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On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator

  • Bui Dai Nghia , Van Tien Nguyen and Le Dinh Long EMAIL logo
Published/Copyright: January 19, 2023
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

In this article, we considered the pseudo-parabolic equation with Caputo-Fabrizio fractional derivative. This equation has many applications in different fields, such as science, technology, and so on. In this article, we gave the formula of mild solution, which is represented in the form of Fourier series by some operators . In the linear case, we investigated the continuity of the mild solution with respect to the fractional order. For the nonlinear case, we investigated the existence and uniqueness of a global solution. The main proof technique is based on the Banach fixed point theorem combined with some Sobolev embeddings. For more detailed, we obtained two other interesting results: the continuity of mild solution with respect to the derivative order and the convergence of solution as the coefficient k approaches to zero.

Keywords: Caputo-Fabrizio derivative operator; Burger equation; Banach fixed point theory; Sobolev embeddings
MSC 2010: 35R11; 35B65; 26A33

1 Introduction

Recently, fractional differential equations have been extensively used in describing various mathematical models of physical processes and natural phenomena, for example, in mechanics, physics, and engineering sciences, etc. The number of works in this direction is quite abundant and attracts many interested mathematicians. We list here some works using fractional derivative models that closely related to this article, such as [1,2, 3,4,5, 6,7,8, 9,10,11, 12,13,14, 15,16,17, 18,19] and references therein.

Let M be a simply connected and bounded domain in R n with a smooth boundary ∂ M . Let T be a positive real number. In this article, we are interested in considering the pseudo-parabolic equation as follows:

(1.1) D t β CF ( Z + k L Z ) + L p Z = G ( Z ) , in M × ( 0 , T ] , Z ( x , 0 ) = z 0 ( x ) , in M , z ( x , t ) = 0 , in ∂ M ,

where D t β CF is the Caputo-Fabrizio derivative operator of order β , which is defined as (see [20,21]) follows:

D t β CF z ( t ) = Y ( β ) 1 − β ∫ 0 t D β ( t − r ) ∂ z ( r ) ∂ r d r , for t ≥ 0 ,

where D β ( r ) = exp − β 1 − β r and Y ( β ) satisfies Y ( 0 ) = Y ( 1 ) = 1 .

It is easy to observe that if β = 1 , then problem (1.1) turns to the classical pseudo-parabolic equation. The pseudo-parabolic equation describes several important physical and biological phenomena, such as the analysis of unstable processes in [22] and population aggregation in [23]. We encourage the interested reader to the recent work by Tuan and Caraballo [24] on classical pseudo-parabolic equation, which is in the spirit of the Fourier series semigroup. Currently, several models with pseudo-parabolic equations, combined with fractional derivative, have attracted the interest of a number of mathematicians. Luc et al. in [25] derived the pseudo-parabolic equation with Caputo derivative as follows:

(1.2) D t β C ( Z + k L Z ) + L p Z = G ( Z ) , in M × ( 0 , T ] , Z ( x , 0 ) = z 0 ( x ) , in M , Z ( x , t ) = 0 , in ∂ M ,

where 0 < β < 1 , D t β C is a Caputo fractional derivative operator of order β . The authors have obtained the existence of mild solutions in both aspects: local and global sense. The main method is to use the Banach contraction mapping theorem in Hilbert scales space. In [5], Tuan et al. studied problem (1.2) when the source function has a logarithmic form. The technique used in this article is quite interesting with many different embeddings in L p and rather complicated evaluations. In [26], Can et al. focused on the nonlocal pseudo-parabolic equation with linear case. In [27], Tuan et al. considered the pseudo-parabolic equation associated with integral condition under nonlinear case. Shen et al. in [28] studied the fractional pseudo-parabolic equation with the Riemann-Liouville derivative. They obtained the global and local existence of weak solutions by using the Galerkin method.

To our knowledge, there have been quite a few investigations on diffusion equations with the presence of the Caputo-Fabrizio derivative. Let us refer some interesting papers that focus on the existence of the mild solution to some problems, e.g., [29,20,30,31,21]. In [20], Tuan and Zhou established the existence and uniqueness of the mild solution for diffusion equation (1.1) with k = 0 . They also provided the existence of local mild solutions to the problem, and then a blow-up alternative is established. In [30], Tuan studied the Cahn-Hilliard equation with the Caputo-Fabrizio operator as follows:

(1.3) D t β CF Z + Δ 2 Z = Δ ( Z − Z 3 ) , in M × ( 0 , T ] , Z ( x , 0 ) = z 0 ( x ) , in M , Z ( x , t ) = Δ Z = 0 , in ∂ M ,

and he proved the local existence result for problem (1.3). He first provided that the connections of the mild solution to problem (1.3) between the Cahn-Hilliard equation in the case 0 < β < 1 and β = 1 . The main key of the proof is the proficient use of some embeddings between L p spaces and Hilbert scales spaces. Recently, the time-fractional integro-differential equation with the Caputo-Fabrizio type derivative has been considered in [31]. Let us refer the reader to some works in the spirit of mild solution with Caputo Fabrizio derivative, such as [29,32,21,36,37,38, 39,41,42, 43,44,45, 46,47,48, 49,50].

There are two advantages when we investigated the equation using this type of derivative. The most important thing is that we can avoid the singularity kernel, which frequently occurs in the case of the Caputo or Riemann-Liouville derivative. The second advantage is that the model with Caputo-Fabrizio is effectively used in physical models involving an exponential power multiplied to some components of the equations. To the best of our knowledge, the current article is the first study about pseudo-parabolic equations with Caputo-Fabrizio derivative.

The main results of this article are described as follows:

  1. In the linear case, we investigate the continuity of the mild solution respect to the derivative order. This research direction was inspired by an article by Dang et al. [33]. This article focuses on the question: “Does Z β ′ tends to Z β in an appropriate sense as β ′ → β ?”

  2. For the nonlinear case, we prove the existence and uniqueness of global mild solution when the Lipschitz source function is global in Hilbert spaces H p . The main tool is to use Banach’s theorem in suitable spaces. We also obtained continuous dependence of mild solution respect to derivative order for the nonlinear case.

  3. Another interesting contribution of this article is to show that the solution of the pseudo-parabolic equation converges to the solution of the corresponding parabolic problem. The ideas and methods are partially found from our recent paper by Tuan et al. [40] and Tuan [30]. To complete the proofs, we had to overcome many challenges by skillfully evaluating the upper bounds.

This article is organized as follows. Section 2 presents some preliminaries, the formula of the mild solution, and its representation in the operator form. Section 3 is dedicated to the results on continuity of fractional order for the linear problem. In Section 4, we showed the global existence of the mild solution in the nonlinear case. We also obtained the continuity results of fractional order in the case where the function G is globally Lipschitz function. In Section 5, we investigate the convergence of mild solutions of the problem (1.1) when k → 0 + .

2 Preliminaries, mild solution, and solution operators

The Hilbert scale space H m ( M ) is defined in the following line:

H m ( M ) = θ ∈ L 2 ( M ) , ∑ n = 1 ∞ λ n 2 m ∫ M θ ( x ) ψ n ( x ) d x 2 < ∞ ,

for some m ≥ 0 . The norm of H m ( M ) is also given as follows:

‖ θ ‖ H m ( M ) = ∑ n = 1 ∞ λ n 2 m ∫ M θ ( x ) ψ n ( x ) d x 2 , θ ∈ H m ( M ) .

This Hilbert scales space plays an important role in investigating the properties of regularity for mild solutions.

