Estimates for the Approximation and Eigenvalues of the Resolvent of a Class of Singular Operators of Parabolic Type
Round 1
Reviewer 1 Report
Comments and Suggestions for Authors
The article considers the properties of parabolic equations with unbounded coefficients defined in infinite strip. The authors show that the equations admit compact resolvent under not too strong conditions. Even though I the statements of the article seem to be correct, it is very hard to recommend it for publication. The manuscript lacks such obligatory parts such as introduction (with the literature review, and motivation for the study) and conclusion, the body of the text is just a list of theorems and lemmas, without illustrations of results and their applications to the actual equations. Given a variety of textbooks with reader-friendly text on the subject, I do not think that current article can be interesting to the readers. I do not recommend publication in current form; the revised version of the text can be reconsidered.
Author Response
Dear reviewer!
We agree with your comments and take this opportunity to thank you for your useful advice to improve the content of the manuscript.
We have supplemented the introduction. In the introduction we introduced answers to the following questions:
- about the applications of the considered operator;
- compared the results of this paper with the results of other authors;
- we have introduced the following proposals on the use of estimates of approximation numbers in the application: “Here we note that estimates of approximation numbers show the speed of approach of approximate solutions of the equation to the exact one. Thus, by estimating approximation numbers, we not only study the spectral properties of the inverse operator, but come into closer contact with application issues.”
We have also added the section “Conclusion”.
We also edited the text of the article to correct the grammar.
The references has been redone and links to them.
Best wishes
Authors
Reviewer 2 Report
Comments and Suggestions for Authors
The paper has interesting contributions, but there are details to improve:
1) The Introduction is very brief and does not provide sufficient motivation for the results that will be presented.
2) If Lemma 6 is proven in [21], it is not necessary to prove it again.
3) It is important to clearly determine what the differences are in the contributions of references [7], [8], [17], [18] and [20] with this paper. Especially with reference [17].
4) Applications to other areas of mathematics are not mentioned.
5) There are no conclusions and no future work is mentioned.
Comments on the Quality of English Language
Moderate editing of English language required.
Author Response
Dear reviewer!
We agree with your comments and take this opportunity to thank you for your useful advice to improve the content of the manuscript.
We have supplemented the introduction. In the introduction we introduced answers to the following questions:
- about the applications of the considered operator;
- compared the results of this paper with the results of other authors;
- we have introduced the following proposals on the use of estimates of approximation numbers in the application: “Here we note that estimates of approximation numbers show the speed of approach of approximate solutions of the equation to the exact one. Thus, by estimating approximation numbers, we not only study the spectral properties of the inverse operator, but come into closer contact with application issues.”
We have also added the section “Conclusion”.
We also edited the text of the article to correct the grammar.
The references has been redone and links to them.
Best wishes
Authors
Round 2
Reviewer 1 Report
Comments and Suggestions for Authors
The authors made significant improvements but the proposed changes are done only partially. The conclusion just summarises results, instead of discussion of potential applications. Also, I see missprints "[?]" references on p.3 and p.4. I recommend minor revision before publication.
Author Response
Dear reviewer!
We have fixed the error on p.3 and p.4
In conclusion we have added the following suggestions:
Here we note that in applications often appear equations in an unbounded domain with unbounded coefficients at infinity [1, 2, 3, 9, 10, 11, 12, 13, 15]. In this regard, there is a need to study the spectral and approximation properties of differential operators with unbounded coefficients.
The results and methods used in this paper allow us to study the following questions for a class of parabolic differential operators defined in an unbounded domain with strongly increasing coefficients at infinity::
- existence of a resolvent;
- discreteness of the spectrum and estimates of eigenvalues;
- estimates of approximation numbers, which play an important role in approximation theory.
Best wishes
Authors
Reviewer 2 Report
Comments and Suggestions for Authors
The authors have made the suggested changes and the manuscript has improved substantially.
Comments on the Quality of English Language
The authors have made the suggested changes and the manuscript has improved substantially. From my point of view the manuscript can be accepted.
Author Response
In conclusion, taking into account the reviewer’s comments, we have added the following suggestions:
Here we note that in applications often appear equations in an unbounded domain with unbounded coefficients at infinity [1, 2, 3, 9, 10, 11, 12, 13, 15]. In this regard, there is a need to study the spectral and approximation properties of differential operators with unbounded coefficients.
The results and methods used in this paper allow us to study the following questions for a class of parabolic differential operators defined in an unbounded domain with strongly increasing coefficients at infinity::
- existence of a resolvent;
- discreteness of the spectrum and estimates of eigenvalues;
- estimates of approximation numbers, which play an important role in approximation theory.
Best wishes
Authors
