The Second Critical Exponent for a Time-Fractional Reaction-Diffusion Equation

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please see attachment.

Comments for author File: Comments.pdf

Author Response

Comments 1: Why do you study the second critical exponent? What is the importance?
Response 1: Thank you for pointing this out. I agree with these comments. Therefore, I have added these points from line number 150 to line number 157 on page 5. 

Comments 2: The paper considered the same PDE model, methods and topics (e.g., blow-up and global existence) with the ref.[21]. In sections 3-4, the author should give some discussions by compared to the main results (e.g., Theorem 4.4) in [21]
Response 2: Thank you for pointing this out. I agree with this comment. Therefore, I have added Remark 3 on page 11 and Remark 4 on page 12 to emphasize this point.

Comments 3: Could you give some numerical example to verify the feasibility of the main result?
Response 3: I have added Example 1 on page 13. 

Reviewer 2 Report

Comments and Suggestions for Authors

- The proposed results seem very interesting to but you need to detail your results a little more in relation to what exists. 

- The Difference between your results and those of :

1. Sunday A. Asogwa, Mohammud Foondun, Jebessa B. Mijena and Erkan Nane ;
Critical parameters for reaction–diffusion equations involving
space–time fractional derivatives. Nonlinear Differ. Equ. Appl. (2020)

2. Masamitsu Suzuki; Local existence and nonexistence for fractional in time
reaction–diffusion equations and systems with rapidly growing
nonlinear terms. Nonlinear Analysis 2022, 112909. 

- Ajouter par exemple les application de ces types d equation.

Author Response

Comments 1: The proposed results seem very interesting to but you need to detail your results a little more in relation to what exists. 

Response 1: Thank you for pointing this out. I agree with this comment. Therefore, I have added Remark 3 on page 11 and Remark 4 on page 12 to emphasize this point. 

Comments 2: The Difference between your results and those of :

1. Sunday A. Asogwa, Mohammud Foondun, Jebessa B. Mijena and Erkan Nane ; Critical parameters for reaction–diffusion equations involving space–time fractional derivatives. Nonlinear Differ. Equ. Appl. (2020)

2. Masamitsu Suzuki; Local existence and nonexistence for fractional in time reaction–diffusion equations and systems with rapidly growing nonlinear terms. Nonlinear Analysis 2022, 112909.

Ajouter par exemple les application de ces types d equation.

Response 2: Thank you for pointing this out. I agree with these comments. Therefore, I have added these points from line number 336 to line number 349 on page 13. 

Reviewer 3 Report

Comments and Suggestions for Authors

Reviewers’ Comments
Manuscript ID: mathematics-3142447
Title of Paper: The Second Critical Exponent for a Time Fractional Reaction-Diffusion Equation
Author(s): Takefumi Igarashi
I have reviewed the manuscript in a fairly detailed way. The author of the manuscript studied the Cauchy problem of a time fractional nonlinear diffusion equation. According to the Kaplan’s first eigenvalue method, the author proved the blow-up of the solutions in finite time for some sufficient conditions. In addition, he provided sufficient conditions for the existence of global solutions by using the result of Zhang and Sun. The results are positive and promising, and it appears that the research followed accepted scientific procedures. Furthermore, I would like to draw attention to the following issues:
1. Motivation is not sufficiently stated in the introduction part. It should be clarified why they consider this problem and what are the advantages of the proposed methods?
2. What are the key features of the proposed methods? (Properties, characteristics, and weaknesses).
3. Highlight the main novelty of this paper and the significance of the results.
4. The equations in the examples should be cited to present where they are taken from.
5. Please check entire manuscript carefully for grammatical errors and typos.
6. Authors should better explain the advantages (also disadvantages) of the present method over the other methods.
7. Conclusion part of the manuscript can be developed and more explanation about the results can be added.
8. The reaction-diffusion equation is used in a wide range of applications. Please explain why a fractional modification is needed ''generalized Caputo derivative''. A geometrical explanation of the fractional calculus should be given.
9. In order to have the introduction of the paper more meaningful, the following references to be added in the list of references and appropriately incorporated in the introduction of the paper:
https://doi.org/10.1016/j.chaos.2023.114309
DOI: 10.1016/j.aml.2018.12.021
Doi: 10.3934/dcdss.2023062
10. Finally, please check the entire manuscript carefully about the above mentioned suggestions.
Last of all, this manuscript contains significant methodological content. The topic content of this paper is consistent with the goals of your journal. So my recommendation is acceptance of the manuscript after the minor changes. It will hopefully add value to the existing literature. So, I want to read speedily the revised version of paper before publishing if it is possible for you.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

The language of the manuscript should be modified carefully.

Author Response

Comments 1: Motivation is not sufficiently stated in the introduction part. It should be clarified why they consider this problem and what are the advantages of the proposed methods?

Response 1: Thank you for pointing this out. I agree with these comments. Therefore, I have added these points from line number 17 on page 1 to lines 26 on page 2 and from line number 150 to line number 157 on page 5.

Comments 2: What are the key features of the proposed methods? (Properties, characteristics, and weaknesses).

Response 2: Thank you for pointing this out. I agree with this comment. Therefore, I have added Remark 2 on page 11 to emphasize this point.

Comments 3: Highlight the main novelty of this paper and the significance of the results.

Response 3: I have added the main novelty of this paper in Remark 3 on page 11 and the significance of the results from line number 320 to line number 324 on page 13.

Comments 4: The equations in the examples should be cited to present where they are taken from.

Response 4: I have added this point in line number 103 on page 4.

Comments 5: Please check entire manuscript carefully for grammatical errors and typos.

Response 5: I have checked the entire manuscript using a Grammar Checker (wordcount.com).

Comments 6: Authors should better explain the advantages (also disadvantages) of the present method over the other methods.

Response 6: I have added them from line number 206 on page 7 to line number 210 on page 8.

Comments 7: Conclusion part of the manuscript can be developed and more explanation about the results can be added.

Response 7: I have added them from line number 320 to line number 349 on page 13.

Comments 8: The reaction-diffusion equation is used in a wide range of applications. Please explain why a fractional modification is needed ''generalized Caputo derivative''. A geometrical explanation of the fractional calculus should be given.  

Response 8: I have added them from line number 18 to line number 23 on page 1.

Comments 9: In order to have the introduction of the paper more meaningful, the following references to be added in the list of references and appropriately incorporated in the introduction of the paper:

https://doi.org/10.1016/j.chaos.2023.114309

DOI: 10.1016/j.aml.2018.12.021

Doi: 10.3934/dcdss.2023062

Response 9: I have added DOI: 10.1016/j.aml.2018.12.021 and https://doi.org/10.1016/j.chaos.2023.114309 to the list of references [25] and [27], and incorporated them from lines 111 on page 4 to lines 149 on page 5.  I could not get Doi: 10.3934/dcdss.2023062 because it is not subscribed to by Nihon University, but I have added reference [26] instead.

Comments 10: Finally, please check the entire manuscript carefully about the above mentioned suggestions.

Response 10: Finally, I have checked the entire manuscript.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Accept in present form

Author Response

Thank you so much for accepting it in its present form.

Reviewer 2 Report

Comments and Suggestions for Authors After modifications. The article seems well written and interesting to me and deserves to be published.

Author Response

Thank you so much for agreeing to be published.