Lemma 2.1

(See [4]) There hold that

  1. H m ∕ 2 ( M ) ↪ L p ( M ) , for m = N 2 , 1 ≤ p < ∞ , or 0 ≤ m < N 2 , 1 ≤ p ≤ 2 N N − 2 m ;

  2. L p ( M ) ↪ H m ∕ 2 ( M ) , for − N 2 < m ≤ 0 , p ≥ 2 N N − 2 m .

Definition 2.2

(See [35]) Let Y a , d ( ( 0 , T ] ; B ) denote the weighted space of all functions v ∈ C ( ( 0 , T ] ; B ) such that

(2.4) ‖ f ‖ Y a , d ( ( 0 , T ] ; B ) ≔ sup t ∈ ( 0 , T ] t a e − d t ‖ f ( t , ⋅ ) ‖ B < ∞ .

The space Y a , d ( ( 0 , T ] ; B ) is used in proving the global solution of the nonlinear problem. In the following, we introduce the formula of mild solution to our problem. Let us assume that

Z ( x , t ) = ∑ n = 1 ∞ ∫ M Z ( x , t ) ψ n ( x ) d x ψ n ( x ) .

Then we obtain the following differential equation:

(2.5) D t β CF ∫ M Z ( x , t ) ψ n ( x ) d x + λ n p 1 + k λ n ∫ M Z ( x , t ) ψ n ( x ) d x = 1 1 + k λ n ∫ M G ( x , t ) ψ n ( x ) d x .

By solving the aforementioned differential equation, the mild solution is given by

(2.6) Z ( t ) = Q β , p ( t ) z 0 + ∫ 0 t J β , p ( t − ν ) G ( ν ) d ν ,

where Q β , p ( t ) and J β , p ( t ) are defined via the Fourier series as follows:

(2.7) Q β , p ( t ) f ( x ) ≔ ∑ n = 1 ∞ 1 + k λ n 1 + k λ n + ( 1 − β ) λ n p exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p f n ψ n ( x ) ,

(2.8) J β , p ( t ) f ( x ) ≔ ∑ n = 1 ∞ β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p f n ψ n ( x ) ,

for any f ∈ L 2 ( M ) . Here, f n is Fourier coefficient of f , which is defined by

(2.9) f n = ∫ M f ( x ) ψ n ( x ) d x .

The following lemma is introduced to play an important role in investigating some properties of solutions.

Lemma 2.3

Let m be any real number and θ ∈ H m − p ( M ) . Then we obtain

(2.10) ∥ Q β , p ( t ) θ ∥ H m ( M ) ≤ 1 1 − β ‖ θ ‖ H m − p ( M )

and

(2.11) ∥ J β , p ( t ) θ ∥ H m ( M ) ≤ 2 β 1 − β ‖ θ ‖ H m − p ( M ) .

Proof

In view of Parseval’s equality and noting that e − z ≤ 1 for z ≥ 0 , we obtain that

∥ Q β , p ( t ) θ ∥ H m ( M ) 2 = ∑ n = 1 ∞ 1 + k λ n 1 + k λ n + ( 1 − β ) λ n p 2 λ n 2 m exp − 2 β λ n p t 1 + k λ n + ( 1 − β ) λ n p θ n 2 ≤ ∑ n = 1 ∞ 1 ( 1 − β ) λ n p 2 λ n 2 m exp − 2 β λ n p t 1 + k λ n + ( 1 − β ) λ n p θ n 2 = 1 ( 1 − β ) 2 ∑ n = 1 ∞ λ n 2 m − 2 p θ n 2 = 1 ( 1 − β ) 2 ‖ θ ‖ H m − p ( M ) 2 ,

where θ n is given by θ n = ∫ M θ ( x ) ψ n ( x ) d x . By a similar explanation as mentioned earlier, we find that

∥ J β , p ( t ) θ ∥ H m ( M ) 2 = ∑ n = 1 ∞ β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 2 λ n 2 m exp − 2 β λ n p t 1 + k λ n + ( 1 − β ) λ n p θ n 2 .

By using the Cauchy inequality ( a + b ) 2 ≥ 4 a b for any a , b ≥ 0 , we find that

( 1 + k λ n + ( 1 − β ) λ n p ) 2 ≥ 4 ( 1 − β ) ( 1 + k λ n ) λ n p .

Hence, we derive that

(2.12) β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 ≤ β 4 ( 1 − β ) λ n − p ≤ 2 β ( 1 − β ) λ n − p .

Hence, by using this inequality, we infer that

∥ J β , p ( t ) θ ∥ H m ( M ) 2 ≤ ∑ n = 1 ∞ 2 β 1 − β λ n − p 2 λ n 2 m exp − 2 β λ n p t 1 + k λ n + ( 1 − β ) λ n p θ n 2 = 4 β 2 ( 1 − β ) 2 ∑ n = 1 ∞ λ n 2 m − 2 p θ n 2 = 4 β 2 ( 1 − β ) 2 ‖ θ ‖ H m − p ( M ) 2 .

So, the proof of Lemma 2.3 is completed. Let us continue to introduce the following lemma, showing the continuity of the derivative order for mild solutions.

Lemma 2.4

Let γ > 0 and μ > 0 . Then we obtain the following estimate

(2.13) ∥ Q β , p ( t ) f − Q β ′ , p ( t ) f ∥ H m ( M ) ≤ C 6 ( t γ ∣ β − β ′ ∣ γ ∥ f ∥ H m − p + p γ − γ ( M ) + ∣ β − β ′ ∣ t − μ ‖ f ‖ H m + μ − μ p ( M ) ) .

Remark 2.1

Since the fact that m − p + p γ − γ ≤ m + μ − μ p , we deduce that the following Sobolev embedding

(2.14) H m + μ − μ p ( M ) ↪ H m − p + p γ − γ ( M ) .

This follows from Lemma 2.4 that

(2.15) ∥ Q β , p ( t ) f − Q β ′ , p ( t ) f ∥ H m ( M ) ≤ C ¯ 6 t − μ ( ∣ β − β ′ ∣ γ + ∣ β − β ′ ∣ ) ‖ f ‖ H m + μ − μ p ( M ) .

Remark 2.2

Since the proof of Lemma 2.4, it is easy to obtain the following estimate

(2.16) ∥ Q β , p ( t ) f − Q β ′ , p ( t ) f ∥ H m ( M ) ≲ ∣ β − β ′ ∣ ‖ f ‖ H m ( M )

for any f ∈ H m ( M ) .

Proof

From (2.7), we derive that

(2.17) Q β , p ( t ) f ( x ) − Q β ′ , p ( t ) f ( x ) = ∑ n = 1 ∞ 1 + k λ n 1 + k λ n + ( 1 − β ) λ n p exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p − exp − β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p f n ψ n ( x ) + ∑ n = 1 ∞ 1 + k λ n 1 + k λ n + ( 1 − β ) λ n p − 1 + k λ n 1 + k λ n + ( 1 − β ′ ) λ n p exp − β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p f n ψ n ( x ) = A 1 ( t , x ) + A 2 ( t , x ) .

The first term A 1 is bounded by

(2.18) ∥ A 1 ( t , . ) ∥ H m ( M ) 2 = ∑ n = 1 ∞ λ n 2 m 1 + k λ n 1 + k λ n + ( 1 − β ) λ n p 2 exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p − exp − β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p 2 ∣ f n ∣ 2 .

It is obvious to see that

(2.19) 1 + k λ n 1 + k λ n + ( 1 − β ) λ n p 2 ≤ λ n − 2 p ( 1 − β ) 2 .

