _N}\left( \left\| u^N(s)-u(s)\right\| \right. \right. \\&\quad \left. \left. +\int _s^\tau \left\| u^N(\xi )-u(\xi )\right\| \,d\xi \right) \,ds\right) . \end{aligned}$$

Now, note that

$$\begin{aligned} \int _t^\tau&e^{s\lambda _N}\int _s^\tau \left\| u^N(\xi )-u(\xi )\right\| \,d\xi \,ds\\&\quad =\int _t^\tau e^{s\lambda _N}\int _s^\tau e^{\xi \lambda _N}e^{-\xi \lambda _N}\left\| u^N(\xi )-u(\xi )\right\| \,d\xi \,ds\\&\quad \le \int _t^\tau e^{s\lambda _N}\int _s^\tau e^{-s\lambda _N}e^{\xi \lambda _N}\left\| u^N(\xi )-u(\xi )\right\| \,d\xi \,ds\\&\quad =\int _t^\tau \int _s^\tau e^{\xi \lambda _N}\left\| u^N(\xi )-u(\xi )\right\| \,d\xi \,ds. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} e^{t\lambda _N}\left\| u(t)-u^N(t)\right\|&\le \lambda _N^{-p}\left\| u\right\| _{L^\infty (0,\tau ;{\mathbb {G}}_{p,\tau })}+\kappa _0\left( \int _t^\tau e^{s\lambda _N}\left\| u(s)-u^N(s)\right\| \,ds\right. \\&\left. \quad +\int _t^\tau \int _s^\tau e^{\xi \lambda _N}\left\| u(\xi )-u^N(\xi )\right\| \,d\xi \,ds\right) . \end{aligned}$$

Therefore, by Lemma 3.1, we obtain

$$\begin{aligned} e^{t\lambda _N}\left\| u(t)-u^N(t)\right\| \le \lambda _N^{-p}\left\| u\right\| _{L^\infty (0,\tau ;{\mathbb {G}}_{p,\tau })}e^{\left( 1+\kappa _0\right) \left( \tau -t\right) }. \end{aligned}$$

Thus,

$$\begin{aligned} \left\| u(t)-u^N(t)\right\| \le e^{\left( 1+\kappa _0\right) \left( \tau -t\right) }\left\| u\right\| _{L^\infty (0,\tau ;{\mathbb {G}}_{p,\tau })}\lambda _N^{-p}e^{-\lambda _Nt}. \end{aligned}$$

This completes the proof of (i).

(ii) Let \(u\in L^\infty (0,\tau ;{\mathbb {G}}_{0,q+\tau }).\) Then similar to the above argument we obtain

$$\begin{aligned} \left\| u(t)-u^N(t)\right\|&\le \sqrt{\sum _{j=N+1}^\infty e^{2(q+t)\lambda _j}e^{-2(q+t)\lambda _j}\left| u_j(t)\right| ^2}\\&\quad +\kappa \int _t^\tau e^{\lambda _N(s-t)}\left\| u^N(s)-u(s)\right\| \,ds\\&\quad + \int _t^\tau e^{\lambda _N(s-t)}\int _s^\tau \left\| u^N(\xi )-u(\xi )\right\| \,d\xi \,ds\\&\le e^{-(q+t)\lambda _N}\left\| u(t)\right\| _{{\mathbb {G}}_{0,q+\tau }}+\kappa _0\left( \int _t^\tau e^{\lambda _N(s-t)}\left( \left\| u^N(s)-u(s)\right\| \right. \right. \\&\left. \left. \quad +\int _s^\tau \left\| u^N(\xi )-u(\xi )\right\| \,d\xi \right) \,ds\right) , \end{aligned}$$

and hence

$$\begin{aligned} e^{t\lambda _N}\left\| u(t)-u^N(t)\right\|&\le e^{-q\lambda _N}\left\| u\right\| _{L^\infty (0,\tau ;{\mathbb {G}}_{0,q+\tau })}+\kappa _0\left( \int _t^\tau e^{s\lambda _N}\left\| u(s)-u^N(s)\right\| \,ds\right. \\&\left. \quad +\int _t^\tau \int _s^\tau e^{\xi \lambda _N}\left\| u(\xi )-u^N(\xi )\right\| \,d\xi \,ds\right) . \end{aligned}$$

Therefore, by Lemma 3.1, we have

$$\begin{aligned} \left\| u(t)-u^N(t)\right\| \le e^{\left( 1+\kappa _0\right) \left( \tau -t\right) }\left\| u\right\| _{L^\infty (0,\tau ;{\mathbb {G}}_{0,q+\tau })}e^{-(q+t)\lambda _N}. \end{aligned}$$

\(\square \)

We now analyse the case when we have a noisy final value in place of the exact final value. For \(\delta >0,\) let \(g^\delta \in L^2(\Omega )\) be the noisy final value satisfying (2).

