Introduction
In this paper, we consider the following problem for the space-time fractional evolution equation:
1
where , , , . The nonlocal fractional Laplacian operator is realized as a Fourier multiplier with symbol : under Fourier transform, is the Caputo fractional derivative operator, and denotes the Riemann–Liouville (R–L) fractional integral operator; they are defined respectively aswhere is the Euler gamma function. is the normal space in which all continuous functions decay to zero as they approach infinity.In the past decades, this kind of system has attracted a lot of attentions [1, 2], which is widely used in the fields such as fluid mechanics [3], control theory [4], engineering applications [5], and life sciences [6]. The most recent application encompasses biomedical therapy [7, 8], anomalous diffusion [9], and signal processing [10].
For this, Nagasawa [11], Kobayashi [12], Guedda and Kirane [13] considered the evolution equation involving fractional diffusion
2
Additionally, Cazenave, Dickstein and Weissler [14] proved the sharp blow-up, global existence results for the heat equation with nonlinear memory,3
Fino and Kirane [15] generalized and solved the associated problems of the following equations based on article [14]:4
Using an existence-uniqueness test, they confirmed the validity of the equation and proved the existence of a blow-up solution. Whenthe solution to problem (4) is blow-up in finite time. In addition, the conditions for local or global solutions have also been established.Different from previous work, in this paper, we focus on the evolution equations with two fractional forms of the Caputo time and the fractional diffusion. Our interest in this problem is motivated by the studies above to develop a general blow-up theory for (1)with R–L type nonlinear term. For this, we perform a fixed point argument to establish the local well-posedness. Our proof relies on the properties of the space-time fractional operators derived from Mittag–Leffler functions [19]. In addition, by contradiction argument, we obtain sufficient conditions for blow-up. Usually we need to define the mild solution as follows:
Definition 1.1
Let , , , , . Assume that satisfies the integral equation given below,
5
then is a mild solution to the problem (1). For the specific definition of , see the preliminaries.However, due to the lack of space-time estimates for , we cannot directly derive the blow-up of the solutions. In order to overcome the technical difficulty, we introduce an integral test function which allows us to deal with the nonlinearity properly, and then use the relation of the R–L type operators and the Mittag–Leffler operators to show the equivalence between mild solutions and weak solutions. It turns out that weak solutions work well for achieving our goal and obtaining the threshold of p. It is worth noticing that this threshold will tend to the one obtained in [15] as . Moreover, we shall present the upper bound of the lifespan of solutions for some special initial data, the proof of this point is standard.
For simplicity, we use to denote , where C may have different values in different lines. Our main results are summarized below.
Theorem 1.2
Let , . Then there exists a positive maximal time such that problem (1) has a unique mild solution . Furthermore, either or and as . In particular, if , , , then is positive.
Definition 1.3
Let , , and . If and for any function , satisfies the following equation
6
then is a weak solution to the problem (1), where , is Schwartz space, for all , , , q is the conjugate of p.Theorem 1.4
Let . Suppose is a mild solution of the problem (1), then is also a weak solution.
Theorem 1.5
Let , , and . If , or , then the solution of (1) blows up at a finite time.
Theorem 1.6
Let , , or . Given with , whenholds, the lifespan of the solution of the problem (1) has the following upper bound:
Corollary 1.7
Let , , or . Suppose where , ifis holding for some positive constant . Then, for any , the lifespan of the solution to the problem (1) satisfies the following bound:
The remainder of this paper is divided into two parts. In part 2, we collect the necessary definitions and Lemmas. The part 3 is devoted to the proof of our main conclusions.
Preliminaries
In this section, we outline and review the relevant properties about the evolution operators, which are essential to prove our main conclusions.
Given , where is a self-adjoint operator on , it follows that Z(t) is a strongly continuous semigroup on generated by (see [16]). , where stands for convolution, andUsing the self-similar form of and convolution by Young’s inequality, we can conclude
7
for all , . Besides, since is a self-adjoint operator,8
holds for all .Regarding the representation (5) of the mild solution, it contains the relevant content of the Mainardi’s and Mittag–Leffler functions. In the following, we will present their definitions.
A function of the Mittag–Leffler form with two parameters is defined asThe semigroup Z(t) whose Mittag–Leffer operators forms are defined as follows:andwhere is a Mainardi’s function defined by for all and satisfies the following properties:
9
10
From this we can derive the following results.Lemma 2.1
11
12
Proof
Using (7), (9), and (10), which yieldsand
In addition, we review the previous conclusions concerning time fractional operators.