By using the inequality ∣ e − a − e − b ∣ ≤ C γ ∣ a − b ∣ γ for any γ > 0 , we know that

(2.20) exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p − exp − β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p ≤ C γ β λ n p t 1 + k λ n + ( 1 − β ) λ n p − β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p γ = C γ t γ λ n p γ ∣ β − β ′ ∣ γ 1 + k λ n + λ n p ( 1 + k λ n + ( 1 − β ) λ n p ) ( 1 + k λ n + ( 1 − β ′ ) λ n p ) γ .

In view of the two following observations:

(2.21) 1 + k λ n + λ n p 1 + k λ n + ( 1 − β ) λ n p γ ≤ 1 ( 1 − β ) γ , 1 1 + k λ n + ( 1 − β ′ ) λ n p γ ≤ k − γ λ n − γ ,

we find that

(2.22) exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p − exp − β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p ≤ C ( k , γ ) ( 1 − β ) γ t γ λ n p γ − γ ∣ β − β ′ ∣ γ .

By combining (2.18), (2.19), and (2.22), we obtain that

(2.23) ∥ A 1 ( t , . ) ∥ H m ( M ) 2 ≤ ∣ C ( k , γ ) ∣ 2 ( 1 − β ) 2 γ + 2 t 2 γ ∣ β − β ′ ∣ 2 γ ∑ n = 1 ∞ λ n 2 m − 2 p + 2 p γ − 2 γ f n 2 .

Therefore, we arrive at the following estimate:

(2.24) ∥ A 1 ( t , . ) ∥ H m ( M ) ≤ C ( k , γ ) ( 1 − β ) γ + 1 t γ ∣ β − β ′ ∣ γ ∥ f ∥ H m − p + p γ − γ ( M ) .

The second term A 2 can be bounded as follows. By using Parseval’s equality, we derive that

(2.25) ∥ A 2 ( t , . ) ∥ H m ( M ) 2 = ∑ n = 1 ∞ λ n 2 m 1 + k λ n 1 + k λ n + ( 1 − β ) λ n p − 1 + k λ n 1 + k λ n + ( 1 − β ′ ) λ n p 2 exp − 2 β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p ∣ f n ∣ 2 .

It is not difficult to see that

(2.26) 1 + k λ n 1 + k λ n + ( 1 − β ) λ n p − 1 + k λ n 1 + k λ n + ( 1 − β ′ ) λ n p = ∣ β − β ′ ∣ ( 1 + k λ n ) λ n p ( 1 + k λ n + ( 1 − β ) λ n p ) ( 1 + k λ n + ( 1 − β ′ ) λ n p ) .

By using the Cauchy inequality a + b ≥ 2 a b for any a , b ≥ 0 , we obtain that

(2.27) 1 + k λ n + ( 1 − β ) λ n p ≥ 2 1 + k λ n 1 − β λ n p ∕ 2

and

(2.28) 1 + k λ n + ( 1 − β ′ ) λ n p ≥ 2 1 + k λ n 1 − β ′ λ n p ∕ 2 .

By summarizing the aforementioned three results, we immediately have

(2.29) 1 + k λ n 1 + k λ n + ( 1 − β ) λ n p − 1 + k λ n 1 + k λ n + ( 1 − β ′ ) λ n p ≤ ∣ β − β ′ ∣ 4 1 − β 1 − β ′ .

By using the inequality e − z ≤ C μ z − μ and in view of ( a + b ) μ ≤ C μ ( a μ + b μ ) for any μ > 0 , we find that

(2.30) exp − β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p ≤ C μ β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p − μ ≤ C 1 , μ t − μ ( 1 − β ′ + λ 1 − p + k λ n 1 − p ) μ ≤ t − μ ( C 2 + C 3 λ n μ ( 1 − p ) ) ,

where C 2 depends on μ , β ′ , λ 1 , and p and C 3 depends on μ , k . By combining (2.25), (2.29), and (2.30), we derive that for any μ > 0

(2.31) ∥ A 2 ( t , . ) ∥ H m ( M ) 2 ≤ ∣ β − β ′ ∣ 2 8 ( 1 − β ) ( 1 − β ′ ) ∑ n = 1 ∞ λ n 2 m ( C 2 2 t − 2 μ + C 3 2 t − 2 μ λ n 2 μ − 2 μ p ) ∣ f n ∣ 2 .

This estimate and Parseval’s equality implies immediately that

(2.32) ∥ A 2 ( t , . ) ∥ H m ( M ) 2 ≤ C 4 2 ∣ β − β ′ ∣ 2 t − 2 μ ( ‖ f ‖ H m ( M ) 2 + ‖ f ‖ H m + μ − μ p ( M ) 2 ) ,

where C 4 = max ( C 2 , C 3 ) and depends on μ , β ′ , λ 1 , p , and k . By Sobolev embedding H m + μ − μ p ( M ) ↪ H m ( M ) , it follows from (2.32) that

(2.33) ∥ A 2 ( t , . ) ∥ H m ( M ) ≤ C 5 ∣ β − β ′ ∣ t − μ ‖ f ‖ H m + μ − μ p ( M ) ,

where C 5 depends on μ , β ′ , λ 1 , p , k , and m . By combining (2.17), (2.24), and (2.33), we derive that

(2.34)□ ∥ Q β , p ( t ) f − Q β ′ , p ( t ) f ∥ H m ( M ) ≤ ∥ A 1 ( t , . ) ∥ H m ( M ) + ∥ A 2 ( t , . ) ∥ H m ( M ) ≤ C 6 ( t γ ∣ β − β ′ ∣ γ ∥ f ∥ H m − p + p γ − γ ( M ) + ∣ β − β ′ ∣ t − μ ‖ f ‖ H m + μ − μ p ( M ) ) .

Lemma 2.5

Let any μ > 0 . Then we obtain the following estimate:

(2.35) ∥ J β , p ( t ) f − J β ′ , p ( t ) f ∥ H m ( M ) ≲ t − μ ∣ β − β ′ ∣ ‖ f ‖ H m + μ − μ p ( M ) ,

where the hidden constant depends on k , μ , p , λ 1 , m , β , and β ′ .

Remark 2.3

Since the proof of Lemma 2.5, it is easy to obtain that the following estimate:

(2.36) ∥ J β , p ( t ) f − J β ′ , p ( t ) f ∥ H m ( M ) ≲ ∣ β − β ′ ∣ ‖ f ‖ H m ( M ) ,

for any f ∈ H m ( M ) .

Proof

In view of the two following equalities J β , p ( t ) and J β ′ , p ( t ) , we have

(2.37) J β , p ( t ) f ( x ) ≔ ∑ n = 1 ∞ β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p f n ψ n ( x )

and

(2.38) J β ′ , p ( t ) f ( x ) ≔ ∑ n = 1 ∞ β ′ ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ′ ) λ n p ) 2 exp − β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p f n ψ n ( x ) .

Two aforementioned equalities provide us that

(2.39) J β , p ( t ) f ( x ) − J β ′ , p ( t ) f ( x ) = ∑ n = 1 ∞ β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p − exp − β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p f n ψ n ( x ) + ∑ n = 1 ∞ β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 − β ′ ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ′ ) λ n p ) 2 exp − β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p f n ψ n ( x ) = B 1 ( t , x ) + B 2 ( t , x ) .

In view of (2.22), we have the following bound for the term B 1

(2.40) ∥ B 1 ( t , . ) ∥ H m ( M ) 2 = ∑ n = 1 ∞ λ n 2 m β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 2 × exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p − exp − β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p 2 ∣ f n ∣ 2 ≤ C ( k , γ ) ( 1 − β ) γ 2 t 2 γ ∣ β − β ′ ∣ 2 γ ∑ n = 1 ∞ λ n 2 m + 2 p γ − 2 γ β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 2 .