For \(v\in C([0,\tau ];L^2(\Omega ))\), let

$$\begin{aligned} {\mathfrak {T}}_\delta v={\mathcal {S}}_N(\tau -t)g^\delta -\int _t^\tau {\mathcal {S}}_N(s-t)F(s,v(s))\,ds-\int _t^\tau {\mathcal {S}}_N(s-t)\int _s^\tau v(\xi )\,d\xi \,ds. \end{aligned}$$
(13)

Then following the arguments of Theorems 3.4 and 3.5 it is obvious that \({\mathfrak {T}}_\delta :C([0,\tau ];L^2(\Omega ))\rightarrow C([0,\tau ];L^2(\Omega ))\) is well-defined and there exists a unique \(u_\delta ^N\in C([0,\tau ];L^2(\Omega ))\) such that

$$\begin{aligned} {\mathfrak {T}}_\delta u^N_\delta =u^N_\delta , \end{aligned}$$
(14)

that is,

$$\begin{aligned} u^N_\delta (t)&={\mathcal {S}}_N(\tau -t)g^\delta -\int _t^\tau {\mathcal {S}}_N(s-t)F(s,u^N_\delta (s))\,ds\\&\quad -\int _t^\tau {\mathcal {S}}_N(s-t)\int _s^\tau u^N_\delta (\xi )\,d\xi \,ds. \end{aligned}$$

Having obtained \(u^N_\delta ,\) we now obtain estimate for \(\left\| u^N(t)-u^{N}_\delta (t)\right\| .\)

Theorem 4.2

Let \(F:[0,\tau ]\times L^2(\Omega )\rightarrow L^2(\Omega )\) satisfies (F1) and (F2). Let \(\kappa \) be as in (F1) and \(\kappa _0=\max \{\kappa ,1\}.\) Let \(u^N, u^N_\delta \) be as mentioned in Remark 3.2 and (14), respectively. Then

$$\begin{aligned} \left\| u^N(t)-u^N_\delta (t)\right\| \le \delta e^{\lambda _N(\tau -t)}e^{(1+\kappa _0)(\tau -t)},\quad 0\le t\le \tau . \end{aligned}$$

Proof

We have

$$\begin{aligned} \left\| u^N(t)-u^N_\delta (t)\right\|&\le \left\| {\mathcal {S}}_N(\tau -t)\left( g-g^\delta \right) \right\| \\&\quad +\int _t^\tau \left\| {\mathcal {S}}_N(s-t)\left( F(s,u^N(s))-F(s,u^N_\delta (s))\right) \right\| \,ds\\&\quad + \int _t^\tau \left\| {\mathcal {S}}_N(s-t)\int _s^\tau \left( u^N(\xi )-u^N_\delta (\xi )\right) d\xi \right\| \,ds\\&\le e^{\lambda _N\left( \tau -t\right) }\delta +\int _t^\tau \kappa e^{\lambda _N(s-t)}\left\| u^N(s)-u^N_\delta (s)\right\| \,ds \\&\quad + \int _t^\tau e^{\lambda _N(s-t)}\int _s^\tau \left\| u^N(\xi )-u^N_\delta (\xi )\right\| \,d\xi \,ds\\&\le e^{\lambda _N\left( \tau -t\right) }\delta +\kappa _0\left( \int _t^\tau e^{\lambda _N(s-t)}\left\| u^N(s)-u^N_\delta (s)\right\| \,ds \right. \\&\left. \quad + \int _t^\tau e^{\lambda _N(s-t)}\int _s^\tau \left\| u^N(\xi )-u^N_\delta (\xi )\right\| \,d\xi \,ds\right) . \end{aligned}$$

