Remarks on parabolic equation with the conformable variable derivative in Hilbert scales

1.   Introduction

Fractional calculus has recently attracted the attention of many researchers and has become an attractive field of study with its different application areas. Some researchers have discovered that fractional differential equations with different singular or non-singular kernel need to be determined by real-world problems in the fields of engineering and science. Some definitions/approaches, for example, Riemann-Liouville, Hadamard, Katugampola, Riesz, Caputo-Fabrizio, and Atangana-Baleanu operators, were presented and tested using a variety of theories. Many important analytical methods have been used to achieve analytical solutions to fractional diffusion equations. By replacing many differential operators of fractional order with different PDE types of integer order, we form different types of boundary value problems with fractional order. However, the types of diffusion equations with fractional derivatives in Hilbert scales space are not really abundant because of their difficulty. We can list a few interesting works on PDEs with fractional derivatives, for example, ^{[7,8,9,12,13,14,16,21,22,27,28,29]} and the references therein.

Let $T$ be a positive number. In this paper, we consider the initial value problem for the conformable heat equation (or called parabolic equation with conformable operator)

where $\frac{ \partial^{\beta(t)} }{\partial t^{\beta(t)}} y = T^{0}_{\beta (t)} y(t)$ is defined in Definition (2.3). Here $\Omega \subset\mathbb{R}^N$ ($N \ge 1$) is a bounded domain with the smooth boundary $\partial \Omega$. We are interested to study two following conditions

or nonlocal in time condition

The condition (1.2) is also known as initial conditions, which is familiar to mathematicians in the field of PDEs. Let us provide some remarks on the condition (1.3). Non-local conditions present and explain some more realistic perspectives for some particular phenomena for which usual initial conditions are replaced by multi-time point data such as studying atomic reactors ^[1,2,26]. In terms of mathematical aspect, since these conditions provide different data from the usual initial/terminal conditions problems with associated nonlocal conditions possess particular properties. In particular, it is well-known that while the problem for the usual parabolic equation is well-posed with the initial Cauchy condition at $t = 0$ and such problem is ill-posed with given data at terminal time $t = T > 0,$ the well-posedness can be witnessed for the problems involving forward parabolic equations with non-local in time conditions connecting the values at different times ^[5]. In fact, throughout this work, we can see that the techniques to derive well-posed results for the initial value problem and the nonlocal in time problem are quite different. The above remarks play an important role in our motivation for deciding to carry out this study. As far as we know, there is very little documentation on the solution connection boundary conditions at different points in time, for example, at the beginning and at the end. Consideration of non-local initial conditions or non-local final conditions derived from actual processes.

Before we cover our problem, we give some background on conformable derivatives. A Conformable derivative can be first stated by Khalil and his colleagues ^[3] for functions $f: [0, \infty] \to +\infty$, it can be considered as the general form of the classical derivative and follows the same properties as the classical derivative. Furthermore, the physical meaning of the conformable derivative is assumed to be a modification of the classical derivative of direction and magnitude. More precisely, the general conformable derivative possesses similar physical and geometrical interpretations of Newton's derivative. However, while Newton's derivative describes the velocity of a particle or slope of a tangent, the general conformable derivative can be regarded as a special velocity, its direction and strength rely on a particular function ^[23].

Let us take $M$ as a Banach space, and the function $f:[0, \infty) \to M$ and $\frac{{ }^{\mathscr C}\partial^\beta}{\partial t^\beta}$ be the conformable derivative of order $0 < \beta \le 1$ locally defined by

for each $t > 0$. For additive information about the above definition, we refer the reader to ^{[3,4,6,10,11,20]}. An easy observation is that if $\beta = 1$ then the definition given above is the definition of the classical derivative. To further understand the relationship between conformable and classical derivatives, we direct the reader to the interesting paper ^[15]. This paper can be considered as one of the first works to investigate diffusion equations with conformable derivative in the Sobolev space. According to natural development, based on the conformable derivative, mathematicians have built a good theory for conformable derivative with orders dependent on a variable.

For the reader to better understand the history of this problem, we present a number of related works. Let us provide the comments of some fractional diffusion equations associated with fractional derivative whose order is a constant, i.e., $\beta(t) = \beta$.