By using Cauchy inequality, we find that

(2.41) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 ≥ 2 ( 1 − β ) ( 1 + k λ n ) λ n p .

Thus, it also implies that

(2.42) β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 2 ≤ β 2 4 ( 1 − β ) 2 λ n − 2 p .

By combining (2.40) and (2.42), we derive that

(2.43) ∥ B 1 ( t , . ) ∥ H m ( M ) 2 ≤ C 7 2 t 2 γ ∣ β − β ′ ∣ 2 γ ∑ n = 1 ∞ λ n 2 m − 2 p + 2 p γ − 2 γ ∣ f n ∣ 2 ,

where C 7 depends on k , γ , and β . Hence, we derive that the following estimate:

(2.44) ∥ B 1 ( t , . ) ∥ H m ( M ) ≤ C 7 ∣ β − β ′ ∣ γ t γ ∥ f ∥ H m − p + p γ − γ ( M ) .

Let us move to the consideration for the component B 2 . By using Parseval’s equality, we also find that

(2.45) ∥ B 2 ( t , . ) ∥ H m ( M ) 2 = ∑ n = 1 ∞ λ n 2 m β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 − β ′ ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ′ ) λ n p ) 2 2 exp − 2 β ′ λ n p t 1 + k λ n + ( 1 − β ′ ) λ n p ∣ f n ∣ 2 .

It is obvious to see that

(2.46) β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 − β ′ ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ′ ) λ n p ) 2 ≤ β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 − β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ′ ) λ n p ) 2 + ∣ β − β ′ ∣ ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ′ ) λ n p ) 2 = D 1 + D 2 .

Let us assume that β ≥ β ′ . The term D 1 is equal to

(2.47) D 1 = β ∣ β − β ′ ∣ λ n p ( 1 + k λ n ) ( 2 + 2 k λ n + ( 2 − β − β ′ ) λ n p ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 ( 1 + k λ n + ( 1 − β ′ ) λ n p ) 2 ⋅

It is easy to verify that

(2.48) 2 + 2 k λ n + ( 2 − β − β ′ ) λ n p 1 + k λ n + ( 1 − β ′ ) λ n p ≤ 2 − β − β ′ 1 − β ′ .

By using Cauchy inequality, we find that

(2.49) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 ≥ 2 ( 1 − β ) ( 1 + k λ n ) λ n p .

From three aforementioned observations, we obtain

(2.50) D 1 ≤ β ∣ β − β ′ ∣ ( 2 − β − β ′ ) 2 ( 1 − β ) ( 1 − β ′ ) ( 1 + k λ n + ( 1 − β ′ ) λ n p ) .

It is easy to obtain the following bound for D 2

(2.51) D 2 ≤ ∣ β − β ′ ∣ ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ′ ) λ n p ) 2 ≤ ∣ β − β ′ ∣ 1 + k λ n + ( 1 − β ′ ) λ n p .

By combining (2.46), (2.50), and (2.51), we derive that

(2.52) β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 − β ′ ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ′ ) λ n p ) 2 ≤ ∣ β − β ′ ∣ 1 + k λ n + ( 1 − β ′ ) λ n p 1 + β ( 2 − β − β ′ ) 2 ( 1 − β ) ( 1 − β ′ ) ≤ C 8 ∣ β − β ′ ∣ .

Here, C 8 depends on β and β ′ . In view of (2.30) and (2.45) and by a similar explanation as in (2.33), we can claim that

(2.53) ∥ B 2 ( t , . ) ∥ H m ( M ) ≤ C 9 ∣ β − β ′ ∣ t − μ ‖ f ‖ H m + μ − μ p ( M ) ,

where C 9 depends on μ , β ′ , λ 1 , p , k and m . By combining (2.44) and (2.53), we deduce that

∥ J β , p ( t ) f − J β ′ , p ( t ) f ∥ H m ( M ) ≤ ∥ B 1 ( t , . ) ∥ H m ( M ) + ∥ B 2 ( t , . ) ∥ H m ( M ) ≤ C 7 t γ ∣ β − β ′ ∣ γ ∥ f ∥ H m − p + p γ − γ ( M ) + C 9 ∣ β − β ′ ∣ t − μ ‖ f ‖ H m + μ − μ p ( M ) .

By a similar claim as shown in Remark 2.1, we complete the proof of Lemma 2.5.□

3 Linear inhomogeneous case

In this section, we will investigate the continuity of solutions of linear problem with respect to β – the fractional order.

Theorem 3.1

Let the initial datum z 0 ∈ H m ( M ) and G ∈ L ∞ ( 0 , T ; H m + μ − μ p ( M ) ) for any 0 < μ < 1 . Then we have

(3.54) ∥ Z β ( t , . ) − Z β ′ ( t , . ) ∥ H m ( M ) ≲ ∣ β − β ′ ∣ ‖ z 0 ‖ H m ( M ) + T 1 − μ 1 − μ ∣ β − β ′ ∣ ∥ G ∥ L ∞ ( 0 , T ; H m + μ − μ p ( M ) ) .

Proof

From (2.6), we obtain the formula of two following mild solutions as follows:

(3.55) Z β ( t , x ) = Q β , p ( t ) z 0 + ∫ 0 t J β , p ( t − ν ) G ( ν ) d ν

and

(3.56) Z β ′ ( t , x ) = Q β ′ , p ( t ) z 0 + ∫ 0 t J β ′ , p ( t − ν ) G ( ν ) d ν .

By subtracting (3.55) from (3.56) on each side, we obtain

(3.57) Z β ( t , x ) − Z β ′ ( t , x ) = ( Q β , p ( t ) − Q β ′ , p ( t ) ) z 0 + ∫ 0 t ( J β , p ( t − ν ) − J β ′ , p ( t − ν ) ) G ( ν ) d ν .

This implies the following bound:

(3.58) ∥ Z β ( t , . ) − Z β ′ ( t , . ) ∥ H m ( M ) ≤ ∥ ( Q β , p ( t ) − Q β ′ , p ( t ) ) z 0 ∥ H m ( M ) + ∫ 0 t ( J β , p ( t − ν ) − J β ′ , p ( t − ν ) ) G ( ν ) d ν H m ( M ) .

Case 1. z 0 ∈ H m ( M ) . In this case, we use Remark 2.2 to obtain the following equation:

(3.59) ∥ ( Q β , p ( t ) − Q β ′ , p ( t ) ) z 0 ∥ H m ( M ) ≲ ∣ β − β ′ ∣ ‖ z 0 ‖ H m ( M ) .

In view of Lemma 2.5, we bound the second term on the right-hand side of (3.58) as follows:

(3.60) ∫ 0 t ( J β , p ( t − ν ) − J β ′ , p ( t − ν ) ) G ( ν ) d ν H m ( M ) ≲ ∣ β − β ′ ∣ ∫ 0 t ( t − ν ) − μ ∥ G ( ν ) ∥ H m + μ − μ p ( M ) d ν ≲ ∣ β − β ′ ∣ ∥ G ∥ L ∞ ( 0 , T ; H m + μ − μ p ( M ) ) ∫ 0 t ( t − ν ) − μ d ν .

Since μ < 1 , we know that the proper integral ∫ 0 t ( t − ν ) − μ d ν is convergent. Therefore, we obtain

(3.61) ∫ 0 t ( J β , p ( t − ν ) − J β ′ , p ( t − ν ) ) G ( ν ) d ν H m ( M ) ≤ t 1 − μ 1 − μ ∣ β − β ′ ∣ ∥ G ∥ L ∞ ( 0 , T ; H m + μ − μ p ( M ) ) .