Thus,

$$\begin{aligned} e^{\lambda _Nt}\left\| u^N(t)-u^N_\delta (t)\right\|&\le e^{\lambda _N\tau }\delta +\kappa _0\left( \int _t^\tau e^{\lambda _Ns}\left\| u^N(s)-u^N_\delta (s)\right\| \,ds \right. \\&\left. \quad + \int _t^\tau e^{\lambda _Ns}\int _s^\tau \left\| u^N(\xi )-u^N_\delta (\xi )\right\| \,d\xi \,ds\right) \\&\le e^{\lambda _N\tau }\delta +\kappa _0 \left( \int _t^\tau e^{\lambda _Ns}\left\| u^N(s)-u^N_\delta (s)\right\| \,ds\right. \\&\left. \quad + \int _t^\tau \int _s^\tau e^{\lambda _N\xi }\left\| u^N(\xi )-u^N_\delta (\xi )\right\| \,d\xi \,ds\right) . \end{aligned}$$

Therefore, by Lemma 3.1, we obtain

$$\begin{aligned} e^{\lambda _Nt}\left\| u^N(t)-u^N_\delta (t)\right\| \le \delta e^{\lambda _N\tau }e^{(1+\kappa _0)(\tau -t)}, \end{aligned}$$

that is,

$$\begin{aligned} \left\| u^N(t)-u^N_\delta (t)\right\| \le \delta e^{\lambda _N(\tau -t)}e^{(1+\kappa _0)(\tau -t)}. \end{aligned}$$

\(\square \)

Therefore combining Theorems 4.1 and 4.2, we have the following result.

Theorem 4.3

Let \(p,q>0,\) and \(\kappa _0=\max \{\kappa ,1\},\) where \(\kappa \) is as in (F1). For \(N\in {{\mathbb {N}}},\) let \(u^N,u^N_\delta \) be as in Theorem 4.2. Then the following hold:

  1. (i)

    If \(u\in L^\infty (0,\tau ;{\mathbb {G}}_{p,\tau })\) then for \(0\le t\le \tau ,\)

    $$\begin{aligned} \left\| u(t)-u^N_\delta (t)\right\|&\le e^{\left( 1+\kappa _0\right) \left( \tau -t\right) }\left\| u\right\| _{L^\infty (0,\tau ;{\mathbb {G}}_{p,\tau })}\lambda _N^{-p}e^{-\lambda _Nt} \\&\quad + \delta e^{\lambda _N(\tau -t)}e^{(1+\kappa _0)(\tau -t)}. \end{aligned}$$
  2. (ii)

    If \(u\in L^\infty (0,\tau ;{\mathbb {G}}_{0,q+\tau })\) then for \(0\le t\le \tau ,\)

    $$\begin{aligned} \left\| u(t)-u^N_\delta (t)\right\|&\le e^{\left( 1+\kappa _0\right) \left( \tau -t\right) }\left\| u\right\| _{L^\infty (0,\tau ;{\mathbb {G}}_{0,q+\tau })}e^{-(q+t)\lambda _N} \\&\quad + \delta e^{\lambda _N(\tau -t)}e^{(1+\kappa _0)(\tau -t)}. \end{aligned}$$

We now provide parameter choice strategies, that is, a strategy to choose the regularization parameter N depending on \(\delta \) in such a way that \(N\rightarrow \infty \) and \(\left\| u(t)-u^N_\delta (t)\right\| \rightarrow 0\) as \(\delta \rightarrow 0.\) We first recall an important result from [23] that will be useful in the upcoming analysis.

Lemma 4.4

[23, Lemma 3.3] Let \(0<a<1, b,c,d\) be positive constants. Let \(\zeta :(0,a]\rightarrow {{\mathbb {R}}}\) be given by \(\zeta (s):=s^b\left( d\ln \frac{1}{s}\right) ^{-c}.\) Then for the inverse function \(\zeta ^{-1}(s),\) we have

$$\begin{aligned} \zeta ^{-1}(s)=s^{1/b}\left( \frac{d}{b}\ln \frac{1}{s}\right) ^{c/b}\left( 1+o(1)\right) \quad \text {for}\quad s\rightarrow 0. \end{aligned}$$

