On Blow-Up Solutions for the Fourth-Order Nonlinear Schrödinger Equation with Mixed Dispersions
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis is the review report of the paper "On the Blow-up Solutions for the Fourth-order Nonlinear Schrödinger Equation with the Mixed Dispersions" by the authors Hui-Ling Niu, Abdoulaye Ali Youssouf and Binhua Feng.
In general, I found the paper interesting and it deserves to be published in Axioms. I have some concerns that need to be considered by the authors, hence I will provide major revisions recommendation at this stage:
1. The abstract is poorly constructed. Avoid references and equations in the abstract. Start by putting the problem into a context, provide a short summary of the metodology used, highlight your novelties and provide a short summary of conclusions.
2. Introduce additional justifications for the Eq. 3 and 4.
3. Add a references for Lemma 1.
4. The introduction section is confusing to me. The authors start by the Eq. (1), and then they say that this equation has not the scaling invariant property. Afterward, they provide some initial bounds and critical exponents. Right after, the authors provide energy formulations for Eq. (1). At some moment, I get confused. My recommendation is to clearly state what part of the introduction discusses the case of mu = 0 and then other distinct part of the introduction should be focused on mu =/ 0.
5. The theory of blow-up in higher order operators is not new. I recommend the author to increase their literature review by adding more theoretical references on perhaps less complex problems. In addition, the relevance of higher-order operators in science and the theory of parabolic operators should be discussed further and extended to more works not only connected with the authors problem. In this way, the reader will have a comprenhensive view of the importance of higher order operators in science and mathematics. My recommendation is to write a single paragraph in the introduction highlighting the relevance of higher order operators by citing the following references (this is a suggestion of references, the author may consider to search others):
- https://researchportal.bath.ac.uk/en/publications/existence-and-blow-up-for-higher-order-semilinear-parabolic-equat
- https://link.springer.com/article/10.1007/s40314-021-01689-y
- https://www.mdpi.com/2227-7390/9/18/2300
- https://www.sciencedirect.com/science/article/pii/S1631073X02025670?via%3Dihub
- https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.9113
- https://digital.library.txst.edu/items/5a620f12-1fbd-4713-b01b-5274087c6975
Comments on the Quality of English Language
Some minor issues detected, but nothing serious.
Author Response
Please see the attachment.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsThe authors show expertise in studying the dynamical properties of the different cases of nonlinearity in the Schrödinger equation. The current paper presents the case of introducing the term that gives the fourth-order linear equation with mixed dispersion. It is evidenced in Theorem 1 the sharp threshold mass of blow-up and global existence.
From the mathematical point of view, it might present some interest, however, physics starts where the solutions blow-up, as the Feynman Propagator theory in quantum fields shows. Thus, if the physics is mentioned more resembled examples will be an added value to the work.
The paper is well written and meets the standards for publication.
Comments for author File: Comments.pdf
Author Response
Thank you very much for taking the time to review this manuscript. We have revised the abstract and introduction section to improve the manuscript, and also the references are updated, see the revised manuscript for details. Thank you again.
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsNo comments. Accept. Good luck with your publication.
Comments on the Quality of English LanguageMinor issues