By combining (3.58), (3.59), and (3.61), we complete the proof of Theorem 3.1.□

4 Globally nonlinearity Lipschitz case

In this section, we established some results about the global existence and continuity of solutions for Problem (1.1). The main results are given in Theorems 4.1 and 4.3.

4.1 Global existence

Let G : H q ( M ) → H s ( M ) such that G ( 0 ) = 0 and

(4.62) ‖ G ( θ 1 ) − G ( θ 2 ) ‖ H s ( M ) ≤ K ‖ θ 1 − θ 2 ‖ H q ( M ) ,

for any θ 1 , θ 2 ∈ H q ( M ) and K is a positive constant. Here, s ≤ q ≤ p + s .

Theorem 4.1

Let G be defined as mentioned earlier. Let the initial datum z 0 ∈ H q − p ( M ) . Then problem (1.1) has a unique solution Z β ∈ L r ( 0 , T ; H q ( M ) ) , where 1 < r < 1 a , 0 < a < 1 . Moreover, we obtain

(4.63) ∥ Z β ( t , . ) ∥ H q ( M ) ≤ 2 T a e d 0 t 1 − β t − a ‖ z 0 ‖ H q − p ( M )

for any d 0 > 0 large enough.

Proof

The key idea of this proof is referenced by us in the article of the second author with his colleagues [40]. To use the Banach fixed-point theorem, we need to define the following operator P : Y a , d ( ( 0 , T ] ; H q ( M ) ) → Y a , d ( ( 0 , T ] ; H q ( M ) ) with d > 0 , by

(4.64) P θ ( t ) ≔ Q β , p ( t ) z 0 + ∫ 0 t J β , p ( t − ν ) G ( θ ( ν ) ) d ν .

Since the property G ( 0 ) = 0 and combine with Lemma 2.3, we infer that

(4.65) ∥ P ( θ ( t ) = 0 ) ∥ H q ( M ) = ∥ Q β , p ( t ) z 0 ∥ H q ( M ) ≤ 1 1 − β ‖ z 0 ‖ H q − p ( M ) .

By multiplying both sides by t a e − d t , we have

(4.66) t a e − d t ∥ P ( θ ( t ) = 0 ) ∥ H q ( M ) ≤ t a e − d t 1 − β ‖ z 0 ‖ H q − p ( M ) ≤ T a 1 − β ‖ z 0 ‖ H q − p ( M ) .

The latter estimate provides us that

P ( θ ( t ) = 0 ) ∈ Y a , d ( ( 0 , T ] ; H q ( M ) ) ,

for any d > 0 . Given any two elements θ 1 and θ 2 in the space Y a , d ( ( 0 , T ] ; H q ( M ) ) . From equation (4.64), we have the following equality:

(4.67) P θ 1 ( t ) − P θ 2 ( t ) = ∫ 0 t J β , p ( t − ν ) ( G ( θ 1 ( ν ) ) − G ( θ 2 ( ν ) ) ) d ν .

By applying Lemma 2.3, we derive that

(4.68) ∥ P θ 1 ( t ) − P θ 2 ( t ) ∥ H q ( M ) ≤ 2 β 1 − β ∫ 0 t ∥ G ( θ 1 ( ν ) ) − G ( θ 2 ( ν ) ) ∥ H q − p ( M ) d ν .

It is obvious to see that Sobolev embedding

(4.69) H s ( M ) ↪ H q − p ( M ) ,

since we note that q ≤ p + s . This fact combined with the globally Lipschitz property of G yields to

(4.70) ∥ P θ 1 ( t ) − P θ 2 ( t ) ∥ H q ( M ) ≤ 2 β 1 − β ∫ 0 t ∥ G ( θ 1 ( ν ) ) − G ( θ 2 ( ν ) ) ∥ H s ( M ) d ν ≤ 2 β 1 − β ∫ 0 t ∥ θ 1 ( ν ) − θ 2 ( ν ) ∥ H q ( M ) d ν .

Hence, we derive that

(4.71) t a e − d t ∥ P θ 1 ( t ) − P θ 2 ( t ) ∥ H q ( M ) ≤ 2 β 1 − β t a ∫ 0 t ν − a e − d ( t − ν ) ν a e − d ν ∥ θ 1 ( ν ) − θ 2 ( ν ) ∥ H q ( M ) d ν .

It is obvious to see that

∥ θ 1 − θ 2 ∥ Y a , d ( ( 0 , T ] ; H q ( M ) ) = sup 0 ≤ ν ≤ T ν a e − d ν ∥ θ 1 ( ν ) − θ 2 ( ν ) ∥ H q ( M ) .

This implies immediately that

(4.72) t a e − d t ∥ P θ 1 ( t ) − P θ 2 ( t ) ∥ H q ( M ) ≤ 2 β 1 − β t a ∫ 0 t ν − a e − d ( t − ν ) d ν ∥ θ 1 − θ 2 ∥ Y a , d ( ( 0 , T ] ; H q ( M ) ) .

By change variable ν = t ξ , we derive that d ν = t d ξ . Then we have

(4.73) t a ∫ 0 t ν − a e − d ( t − ν ) d ν = t ∫ 0 1 ξ − a e − d t ( 1 − ξ ) d ξ .

Let us provide the following lemma, which is presented in [35], Lemma 8, p. 9.

Lemma 4.2

Let a 1 > − 1 , a 2 > − 1 such that a 1 + a 2 ≥ − 1 , ρ > 0 and t ∈ [ 0 , T ] . For h > 0 , the following limit holds

lim ρ → ∞ sup t ∈ [ 0 , T ] t h ∫ 0 1 ν a 1 ( 1 − ν ) a 2 e − ρ t ( 1 − ν ) d ν = 0 .

Let us look into the right-hand side of (4.73). Since a < 1 , we note that

− a > − 1 .

According to Lemma 4.2, we obtain the following statement:

lim d → + ∞ sup 0 ≤ t ≤ T t ∫ 0 1 ξ − a e − d t ( 1 − ξ ) d ξ = 0 .

Hence, there exists a positive d 0 > 0 such that

(4.74) sup 0 ≤ t ≤ T t a ∫ 0 t ν − a e − d 0 ( t − ν ) d ν ≤ 1 2 ⋅

Combining (4.72) and (4.74), we deduce that

(4.75) t a e − d 0 t ∥ P θ 1 ( t ) − P θ 2 ( t ) ∥ H q ( M ) ≤ 1 2 ∥ θ 1 − θ 2 ∥ Y a , d 0 ( ( 0 , T ] ; H q ( M ) ) .

Thus, we provide the following confirmation:

(4.76) ∥ P θ 1 − P θ 2 ∥ Y a , d 0 ( ( 0 , T ] ; H q ( M ) ) ≤ 1 2 ∥ θ 1 − θ 2 ∥ Y a , d 0 ( ( 0 , T ] ; H q ( M ) ) .