For \(p,q,\varrho >0\), let

$$\begin{aligned} {\mathcal {K}}_{p,\tau }=\left\{ v\in C([0,\tau ];L^2(\Omega )): \left\| v\right\| _{L^\infty (0,\tau ;{\mathbb {G}}_{p,\tau })}\le \varrho \right\} \end{aligned}$$
(15)

and

$$\begin{aligned} {\mathcal {M}}_{0,q}=\left\{ v\in C([0,\tau ];L^2(\Omega )): \left\| v\right\| _{L^\infty (0,\tau ;{\mathbb {G}}_{0,q+\tau })}\le \varrho \right\} . \end{aligned}$$
(16)

Also, let \(\kappa _1=1+\kappa _0,\) where \(\kappa _0\) is as in Theorem 4.3. Let \(u\in {\mathcal {K}}_{p,\tau }.\) Let \(e_1,e_2>0\) be as in (5). Then from Theorem 4.3, we obtain

$$\begin{aligned} \left\| u(t)-u^N_\delta (t)\right\|&\le e^{\kappa _1(\tau -t)}\left( \frac{\varrho }{\lambda _N^pe^{\lambda _Nt}}+\delta e^{\lambda _N(\tau -t)}\right) \\&\le E_1e^{\kappa _1(\tau -t)}\left( \frac{\varrho }{N^{2p/d}\left( e^{N^{2/d}}\right) ^{e_1 t}}+\delta \left( e^{N^{2/d}}\right) ^{e_2(\tau -t)}\right) , \end{aligned}$$

where \(E_1=\max \{e_1^{-p},1\}.\) For \(r>0,\) let \(A(r)=\frac{\varrho }{r^{2p/d}}+\delta \left( e^{r^{2/d}}\right) ^{\eta (t)},\) where \(\eta (t)=e_1t+e_2(\tau -t),\,\,0\le t\le \tau .\) Note that A(r) will attain minimum at \(r_\delta \) if \(\frac{\varrho }{{r_\delta }^{2p/d}}=\delta \left( e^{{r_\delta }^{2/d}}\right) ^{\eta (t)}\) that is, if \(\frac{\delta }{\varrho }=\left( \frac{1}{r_\delta }\right) ^{2p/d}e^{-\eta (t){r_\delta }^{2/d}}.\) Let \(s_\delta =e^{-{r_\delta }^{2/d}}.\) Then it follows that

$$\begin{aligned} \frac{\delta }{\varrho }={s_\delta }^{\eta (t)}\left( \ln \frac{1}{s_\delta }\right) ^{-p}. \end{aligned}$$

Thus, by Lemma 4.4, we have

$$\begin{aligned} s_\delta =\left( \frac{\delta }{\varrho }\right) ^{1/\eta (t)}\left( \frac{1}{\eta (t)}\ln \frac{\varrho }{\delta }\right) ^{p/\eta (t)}\left( 1+o(1)\right) \quad \text {for}\quad \delta \rightarrow 0. \end{aligned}$$

Thus,

$$\begin{aligned} {r_\delta }^{2/d}=\ln \left( \left( \frac{\delta }{\varrho }\right) ^{-1/\eta (t)}\left( \frac{1}{\eta (t)}\ln \frac{\varrho }{\delta }\right) ^{-p/\eta (t)}\right) \left( 1+o(1)\right) \quad \text {for}\quad \delta \rightarrow 0 \end{aligned}$$

and hence

$$\begin{aligned} r_\delta =\left( \ln \left( \left( \frac{\delta }{\varrho }\right) ^{-1/\eta (t)}\left( \frac{1}{\eta (t)}\ln \frac{\varrho }{\delta }\right) ^{-p/\eta (t)}\right) \right) ^{d/2}\left( 1+o(1)\right) \quad \text {for}\quad \delta \rightarrow 0. \end{aligned}$$

