Remarks on Radial Solutions of a Parabolic Gelfand-Type Equation

Round 1

Reviewer 1 Report

1.I would like you to insert more references from this journal!

2. You did not specify what is the practical application of the study .

Author Response

1) I could not find an appropriate reference from this journal. I added 2 references. Could find Section 8 on Page 3 in the attached file?

2) I added the topic about Keller-Segel model in "Introduction" and "Global solution".  Could find Subsection 2.2 on Page 2 and  Section 4 on Page 3 in the attached file?

Author Response File: Author Response.pdf

Reviewer 2 Report

The author investigated the equation with exponential nonlinearity. The radial solutions are the focus of the research. By combining Sobolev embedding and Poincar'e inequalities with diminishing energy, the author established a global solution for the sufficiently small initial value and parameter.

The main results presented in this paper are available and significant. In my opinion, the quality of this paper is good. So I recommend it for publication in Axioms under finishing the following minor revision:

The abstract part needs modification and novel outcomes should be incorporated into the abstract part. 

The motivation of the current work should be improved as well as the introduction part needs serious modification.  

The Proof of Lemma 2 must be rechecked.

Add the conclusive remarks at the end of the manuscript.

Author Response

I expanded "Abstract" and "Introduction". I added the method of the proof and related example(Keller-Segel model).   Could find Sections 1 and 2 on Page 1 in the attached file?

To explain known result and similar result of  nonlocal problem, I moved Sections 4 and 5 in an original manuscript to Section 1. Could find Sections 5 and 6 on Page 3 in the attached file?

I revised the proof of Lemma 2. Could find Sections 3 on Page 3 in the attached file?

I made a remark of Trudinger-Moser inequality at the end of Section 3. Could find Sections 4 on Page 3 in the attached file?

Author Response File: Author Response.pdf

Reviewer 3 Report

The authors analyze radial solutions of a parabolic equation of Gelfand type. They prove the global existence of the radial solution under the assumption of smallness of initial data and parameters.

After careful reading of the manuscript, I have come to the following conclusions. The abstract and introduction should be improved. The bibliography should be supplemented with additional relevant and recent literature. The necessity of including two separate sections containing the full statements of theorems not directly related to the main result of this paper or used in the proof, but only for review purposes, is questionable.

Several specific comments are given below.

1. Some changes are necessary due to the correctness of the English language, especially stylistic errors. For example, when referring to the author of papers from the literature list in the introduction (before Theorem 1), the pronoun they should be used, not he.

2. The abstract should be expanded. The topic covered in the manuscript should be explained better.

3. The introduction should include a description of the problem under consideration in a broader context, supported by citations from the relevant literature. The significance and relevance of the result obtained should also be explained in more detail.

4. More recent references should be cited (within the last 5 years).

5. I do not think that the listing of theorems proved in other papers in Sections 4 and 5 is relevant to this manuscript, since this is not a review article and those results are not used to prove the results in this manuscript. I think it would be more appropriate for the author to explain (rather than state) these results in more detail in the introduction or in a separate section dedicated to the literature review (after the introduction and before the sections containing the main result of this paper).

I think that the paper in its present form is not suitable for publication. However, the paper deals with an interesting topic and the results may be useful for future results and applications. If the authors adequately address the above objections and make the appropriate changes, I think the paper can be considered for publication.

Author Response

1) I checked the English language. Especially, I changed "he" to "they". Could find Subsections 2.3 on Page 2 in the attached file?

2) 3) 5) I expanded "Abstract" and "Introduction". I added the method of the proof, the known result and the corresponding nonlocal problem. I moved Sections 4 and 5 to Introduction. Could find Sections 1 and 2 on Page 1 in the attached file?

4) I could not find the more recent paper. Instead, I added two references.  Could find Section 8 on Page 3 in the attached file?

Author Response File: Author Response.pdf