Lemma 2.2
([17]) Assuming , . Letthen
Lemma 2.3
([17]) For , , then we have ,where .
Next, let us define R–L fractional derivatives and recall several results that will be used in proving the equivalence of the mild solution and the weak one.
Definition 2.4
Let , , AC stands for the space of absolutely continuous functions. Fractional derivatives of order on the left- and right-sides of the R–L are defined as follows:and
Based on this definition, it is not difficult to derive the following relation between Caputo and R–L derivatives
Proposition 2.5
([18]) Let , . Fractional integral formula by partsis valid for every and such that with , where
Proposition 2.6
([19]) Let , . Then we arrive at the following identities:, .
Finally, to prove the blow-up, we need the following lemma.
Lemma 2.7
Let , , we have
13
Proof
Note thatwe haveAs a result,Similarly, we can complete the proof of another equation.
Proof of Main Results
Proof of Theorem 1.2
We prove this conclusion based on the contraction mapping principle. A Banach space is constructed for every ,where . For any given , we define Let , we claim by using (11) and (12), then
14
By choosing T small enough, we havethis implies that . Consequently, we get .For , by using (12), we have the following estimate,
15
due to the inequality16
a choice of small T such thatimplies that is a contraction mapping on . To sum up, we conclude from Banach’s fixed point theorem that there is a mild solution to the problem (1).Concerning the uniqueness issue, we use Gronwall’s inequality to deal with it.
Let be two mild solutions in . Using (12) and (16), we obtain
17
From Gronwall’s inequality, we infer that . In addition, due to the uniqueness, there must be a solution in the maximal interval (see also Fino, Kirane [15]).If and , by (9), (10), we can get directly from (5) that . This closes the proof.
Proof of Theorem 1.4
Equation (5) implies thatBy Lemma 2.2, we getIntegrating the above equation with respect to the variable x yieldsNow, using Lemma 2.3, one hasNext, we construct the time derivative of , let , and obtainBy dominated convergence theorem, we deduce that when h tends to zero, and respectively converge toAfterwards, we consider the estimation of , which can be rewritten as follows:By dominated convergence theorem, we deduce that when h tends to zero, converge toThen, we getUsing the integration by parts formula, we deriveIt followsThe proof is completed.
Proof of Theorem 1.5
Here, we use the contradiction analysis based on the test functions to verify our conclusion. In what follows, we prove this conclusion in two different cases associated with .
Assume is a weak solution to the Eq. (1), then the Eq. (6) holds, , , and .
Let with , , , and , where . The function is smooth, non-increasing, and satisfyingAfter that, we take , and substitute the test function into (6),
18
which implies19
It haswhere we used the estimate for any bounded and continuous function and all [20]. Applying Young’s inequality with , and taking the weight coefficient , we consequently get thatSimilarly, taking , and substituting , , , we have20
If , that is . When , is bounded in , the remaining two terms of the integral are still bounded in . Letting we can obtain that the right terminal term of (20) is zero, while the left terminal term is positive. Therefore, we obtain a contradiction when . While , , let , we can get that is bounded in . Taking T sufficiently large, we obtain a similar result.The proof for the case is similar to the case , we redefine the test function H. Let , where with , , T and R cannot be infinite at the same time. The definition of is the same as in Case 1.
We set , , . Repeating our steps in Case 1, , , , with some details omitted, we also have
21
Since , namely, , let , we conclude thatMoreover, let , then , we get a contradiction.Proof of Theorem 1.6
Let , , we have
22
where . At the same time, we can deduce from (20) that the corresponding inequality holds,23
Consequently, from (22) and (23) we have access to24
where . Thus, it followswhich closes the proof.Proof of Corollary 1.7
Let , and , our estimate is as follows:
25
As in Theorem 1.6, we can get the conclusion that completes the proof.Acknowledgements
The authors wish to acknowledge the referees for their valuable suggestions.
Author Contributions
All authors contribute equally.
Funding
This work was partially supported by National Natural Science Foundation of China (No. 12061040, No. 11701244) and Graduate Quality Course Program of Lanzhou University of Technology.
Data Availability
No data were used for this work.