For \(r\in {{\mathbb {R}}}\), let \(\left[ \left[ r\right] \right] \) denotes the greatest integer not exceeding r. We define \(N_\delta =\left[ \left[ r_\delta \right] \right] .\) Then for this choice of the regularization parameter, from Theorem 4.3, it follows that for each \(t\in [0,\tau ]\) there exists \(\delta _t>0\) such that for \(\delta \le \delta _t,\)

$$\begin{aligned} \left\| u(t)-u^{N_\delta }_\delta (t)\right\|&\le 2E_1e^{\kappa _1(\tau -t)}\,\delta \left( \frac{\delta }{\varrho }\right) ^{-e_2\left( \tau -t\right) /\eta (t)}\left( \frac{1}{\eta (t)}\ln \frac{\varrho }{\delta }\right) ^{-pe_2\left( \tau -t\right) /\eta (t)}\\&= 2 E_1 e^{\kappa _1(\tau -t)}\,\varrho ^{e_2\left( \tau -t\right) /\eta (t)}\,\delta ^{e_1 t/\eta (t)}\left( \frac{1}{\eta (t)}\ln \frac{\varrho }{\delta }\right) ^{-pe_2\left( \tau -t\right) /\eta (t)}. \end{aligned}$$

Now suppose that \(u\in {\mathcal {M}}_{0,q}.\) Let \(e_1,e_2>0\) be as in (5). Then from Theorem 4.3 (ii), we have

$$\begin{aligned} \left\| u(t)-u^N_\delta (t)\right\|&\le e^{\kappa _1(\tau -t)}\left( \frac{\varrho }{e^{(q+t)\lambda _N}}+\delta e^{\lambda _N(\tau -t)}\right) \\&\le e^{\kappa _1(\tau -t)}\left( \frac{\varrho }{e^{(q+t)e_1N^{2/d}}}+\delta e^{(\tau -t)e_2N^{2/d}}\right) . \end{aligned}$$

For \(r>0,\) let \(A(r)=\frac{\varrho }{e^{e_1(q+t)r^{2/d}}}+\delta e^{e_2(\tau -t)r^{2/d}}.\) Clearly A(r) attains it minimum at \(r_\delta \) if

$$\begin{aligned} \frac{\varrho }{\delta }=\left( e^{{r_\delta }^{2/d}}\right) ^{e_1(q+t)+e_2(\tau -t)}, \end{aligned}$$

that is, if

$$\begin{aligned} r_\delta =\left( \frac{1}{e_1(q+t)+e_2(\tau -t)}\ln \frac{\varrho }{\delta }\right) ^{d/2}. \end{aligned}$$

Let \(N_\delta =\left[ \left[ r_\delta \right] \right] .\) Then from Theorem 4.3 (ii), it follows that for each \(t\in [0,\tau ]\)

$$\begin{aligned} \left\| u(t)-u^{N_\delta }_\delta (t)\right\|&\le 2 e^{\kappa _1(\tau -t)}\,\delta \left( \frac{\varrho }{\delta }\right) ^{e_2(\tau -t)/\left( e_1(q+t)+e_2(\tau -t)\right) }\\&=2 e^{\kappa _1(\tau -t)}\,\varrho ^\frac{e_2(\tau -t)}{e_1(q+t)+e_2(\tau -t)}\,\delta ^\frac{e_1(q+t)}{e_1(q+t)+e_2(\tau -t)}. \end{aligned}$$

Thus, we have proved the following result.

Theorem 4.5

For \(p,q,\varrho >0,\) let \({\mathcal {K}}_{p,\tau }\) and \({\mathcal {M}}_{0,q}\) be as defined in (15) and (16), respectively. Let \(e_1,e_2\) be as in (5). Let \(\kappa _0=\max \{\kappa ,1\},\,\,\kappa _1=1+\kappa _0,\) where \(\kappa \) is as in (F1), and \(\eta (t)=e_1 t+e_2(\tau -t),\,\,0\le t\le \tau .\) For \(r\in {{\mathbb {R}}},\) let \(\left[ \left[ r\right] \right] \) denotes the greatest integer not exceeding r. Then the following holds:

  1. (i)

    If \(u\in {\mathcal {K}}_{p,\tau }\) then for each \(t\in [0,\tau ]\) and

    $$\begin{aligned} N_\delta =\left[ \left[ \left( \ln \left( \left( \frac{\delta }{\varrho }\right) ^{-1/\eta (t)}\left( \frac{1}{\eta (t)}\ln \frac{\varrho }{\delta }\right) ^{-p/\eta (t)}\right) \right) ^{d/2}\right] \right] , \end{aligned}$$

    we have

    $$\begin{aligned}&\left\| u(t)-u^{N_\delta }_\delta (t)\right\| \\&\quad =O\left( e^{\kappa _1(\tau -t)}\varrho ^{e_2(\tau -t)/\eta (t)}\,\delta ^{e_1t/\eta (t)}\left( \frac{1}{\eta (t)}\ln \frac{\varrho }{\delta }\right) ^{-pe_2\left( \tau -t\right) /\eta (t)}\right) ,\quad \delta \rightarrow 0. \end{aligned}$$
  2. (ii)