Now, we introduce some previous work mentioned on fractional diffusion equation with variable order. In ^[18], the authors considered the relaxation-type equation with fractional variable order as follows

where $\frac{ \partial^{ \alpha(t)} }{\partial t^{ \alpha(t)}}$ is the left Caputo derivative of order $\alpha(t)$, $B$ is the relaxation coefficient, $f(t)$ denotes the external source term. The authors investigated the cable equation with fractional variable order ^[19]. In ^[24], the authors studied a dynamical system described by the following fractional differential equation with variable order

The authors considered the following dynamical system with variable-order fractional derivative

where $q(t)$ is the variable-order of differentiation ^[25].

To the best of our knowledge, there are not any results for considering the well-posedness of two problem (3.1)–(1.2) and (3.1)–(1.3). We draw attention to the paper ^[17] since it mentioned variable conformable derivative. They investigated the fundamental solutions for initial value problem for linear diffusion differential equations with the conformable variable order derivative. Their techniques are based on upper and lower solutions and monotone iterative method. One difference is that they consider (3.1) on the unbounded domain, while we consider it on the bounded domain. Our approach in this paper is different from ^[17] because we have to learn the ideas of Fourier series. A new point of the current paper is that we carefully examine the well-posedness of our problem.

Let us assert that the problem with the variable conformable derivative is more difficult than the derivatives of constant derivative. The main reason is the appearance of integrals with exponents as functions, for example $\int_0^t r^{\beta(r)-1} dr$ causing many difficulties in calculation and evaluation. To overcome these difficulties, we need to have skillful judgment to control the components containing these singular integrals.

The main objective of this paper is to investigate the existence and regularization of solutions for two problems. With different assumptions of the input functions $F$ and $u_0$, we will show the space containing the solution. As introduced above, we have a challenge with components that contain singular integrals. Another interesting contribution is that we will examine the relationship between the solutions of two problems: nonlocal problem (3.1)–(1.3) and (3.1)–(1.2). The result is proven convergent of the mild solution to (3.1)–(1.3) when $h \to 0^+$. This proof of convergent is understood as a non-trivial task.

The structure of the paper is given as follows. Section 3 examines the well-posedness for the initial value problem (3.1)–(1.2). The existence for the mild solution to (3.1)–(1.3) is investigated in section 4. We also derive that the convergence of the mild solution to problem (3.1)–(1.3) when $h \to 0^-$.

2.   Preliminary results

In this section, we introduce notations and functional settings which will be used throughout this work. Recall that the spectral problem

admits the eigenvalues $0 < \lambda_1 \leq \lambda_2 \leq \dots \leq \lambda_j \leq \dots$ with $\lambda_j \to \infty$ as $j \to \infty$ and the corresponding set of eigenfunctions $\{\psi_j\}_{j\ge1} \subset H_0^1(\Omega)$.

Definition 2.1. We recall the Hilbert scale space as follows

for any $s \geq 0$. It is well-known that $\mathbb Z^s (\Omega)$ is a Hilbert space corresponding to the norm

In the following, we provide definitions of the left integral and the (left) variable order fractional derivative which are taken from ^[17].

Definition 2.2. Let $f: [a, \infty) \to (0, 1]$. The left integral begin at $a$ of variable function $h: (a, \infty) \to \mathbb{R}$ is given by

Definition 2.3. The (left) variable order fractional derivative starting at a of a function $f: [a, \infty)$ of order ${h}:[a, \infty) \to (0, 1]$ is defined by

When $a = 0$, one can write $T_{h(t)}$. Moreover if $T^{a}_{h(t)}f(t)$ exists on $(a, \infty)$ then $T^{a}_{h(t)}f(a) = \lim_{t \to a^{+}} T^{a}_{h(t)}f(t)$.

In addition, if the fractional derivative of order $h(t) \in (0, 1]$ of $f$ exists for all $t \in (a, \infty)$, we simply say $f$ is $h(t)-$ differentiable.