Declarations
Conflict of Interest
The authors declare that they have no conflict interest.
Ethical Approval and Consent to Participate
All authors approve and consent to participate. The authors agree to publication.
References
1. Straka, P; Fedotov, S. Transport equations for subdiffusion with nonlinear particle interaction. J. Theor. Biol.; 2015; 366, pp. 71-83.MathSciNet ID: 3292442[DOI: https://dx.doi.org/10.1016/j.jtbi.2014.11.012]zbMath ID: 1412.92026
2. Viales, AD; Wang, KG; Desposito, MA. Anomalous diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise. Phys. Rev. E; 2009; 80,
3. Fetecau, V. Flow of a generalized Oldroyd-B fluid due to a constantly accelerating plate. Appl. Math. Comput.; 2008; 201,
4. Lanusse, P; Benlaoukli, H; Nelson-Gruel, D et al. Fractional-order control and interval analysis of SISO systems with time-delayed state. IET Control Theory Appl.; 2008; 2,
5. Hristov, J. Linear viscoelastic responses and constitutive equations in terms of fractional operators with non-singular kernels. Eur. Phys. J. Plus; 2019; 134,
6. Diethelm, K. A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dyn.; 2013; 71,
7. Atanacković, T; Konjik, S; Oparnica, L et al. The Cattaneo type space-time fractional heat conduction equation. Continuum Mech. Thermodyn.; 2012; 24,
8. Wang, X; Qi, H; Yang, X et al. Analysis of the time-space fractional bioheat transfer equation for biological tissues during laser irradiation. Int. J. Heat Mass Transf.; 2021; 177, [DOI: https://dx.doi.org/10.1016/j.ijheatmasstransfer.2021.121555]
9. Yu, Y; Deng, W; Wu, Y. Positivity and boundedness preserving schemes for space-time fractional predator-prey reaction-diffusion model. Comput. Math. Appl.; 2015; 69,
10. West, BJ. Colloquium: Fractional calculus view of complexity: A tutorial. Rev. Mod. Phys.; 2014; 86,
11. Nagasawa, M; Sirao, T. Probabilistic treatment of the blowing up of solutions for a nonlinear integral equation. Trans. Am. Math. Soc.; 1969; 139, pp. 301-310.MathSciNet ID: 239379[DOI: https://dx.doi.org/10.1090/S0002-9947-1969-0239379-X]zbMath ID: 0175.40702
12. Kobayashi, K. On some semilinear evolution equations with time-lag. Hiroshima Math. J.; 1980; 10,
13. Guedda, M; Kirane, M. Criticality for some evolution equations. Diff. Equ.; 2001; 37,
14. Cazenave, T; Dickstein, F; Weissler, FB. An equation whose Fujita critical exponent is not given by scaling. Nonlinear Anal. Theory Methods Appl.; 2008; 68,
15. Fino, A; Kirane, M. Qualitative properties of solutions to a time-space fractional evolution equation. Q. Appl. Math.; 2012; 70,
16. Yosida, K.: Functional analysis. Springer Science and Business Media (2012)
17. Zhang, QG; Sun, HR. The blow-up and global existence of solutions of Cauchy problems for a time fractional diffusion equation. Topol. Methods Nonlinear Anal.; 2015; 46,
18. Samko, SG; Kilbas, AA; Marichev, OI. Fractional integrals and derivatives; 1987; Theory and Applications, Gordon and Breach Science Publishers:zbMath ID: 0617.26004
19. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory And Applications of Fractional Differential Equations. Elsevier Science Limited (2006)
20. Samko, SG; Kilbas, AA; Marichev, OI. Fractional integrals and derivatives; 1993; Yverdon-les-Bains, Switzerland, Gordon and breach science publishers, Yverdon:zbMath ID: 0818.26003
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
© The Author(s) 2023. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
Abstract
This paper focuses on the blow-up solutions of the space-time fractional equations with Riemann–Liouville type nonlinearity in arbitrary-dimensional space. Using the Banach mapping principle and the test function method, we establish the local well-posedness and overcome the difficulties caused by the fractional operators to obtain the blow-up results. Furthermore, we get the precise lifespan of blow-up solutions under special initial conditions.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
Details
1 Lanzhou University of Technology, Department of Mathematics, Lanzhou, People’s Republic of China (GRID:grid.411291.e) (ISNI:0000 0000 9431 4158)