From here on, we can conclude that P is the mapping from Y a , d 0 ( ( 0 , T ] ; H q ( M ) ) to itself. By applying the Banach fixed-point theorem, we deduce that P has a fixed point Z β ∈ Y a , d 0 ( ( 0 , T ] ; H q ( M ) ) . It means that Z β is the mild solution to the Problem (1.1). Let us claim the regularity of the mild solution Z β . Indeed, from (4.64) and (4.76), we obtain the following bound:

(4.77) ∥ Z β ∥ Y a , d 0 ( ( 0 , T ] ; H q ( M ) ) = ∥ P Z β ( t , x ) ∥ Y a , d 0 ( ( 0 , T ] ; H q ( M ) ) ≤ sup 0 ≤ t ≤ T t a e − d 0 t ∥ P ( θ ( t ) = 0 ) ∥ H q ( M ) + ∥ P Z β ( t , x ) − P ( θ = 0 ) ∥ Y a , d 0 ( ( 0 , T ] ; H q ( M ) ) ≤ T a 1 − β ‖ z 0 ‖ H q − p ( M ) + 1 2 ∥ Z β ∥ Y a , d 0 ( ( 0 , T ] ; H q ( M ) ) .

This aforementioned inequality implies that

(4.78) ∥ Z β ∥ Y a , d 0 ( ( 0 , T ] ; H q ( M ) ) ≤ 2 T a 1 − β ‖ z 0 ‖ H q − p ( M ) .

Thus, we can provide that

(4.79) ∥ Z β ( t , . ) ∥ H q ( M ) ≤ 2 T a e d 0 t 1 − β t − a ‖ z 0 ‖ H q − p ( M ) ,

which shows that the regularity property

Z β ∈ L r ( 0 , T ; H q ( M ) ) ,

where 1 < r < 1 a , 0 < a < 1 . This completes the proof of our theorem.□

4.2 Continuous dependence

Theorem 4.3

Let z 0 ∈ H q ( M ) and G : H q ( M ) → H q ( M ) such that G ( 0 ) = 0 and

(4.80) ‖ G ( θ 1 ) − G ( θ 2 ) ‖ H q ( M ) ≤ K ‖ θ 1 − θ 2 ‖ H q ( M ) ,

for any θ 1 , θ 2 ∈ H q ( M ) and K is a postive constant. Then we obtain

(4.81) ∥ Z β ( t , . ) − Z β ′ ( t , . ) ∥ H q ( M ) ≲ exp ( K t ) ∣ β − β ′ ∣ ‖ z 0 ‖ H q ( M ) + 2 K e d 0 T ( 1 − β ) ( 1 − a ) ‖ z 0 ‖ H q − p ( M ) .

Proof

Since Z β and Z β ′ are two mild solutions to Problem (1.1) corresponding to β and β ′ . Since (4.64), we have

(4.82) Z β ( t , x ) ≔ Q β , p ( t ) z 0 + ∫ 0 t J β , p ( t − ν ) G ( Z β ( ν ) ) d ν

and

(4.83) Z β ′ ( t , x ) ≔ Q β ′ , p ( t ) z 0 + ∫ 0 t J β ′ , p ( t − ν ) G ( Z β ′ ( ν ) ) d ν .

By combining (4.82) and (4.83), we have that

(4.84) Z β ( t , x ) − Z β ′ ( t , x ) = ( Q β , p ( t ) − Q β ′ , p ( t ) ) z 0 + ∫ 0 t ( J β , p ( t − ν ) − J β ′ , p ( t − ν ) ) G ( Z β ( ν ) ) d ν + ∫ 0 t J β ′ , p ( t − ν ) ( G ( Z β ( ν ) ) − G ( Z β ′ ( ν ) ) ) d ν = J 1 ( t , x ) + J 2 ( t , x ) + J 3 ( t , x ) .

By using Remark 2.2, we have immediately that for any 0 ≤ t ≤ T ,

(4.85) ∥ J 1 ( t , . ) ∥ H q ( M ) = ∥ ( Q β , p ( t ) − Q β ′ , p ( t ) ) z 0 ∥ H q ( M ) ≲ ∣ β − β ′ ∣ ‖ z 0 ‖ H q ( M ) .

Let us now treat the second term J 2 . In view of Remark 2.3, one has

(4.86) ∥ J 2 ( t , . ) ∥ H q ( M ) ≲ ∣ β − β ′ ∣ ∫ 0 t ∥ G ( Z β ( ν ) ) ∥ H q ( M ) d ν .

By using the globally Lipschitz property of G as shown in (4.80) and combining with (4.63), we arrive at

(4.87) ∥ G ( Z β ( ν ) ) ∥ H q ( M ) ≤ K ∥ Z β ( ν ) ∥ H q ( M ) ≤ 2 K T a e d 0 T 1 − β ν − a ‖ z 0 ‖ H q − p ( M )

for any 0 ≤ ν ≤ T . By combining (4.86) and (4.87), we infer that

(4.88) ∥ J 2 ( t , . ) ∥ H q ( M ) ≲ 2 K T a e d 0 T 1 − β ∣ β − β ′ ∣ ‖ z 0 ‖ H q − p ( M ) ∫ 0 t ν − a d ν = 2 K T a e d 0 T ( 1 − β ) ( 1 − a ) ∣ β − β ′ ∣ t 1 − a ‖ z 0 ‖ H q − p ( M ) ≤ 2 K e d 0 T ( 1 − β ) ( 1 − a ) ∣ β − β ′ ∣ ‖ z 0 ‖ H q − p ( M ) .

Finally, we continue to consider the third term J 3 . By using Remark 2.3 and (4.80), we bound it as follows:

(4.89) ∥ J 3 ( t , . ) ∥ H q ( M ) ≤ ∫ 0 t J β ′ , p ( t − ν ) ( G ( Z β ( ν ) ) − G ( Z β ′ ( ν ) ) ) d ν H q ( M ) ≤ ∫ 0 t ∥ G ( Z β ( ν ) ) − G ( Z β ′ ( ν ) ) ∥ H q ( M ) d ν ≤ K ∫ 0 t ∥ Z β ( ν ) − Z β ′ ( ν ) ∥ H q ( M ) d ν .

By combining (4.84), (4.85), (4.86), and (4.89), we derive that

(4.90) ∥ Z β ( t , . ) − Z β ′ ( t , . ) ∥ H q ( M ) ≤ ∥ J 1 ( t , . ) ∥ H q ( M ) + ∥ J 2 ( t , . ) ∥ H q ( M ) + ∥ J 3 ( t , . ) ∥ H q ( M ) ≲ ∣ β − β ′ ∣ ‖ z 0 ‖ H q ( M ) + 2 K e d 0 T ( 1 − β ) ( 1 − a ) ∣ β − β ′ ∣ ‖ z 0 ‖ H q − p ( M ) + K ∫ 0 t ∥ Z β ( ν ) − Z β ′ ( ν ) ∥ H q ( M ) d ν .

By applying Gronwall’s inequality, we obtain that

(4.91)□ ∥ Z β ( t , . ) − Z β ′ ( t , . ) ∥ H q ( M ) ≲ exp ( K t ) ∣ β − β ′ ∣ ‖ z 0 ‖ H q ( M ) + 2 K e d 0 T ( 1 − β ) ( 1 − a ) ‖ z 0 ‖ H q − p ( M ) .

5 The convergence to parabolic diffusion equation

This section consists of two parts. The first part is to prove that the solution of equation (1.1) converges to the solution of the corresponding parabolic equation. The second part examines the continuity of the solution with respect to the order of derivatives.

Theorem 5.1

Let z 0 ∈ H q ( M ) and G : H q ( M ) → H q ( M ) such that G ( 0 ) = 0 and

(5.92) ‖ G ( θ 1 ) − G ( θ 2 ) ‖ H q ( M ) ≤ K ‖ θ 1 − θ 2 ‖ H q ( M ) ,

for any θ 1 , θ 2 ∈ H q ( M ) , and K is a postive constant. Let us assume that 1 2 ≤ p < 1 . Let Z k , β be the mild solution to Problem (1.1) with k > 0 . Let Z ∗ be the mild solution to following parabolic diffusion:

(5.93) D t β CF Z + L p Z = G ( Z ) , in M × ( 0 , T ] , Z ( x , 0 ) = z 0 ( x ) , in M , Z ( x , t ) = 0 , in ∂ M .