3.   Linear inhomogeneous problem with initial condition

In this section, we focus on the initial value problem

where $y_0$ and $F$ will be defined later. Our main purpose in this section is to study the well-posedness of Problem (3.1). We use the Fourier analysis to construct the mild solution. Let us assume that $y(x, t) = \sum_{j = 1}^\infty \langle y(., t), \psi_j \rangle \psi_j (x)$ where $\langle y(., t), \psi_j \rangle: = \int_{\Omega}y(x, t)\psi_j(x)dx$. Taking the inner product $\langle\cdot, \cdot\rangle$ of the main equation of Problem (3.1) with $\psi_j$ gives

By the result in ^[17], we obtain the following equality

where we remind that $\beta : [0, \infty) \to (0, 1]$. By the definition of Fourier series, we have the following formula of the mild solution

Lemma 3.1. Let $m = \min_{0 \le t \le 1 } |\beta(t)|$ and $b = \max_{0 \le t \le 1 } |\beta(t)|$.

i) If $0 \le t \le 1$ then

ii) If $t \ge 1$ then

Proof. We claim $(i)$ as follows. Since $\beta (r) \ge m$ and $0 < \beta (r) \le 1$, we know that $0\le 1-\beta(r) \le 1-m$. Since $0\le r \le t < 1$, we know that $\frac{1}{r} > 1$. It follows that

This implies that

Since $1-\beta (r) \ge 1-b\ge0$, we know that

It implies the following lower bound

We next provide the proof of $(ii)$. Since $t \ge 1$, we derive

Using (3.5) with $t = 1$, we obtain the following upper and lower bound

Our next aim is to consider the term $\int_1^t r^{\beta (r)-1} dr$. It is easy to observe that

From the fact that $0 < \frac{1}{r} < 1$, we get the upper bound below

and also, the lower bound

Connecting all the above inequalities (3.11), (3.12) and (3.13) gives us the assertion (3.6).

Lemma 3.2. Let $m = \min_{0 \le t \le 1 } |\beta(t)|$ and $b = \max_{0 \le t \le 1 } |\beta(t)|$.

i) If $0 \le r \le t \le 1$ then

ii) If $0 < r \le 1 \le t$, we get

iii) If $0\le 1 \le r \le t$ then

Proof. The proof of this lemma is almost the same as that of Lemma (3.1). Our claim is divided into three cases.

$\bullet$ The case $0 < t \le 1$. For this case, it is easy to see that

This implies that

It is easy to verify that

and

Hence, we obtain that for any $0 < r \le t \le 1$

$\bullet$ The case $0 \le r \le t \le 1$. For this case, we get the following identity

By setting $t = 1$ into (3.20), we arrive at

This implies the following estimate

which allows us to deduce the desired result.

$\bullet$ The case $0 \le 1 \le r \le t$. Under this case, we obtain that if $r \le z \le t$ then

This implies that

Hence, we find that

The well-posedness of Problem (3.1) is described by the following theorem.

Theorem 3.3. i) Let $y_0 \in \mathbb Z^{s- \varepsilon } (\Omega)$ for $\varepsilon > 0$ and $F \in L^\infty (0, T; \mathbb Z^s (\Omega))$. Then we get

ii) Let $y_0 \in \mathbb Z^{s- \varepsilon } (\Omega)$ for $\varepsilon > 0$ and $F \in L^\infty (0, T; \mathbb Z^{s-\delta} (\Omega))$ for any $0 < \delta < \frac{1}{2}$. Let us assume that $2m > b$. Then we obtain

Proof. Let us recall the mild solution

Step 1. Estimate of the term $J_1$. Using the inequality $e^{-a} \le C(\varepsilon) a^{-\varepsilon}$ for any $\varepsilon > 0$, we find that

$\bullet$ If $0 < t\le 1$ in view of (3.5), we obtain

where

By Parseval's equality and using (3.30), we derive that

This implies that for $t \le 1$

$\bullet$ If $t \ge 1$ thanks to (3.6) of Lemma (3.1), we obtain

Since $t\ge 1$, it is obvious to see that the following inequality is satisfied

From the previous observations, we get that

Combining (3.32) and (3.34), we deduce the following estimate for any $0 \le t \le T$

Step 2. Estimate of the term $J_2$.

By Parseval's equality and Hölder's inequality, we find that

Let us now consider possible cases as follows.

Case 1: $0 < t\le 1$ and $F \in L^\infty (0, T; \mathbb Z^s (\Omega))$.

In view of (3.14) and the fact that $\exp \Big(- \lambda_j \int_r^t z^{\beta(z)-1} dz \Big) \le 1$, we derive

It follows from (3.36) that

From (3.14) we obtain that the following estimate

Case 2: $t \ge 1$ and $F \in L^\infty (0, T; \mathbb Z^s (\Omega))$.