    If \(u\in {\mathcal {M}}_{0,q}\) then for each \(t\in [0,\tau ]\) and

    $$\begin{aligned} N_\delta =\left[ \left[ \left( \frac{1}{e_1(q+t)+e_2(\tau -t)}\ln \frac{\varrho }{\delta }\right) ^{d/2}\right] \right] , \end{aligned}$$

    we have

    $$\begin{aligned} \left\| u(t)-u^{N_\delta }_\delta (t)\right\| =O\left( e^{\kappa _1(\tau -t)}\,\varrho ^\frac{e_2(\tau -t)}{e_1(q+t)+e_2(\tau -t)}\,\delta ^\frac{e_1(q+t)}{e_1(q+t)+e_2(\tau -t)}\right) ,\quad \delta \rightarrow 0. \end{aligned}$$

Remark 4.1

From the above Theorem it follows that if \(u\in {\mathcal {K}}_{p,\tau }\) then for \(t=0\) we obtain logarithmic rate of convergence and for \(0<t< \tau \) we obtain a logarithmic-type of convergence. Moreover, if \(u\in {\mathcal {M}}_{0,q}\) then we obtain a Hölder rate of convergence for every \(t\in [0,\tau ].\)

Remark 4.2

This remark is due to the observation made by one of the reviewer. Let U be as in Lemma 3.1. The condition

$$\begin{aligned} U(t)\le c_0+c_1\int _t^\tau \left( U(s)+\int _s^\tau U(\xi )\,d\xi \right) ds,\quad 0\le t\le \tau , \end{aligned}$$

is equivalent to

$$\begin{aligned} U(t)\le c_0+c_1\int _t^\tau (1+\xi -t)U(\xi )\,d\xi . \end{aligned}$$

This is justified by the following:

$$\begin{aligned} \int _t^\tau \left( U(s)+\int _s^\tau U(\xi )\,d\xi \right) ds&= \int _t^\tau U(\xi )\,d\xi + \int _t^\tau \int _s^\tau U(\xi )\,d\xi \,ds\\&= \int _t^\tau U(\xi )\,d\xi +\int _t^\tau \int _t^\xi U(\xi )\,ds\,d\xi \\&= \int _t^\tau (1+\xi -t)U(\xi )\,d\xi . \end{aligned}$$

Remark 4.3

Thanks to the observation made by one of the reviewer which has prompted this remark. Note that instead of considering the non-local term \(\int _t^\tau u(x,s)\,ds\) in (1), if we consider some variant with non-constant kernel, for example, \(\int _t^\tau e^{-(s-t)}u(x,s)\,ds,\) then also we can deduce the results of this paper. This can be observed easily by just repeating the arguments employed in all the proofs. In fact, we can say something more general. If \(K\in C([0,\tau ]\times [0,\tau ]; {{\mathbb {R}}})\) and we consider the non-local term as \(\int _t^\tau K(s,t)u(x,s)\,ds\), then also we can deduce all the results of this paper without any significant modification. Indeed, we have to just repeat all the arguments and then simply take into account the constant \(\Vert K\Vert _\infty \) that additionally comes out.

5 Conclusion

We have considered the problem of recovering the solution of a final value problem for a parabolic equation with a non-linear source and a non-local term, from the knowledge of the final value. We have shown by an example that the considered problem is ill-posed, that is, a small perturbation in the data may lead to a large deviation in the sought solution. Since the problem is ill-posed, we have obtained regularized solutions by solving a non-linear integral equation which is derived by considering the truncated version of Fourier representation of the solution. Under appropriate Gevrey smoothness assumption (i.e., source condition), we have established a logarithmic-type convergence rate for the regularized approximation and with some comparatively higher Gevrey smoothness assumption, Hölder rate of convergence was established for the regularized approximations. We have proposed and proved a new version of Grönwalls’ inequality for iterated integrals, which plays the key role in obtaining the above mentioned rates.