Then we obtain the following estimate:

(5.94) ∥ Z k , β − Z β ∗ ∥ L ∞ ( 0 , T ; H m ( M ) ) ≲ k 1 − θ 2 ∥ z 0 ∥ H m + 1 − θ 2 ( M ) + k 1 − θ 2 + k ∥ Z β ∗ ∥ L ∞ ( 0 , T ; H m ( M ) ) .

Proof

From (2.6), we can rewrite the formula of two mild solutions Z k , β and Z ∗ as follows:

(5.95) Z k , β ( t , x ) = R k , β , p ( t ) z 0 ( x ) + ∫ 0 t F k , β , p ( t − ν ) G ( Z k , β ( ν , x ) ) d ν

and

(5.96) Z β ∗ ( t , x ) = R ¯ k , β , p ( t ) z 0 ( x ) + ∫ 0 t F ¯ β , p ( t − ν ) G ( Z β ∗ ( ν , x ) ) d ν ,

where the aforementioned operators are defined as follows:

(5.97) F k , β , p ( t ) f ( x ) ≔ ∑ n = 1 ∞ β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p f n ψ n ( x ) ,

(5.98) F ¯ β , p ( t ) f ( x ) ≔ ∑ n = 1 ∞ β ( 1 + ( 1 − β ) λ n p ) 2 exp − β λ n p t 1 + ( 1 − β ) λ n p f n ψ n ( x ) ,

(5.99) R k , β , p ( t ) f ( x ) ≔ ∑ n = 1 ∞ 1 + k λ n 1 + k λ n + ( 1 − β ) λ n p exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p f n ψ n ( x ) ,

(5.100) R ¯ k , β , p ( t ) f ( x ) ≔ ∑ n = 1 ∞ 1 1 + ( 1 − β ) λ n p exp − β λ n p t 1 + ( 1 − β ) λ n p f n ψ n ( x ) .

To make it clearer, the proof is divided into three steps as follows.

Step 1. Estimate of ∥ F k , β , p ( t ) f ( x ) − F ¯ β , p ( t ) f ( x ) ∥ H m ( M ) .

From the two aforementioned equalities, we find that

(5.101) F k , β , p ( t ) f ( x ) − F ¯ β , p ( t ) f ( x ) = ∑ n = 1 ∞ β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 − β ( 1 + ( 1 − β ) λ n p ) 2 exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p f n ψ n ( x ) + ∑ n = 1 ∞ β ( 1 + ( 1 − β ) λ n p ) 2 exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p − exp − β λ n p t 1 + ( 1 − β ) λ n p f n ψ n ( x ) = S 1 ( t , x ) + S 2 ( t , x ) .

To consider the term S 1 , we have

(5.102) β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 − β ( 1 + ( 1 − β ) λ n p ) 2 = β ( 1 + k λ n ) ( 1 + ( 1 − β ) λ n p ) 2 − β ( 1 + k λ n + ( 1 − β ) λ n p ) 2 ( 1 + k λ n + ( 1 − β ) λ n p ) 2 ( 1 + ( 1 − β ) λ n p ) 2 ≤ k λ n ( ( 1 − β ) 2 λ n 2 p + 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 ( 1 + ( 1 − β ) λ n p ) 2 .

Since p ≥ 1 2 , we know that

(5.103) k λ n ( ( 1 − β ) 2 λ n 2 p + 1 + k λ n ) = k λ n 1 + 2 p ( ( 1 − β ) 2 + λ n − 2 p + k λ n 1 − 2 p ) ≤ k λ n 1 + 2 p ( ( 1 − β ) 2 + λ 1 − 2 p + k λ 1 1 − 2 p ) .

By using the inequality 1 + y > y θ 2 for any y > 0 and 1 < θ < 2 , we obtain that

(5.104) ( 1 − β ) 2 λ n 2 p + 1 + k λ n > 1 + k λ n > k θ 2 λ n θ 2 .

Thus, we derive that

(5.105) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 ( 1 + ( 1 − β ) λ n p ) 2 ≥ k θ 2 λ n θ 2 ( 1 + k λ n + ( 1 − β ) λ n p ) ( 1 + ( 1 − β ) λ n p ) 2 ≥ k θ 2 λ n θ 2 ( 1 − β ) λ n p ( 1 − β ) 2 λ n 2 p = k θ 2 ( 1 − β ) 3 λ n 3 p + θ 2 .

By combining (5.103) and (5.105), we derive that

(5.106) The right-hand side of (5.102) ≤ k 1 − θ 2 ( 1 − β ) 2 + λ 1 − 2 p + k λ 1 1 − 2 p ( 1 − β ) 3 λ n 1 − p − θ 2 ≤ k 1 − θ 2 ( 1 − β ) 2 + λ 1 − 2 p + k λ 1 1 − 2 p ( 1 − β ) 3 λ 1 1 − p − θ 2 ,

where in the last line, we note that 1 − p ≤ 1 2 < θ 2 . By using Parseval’s equality, we infer that

(5.107) ∥ S 1 ( t , . ) ∥ H m ( M ) 2 = ∑ n = 1 ∞ λ n 2 m β ( 1 + k λ n ) ( 1 + k λ n + ( 1 − β ) λ n p ) 2 − β ( 1 + ( 1 − β ) λ n p ) 2 2 exp − 2 β λ n p t 1 + k λ n + ( 1 − β ) λ n p ∣ f n ∣ 2 ≲ k 2 − θ ∑ n = 1 ∞ λ n 2 m ∣ f n ∣ 2 = k 2 − θ ∥ f ∥ H m ( M ) 2 .

To treat the second term S 2 , we use the inequality ∣ e − a − e − b ∣ ≤ ∣ a − b ∣ to obtain that

(5.108) exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p − exp − β λ n p t 1 + ( 1 − β ) λ n p ≤ β λ n p t 1 + k λ n + ( 1 − β ) λ n p − β λ n p t 1 + ( 1 − β ) λ n p ≤ β k t λ n 1 + p ( 1 + k λ n + ( 1 − β ) λ n p ) ( 1 + ( 1 − β ) λ n p ) ≤ β k t λ n 1 − p .

This estimate together with p > 1 2 yield that

(5.109) ∥ S 2 ( t , . ) ∥ H m ( M ) 2 ≤ ∑ n = 1 ∞ β λ n m ( 1 + ( 1 − β ) λ n p ) 2 2 exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p − exp − β λ n p t 1 + ( 1 − β ) λ n p 2 ∣ f n ∣ 2 ≤ β 4 k 2 T 2 ( 1 − β ) 4 ∑ n = 1 ∞ λ n 2 m λ n 2 − 6 p ∣ f n ∣ 2 ≤ β 4 k 2 T 2 λ 1 2 − 6 p ( 1 − β ) 4 ∥ f ∥ H m ( M ) 2 .

Combining (5.107) and (5.109), we deduce that

(5.110) ∥ F k , β , p ( t ) f ( x ) − F ¯ β , p ( t ) f ( x ) ∥ H m ( M ) ≤ ∥ S 1 ( t , . ) ∥ H m ( M ) + ∥ S 2 ( t , . ) ∥ H m ( M ) ≲ k 1 − θ 2 + k ∥ f ∥ H m ( M ) .

Step 2. Estimate of ∥ R k , β , p ( t ) f ( x ) − R ¯ β , p ( t ) f ( x ) ∥ H m ( M ) .