Using (3.38) and by a similar claim in case 1, we get that

In view of (3.6), we obtain

Since $b \ge m$, we have the following inequality

Therefore, we derive that for any $t \ge 1$

Combining case 1 and case 2, we get the following estimate for any $t > 0$ and $F \in L^\infty (0, T; \mathbb Z^s (\Omega))$

Case 3: $0 < t\le 1$ and $F \in L^\infty (0, T; \mathbb Z^{s-\delta} (\Omega))$.

Using the inequality $e^{-a} \le C(\delta) a^{-\delta}$ for any $\delta > 0$, we obtain that

From the fact that $t \le 1$, we use (3.14) to get

Hence, we get the following estimate

where we have used (3.7). Let us now treat the integral term on the right hand side of (3.46). By applying Hölder inequality and noting that $2m > b$, we derive that

Set $r' = r^b$, then $dr' = br^{b-1} dr$. Then, since $2 \delta < 1$, we have

Combining (3.46), (3.47) and (3.48), we get the following estimate for $t\le 1$

where

This inequality together with (3.36) yields

where $\overline C_2 = \overline C_1 C(\delta, b)$. It is obvious to see that

In the previous claim, we showed that

Combining (3.50), (3.51) and (3.52), we obtain that for any $0 < t\le 1$

where we denote by

Case 4: $t \ge 1$ and $F \in L^\infty (0, T; \mathbb Z^{s-\delta} (\Omega))$.

We need to deal with the integral term

To this end, we derive the following equality

For the term $\mathrm{I}_1$, we put $t = 1$ into (3.49) to obtain

For the second term $\mathrm{I}_2$, we note that $1 \le r \le t$. Hence $r^{\beta(r)-1} \le r^{b-1}$. In view of the inequality $e^{-a} \le C(\delta) a^{-\delta}$ for any $\delta > 0$, we get

Using (3.16) and (3.56), we find that

In view of (3.16), we can check easily that

It follows from (3.57) that

Next, using Hölder inequality to derive that

It is not difficult to compute that

From the above two observations, we find that

where

This combined with (3.58) yields to the following bound

where $\overline C_5 = C(\delta) m^\delta\overline C_4$. Combining (3.54), (3.55) and (3.57) and noting that $1 \le t^{m- b\delta}$ we derive that

Therefore, we obtain that for $t \ge 1$

It is obvious to see that the following inequality holds

where we hace applied (3.53). This together with (3.36) and (3.56) allow us to obtain that

where $\overline C_7 = \overline C_6 C(\delta) m^\delta.$ By looking at the estimate (3.61), we infer the following estimate

where $\overline C_8 = \overline C_7 \overline C_4$. Combining (3.64), (3.65) and (3.67), we derive the following bound

Since $t \ge 1$, we follows from (3.68) that

Summarizing two cases 3 and 4, we provide the following statement

Hence, the proof of (3.26) is finished by combining (3.35) and (3.43). At the same time, the proof of (3.27) is derived from (3.35) and (3.70).

4.   Linear problem with nonlocal in time condition

In this section, we focus the nonlocal value problem

Our main purpose in this section is to study the well-posedness of problem (4.1) and the convergence of the mild solution when $h \to 0^ +$.

Theorem 4.1.

i) Let $y_0 \in \mathbb Z^{s- \varepsilon } (\Omega)$ for $\varepsilon > 0$ and $F \in L^\infty (0, T; \mathbb Z^s (\Omega))$. Then Problem (4.1) has a unique solution $y_h$ such that

where the hidden constant depends on $T, b, \varepsilon, m$.

ii) Let $y_0 \in \mathbb Z^{s- \varepsilon } (\Omega)$ for $\varepsilon > 0$ and $F \in L^\infty (0, T; \mathbb Z^{s-\delta} (\Omega))$ for any $0 < \delta < \frac{1}{2}$. Let us assume that $2m > b$. Then we get

where the hidden constant depends on $T, b, \varepsilon, m, \delta$.

Proof. Let us first establish the fomula of the mild solution to nonlocal problem (4.1). Suppose that Problem (4.1) has a solution $y_h$. From (3.3), we get

By let $t = T$ into the above expression, we see that

From the above two equalities and the nonlocal-in-time condition

we deduce the following equality

This implies that the following equality is satisfied

Combining (4.4) and (4.7), we derive that

By the theory of Fourier series, the mild solution is given by

Let us consider the first term $\mathbb K_1$. By Parseval's equality, using (3.30), (3.35) and noting that $1+ h \exp \Big(- \lambda_j \int_0^T r^{\beta(r)-1} dr \Big) > 1$, we derive

where we have used (3.31). Therefore, we obtain that the following estimate

Proof of i). Suppose $F \in L^\infty (0, T; \mathbb Z^s (\Omega))$.

We deal with the second term $\mathbb K_2$. We first obtain

From (3.43), we can easily to verify that

Combining (4.12) and (4.13), we derive the following bound

Let us now treat the third term $\mathbb K_3$. In view of (3.43), we infer that

Combining (4.9), (4.11), (4.14) and (4.15) yields

Proof of ii). Suppose that $F \in L^\infty (0, T; \mathbb Z^{s-\delta} (\Omega))$.

From (3.70), we obtain the following bound

where we note that $t^{m- b\delta} \le T^{m- b\delta}$. This estimate together with (4.12) yield

In view of (3.70), we infer that

Combining (4.9), (4.11), (4.18) and (4.19), we deduce that

The proof is completed.

The following theorem shows the convergence of the mild solution to (3.1)-(1.3) when $h \to 0^-$.

Theorem 4.2. i) Let $y_0 \in \mathbb Z^{s- \varepsilon } (\Omega)$ and $F \in L^\infty (0, T; \mathbb Z^{s-\varepsilon} (\Omega))$ for any $0 < \varepsilon < \frac{1}{b}$. Then, $h\in(0, 1)$ and $k\in(1, 2)$ we get

where $C$ depends on $T, b, \varepsilon, p$.

ii) Let $y_0 \in \mathbb Z^{s- \varepsilon } (\Omega)$ for $\varepsilon > 0$ and $F \in L^\infty (0, T; \mathbb Z^{s} (\Omega))$ for any $\varepsilon > 0$. Let us assume that $2m > b$. Then we get

where the hidden constant depends on $T, b, \varepsilon, m$.

Proof. First, we focus on the formulas of solutions (4.9) and (3.28). Taking the difference, we get

By a simple transformation, it is easy to verify that

We need to consider the term

We consider the denominator component of the above fraction. In view of the inequality

we get the following inequality

and

Since $k < 2$, the latter three observations infer that

From this result and (4.26), we derive that

Using (3.35), we obtain

It is obvious to check that the following estimate holds

Since $0 < h < 1$, we get

Combining (4.28) and (4.30), we deduce that

Next, we consider $\| \mathbb K_2(., t)\|_{\mathbb Z^s (\Omega) }$ in two cases corresponding to part i) and part ii). We use the results in the proof of Theorem (4.1).

Case 1. Proof of (4.21).

Since $F$ is in the space $L^\infty (0, T; \mathbb Z^{s-\varepsilon} (\Omega))$, we follows from (4.14) that

Combining (4.23), (4.31) and (4.32), we find that

Let us choose $\varepsilon$ such that $0 < \varepsilon < \frac{1}{b}$. Since $1 < p < \frac{1}{b \varepsilon}$, we know that the proper integral $\int_0^T t^{-b \varepsilon p} dt$ is convergent. By a simple computation, we deduce that

where $C$ depends on $T, b, \varepsilon, p$.

Case 2. Proof of (4.22).

From the definition of $\mathbb K_2$ as in (4.23), we derive that

We have the following observation

Combining (4.23), (4.31) and (4.36), we find that

From the right-hand side of the above estimate, we deduce the desired result (4.22). The proof of our theorem is completed.

5.   Conclusions

This work considers a time-fractional parabolic equation with conformable variable derivative. We derive the well-posedness for mild solutions in Hilbert spaces for linear initial problem and linear nonlocal problem. We also shows the convergence of non-local solutions to local solutions. The techniques obtained in this study can be further extended to complicated nonlinear problem.

Acknowledgement

The author Van Tien Nguyen thanks the support from FPT University.

Conflict of interest

The authors declare no conflict of interest.