From the aforementioned equalities, we find that

(5.111) R k , β , p ( t ) f ( x ) − R ¯ k , β , p ( t ) f ( x ) = ∑ n = 1 ∞ 1 + k λ n 1 + k λ n + ( 1 − β ) λ n p − 1 1 + ( 1 − β ) λ n p exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p f n ψ n ( x ) + ∑ n = 1 ∞ 1 1 + ( 1 − β ) λ n p exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p − exp − β λ n p t 1 + ( 1 − β ) λ n p f n ψ n ( x ) = S 3 ( t , x ) + S 4 ( t , x ) .

It is obvious to see that

(5.112) S 3 ( t , x ) = ∑ n = 1 ∞ k ( 1 − β ) λ n 1 + p ( 1 + k λ n + ( 1 − β ) λ n p ) ( 1 + ( 1 − β ) λ n p ) exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p f n ψ n ( x ) .

By a similar approach as in Step 1, we can obtain that

(5.113) k ( 1 − β ) λ n 1 + p ( 1 + k λ n + ( 1 − β ) λ n p ) ( 1 + ( 1 − β ) λ n p ) ≲ k 1 − θ 2 λ n 1 − θ 2 .

Thus, we derive the following estimate:

(5.114) ∥ S 3 ( t , . ) ∥ H m ( M ) 2 = ∑ n = 1 ∞ λ n 2 m k ( 1 − β ) λ n 1 + p ( 1 + k λ n + ( 1 − β ) λ n p ) ( 1 + ( 1 − β ) λ n p ) 2 exp − 2 β λ n p t 1 + k λ n + ( 1 − β ) λ n p ∣ f n ∣ 2 ≲ k 2 − θ ∑ n = 1 ∞ λ n 2 m + 2 − θ ∣ f n ∣ 2 = k 2 − θ ∥ f ∥ H m + 1 − θ 2 ( M ) 2 .

By using (5.108), we obtain

(5.115) ∥ S 4 ( t , . ) ∥ H m ( M ) 2 = ∑ n = 1 ∞ λ n 2 m 1 1 + ( 1 − β ) λ n p 2 exp − β λ n p t 1 + k λ n + ( 1 − β ) λ n p − exp − β λ n p t 1 + ( 1 − β ) λ n p 2 ∣ f n ∣ 2 ≲ k 2 ∑ n = 1 ∞ λ n 2 m + 2 − 4 p ∣ f n ∣ 2 = k 2 ∥ f ∥ H m + 1 − 2 p ( M ) 2 .

It is easy to observe that m + 1 − 2 p < m + 1 − θ 2 . Thus, from (5.111), (5.114), and (5.115), we derive that

(5.116) ∥ R k , β , p ( t ) f ( x ) − R ¯ β , p ( t ) f ( x ) ∥ H m ( M ) ≤ ∥ S 3 ( t , . ) ∥ H m ( M ) + ∥ S 4 ( t , . ) ∥ H m ( M ) ≲ k 1 − θ 2 ∥ f ∥ H m + 1 − θ 2 ( M ) .

Step 3. Estimate of ∥ Z k , β ( t , . ) − Z β ∗ ( t , . ) ∥ H m ( M ) .

From the two formulas (5.95) and (5.96), we can easily obtain the following calculation:

(5.117) Z k , β ( t , x ) − Z β ∗ ( t , x ) = ( R k , β , p ( t ) − R ¯ β , p ( t ) ) z 0 + ∫ 0 t F k , β , p ( t − ν ) ( G ( Z k , β ( ν , x ) ) − G ( Z β ∗ ( ν , x ) ) ) d ν + ∫ 0 t ( F k , β , p ( t − ν ) − F ¯ β , p ( t − ν ) ) G ( Z β ∗ ( ν , x ) ) d ν = J 11 ( t , x ) + J 12 ( t , x ) + J 13 ( t , x ) .

In view of (5.116) and the assumption that z 0 ∈ H m + 1 − θ 2 ( M ) , we have the following bound for J 11

(5.118) ∥ J 11 ( t , . ) ∥ H m ( M ) = ∥ ( R k , β , p ( t ) − R ¯ β , p ( t ) ) z 0 ∥ H m ( M ) ≲ k 1 − θ 2 ∥ z 0 ∥ H m + 1 − θ 2 ( M ) .

For the term J 12 , by using Lemma 2.3 and the globally Lipschitz property of G , we find that

(5.119) ∥ J 12 ( t , . ) ∥ H m ( M ) = ∫ 0 t F k , β , p ( t − ν ) ( G ( Z k , β ( ν , x ) ) − G ( Z β ∗ ( ν , x ) ) ) d ν H m ( M ) ≤ 2 β 1 − β ∫ 0 t ∥ G ( Z k , β ( ν , x ) ) − G ( Z β ∗ ( ν , x ) ) ∥ H m ( M ) d ν ≤ 2 K β 1 − β ∫ 0 t ∥ Z k , β ( ν , x ) − Z β ∗ ( ν , x ) ∥ H m ( M ) d ν .

For the third term J 13 , by using 5.110, we obtain the following estimate:

(5.120) ∥ J 13 ( t , . ) ∥ H m ( M ) ≲ k 1 − θ 2 + k ∫ 0 t ∥ G ( Z β ∗ ( ν , x ) ) ∥ H m ( M ) d ν ≲ K k 1 − θ 2 + k ∫ 0 t ∥ Z β ∗ ( ν , x ) ∥ H m ( M ) d ν ≲ K T k 1 − θ 2 + k ∥ Z β ∗ ∥ L ∞ ( 0 , T ; H m ( M ) ) .

By combining the aforementioned four observations, we deduce that

(5.121) ∥ Z k , β ( t , . ) − Z β ∗ ( t , . ) ∥ H m ( M ) ≲ k 1 − θ 2 ∥ z 0 ∥ H m + 1 − θ 2 ( M ) + K T k 1 − θ 2 + k ∥ Z β ∗ ∥ L ∞ ( 0 , T ; H m ( M ) ) + 2 K β 1 − β ∫ 0 t ∥ Z k , β ( ν , x ) − Z β ∗ ( ν , x ) ∥ H m ( M ) d ν .

By applying Gronwall’s inequality, we can conclude that

(5.122) ∥ Z k , β ( t , . ) − Z β ∗ ( t , . ) ∥ H m ( M ) ≲ k 1 − θ 2 ∥ z 0 ∥ H m + 1 − θ 2 ( M ) + k 1 − θ 2 + k ∥ Z β ∗ ∥ L ∞ ( 0 , T ; H m ( M ) ) ,

which allows us to complete (5.94).□

Acknowledgement

The authors gratefully acknowledge Nong Lam University Ho Chi Minh City for financial support for facilities provides through project ID: CS-CB22-KH-01.

  1. Funding information: This work was supported by Nong Lam University Ho Chi Minh City, Vietnam under Project ID: CS-CB22-KH-01.

  2. Author contributions: All authors have accepted responsibility for the entire content of this article and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2022-08-21
Revised: 2022-11-07
Accepted: 2022-11-15
Published Online: 2023-01-19

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2022-0180
Keywords for this article
Caputo-Fabrizio derivative operator; Burger equation; Banach fixed point theory; Sobolev embeddings
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  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
  26. Feature fusion-based text information mining method for natural scenes
  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
  34. Small perturbations of critical nonlocal equations with variable exponents
  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
  81. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives
  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
  88. Petri net analysis of a queueing inventory system with orbital search by the server
  89. Asymptotic stability of an epidemiological fractional reaction-diffusion model
  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
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