Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 7480676 | https://doi.org/10.1155/2020/7480676

Baiping Ouyang, Wei Fan, Yiwu Lin, "Lower Bound for the Blow-Up Time for the Nonlinear Reaction-Diffusion System in High Dimensions", Discrete Dynamics in Nature and Society, vol. 2020, Article ID 7480676, 7 pages, 2020. https://doi.org/10.1155/2020/7480676

Lower Bound for the Blow-Up Time for the Nonlinear Reaction-Diffusion System in High Dimensions

Academic Editor: Rodica Luca
Received06 Mar 2020
Accepted12 May 2020
Published05 Jun 2020

Abstract

In this paper, we study the blow-up phenomenon for a nonlinear reaction-diffusion system with time-dependent coefficients under nonlinear boundary conditions. Using the technique of a first-order differential inequality and the Sobolev inequalities, we can get the energy expression which satisfies the differential inequality. The lower bound for the blow-up time could be obtained if blow-up does really occur in high dimensions.

1. Introduction

During the past decades, the blow-up phenomena for the solutions to the parabolic problems have been widely concerned. It is important in practice that how to determine the bound of the blow-up time of the solutions about the parabolic equations and systems. Their applications are included in physics, chemistry, astronomy, biology, and population dynamics [1, 2]. Actually, when the blow-up occurs at , it is difficult to get the exact value of . We mainly focus on estimating its bounds. At present, the studies on the blow-up phenomena of parabolic problems mainly focus on homogeneous Dirichlet boundary condition and homogeneous Neumann and Robin boundary conditions [312]. There are also some works under nonlinear boundary conditions [1315]. Most of these articles are focused on . There are only a few papers dealing with a lower bound for the blow-up time in high dimensions (see [1618]). Recently, some scholars have started to investigate the blow-up problems with time-dependent coefficients [1921]. In paper [21], the authors considered the following nonlinear reaction-diffusion system with time-dependent coefficients:

The authors obtained the lower and upper bounds for the blow-up time when the blow-up occurred. In this paper, we further consider the blow-up phenomena for the following system with time-dependent coefficients under nonlinear boundary conditions in high dimensions:

We assume that are continuous, and , and satisfywhere is a positive constant to be defined later.

Our goal in this paper is to obtain a lower bound for the blow-up time of the solutions to systems (2) and (3) in for any . The nonlinear terms and and the boundary conditions are difficult to tackle. We cannot get the result by following the method proposed in [21], so we must use a new method to overcome these difficulties. To the best of our knowledge, no results exist in that direction, and we think our result is new and interesting.

In the further discussions, we will use the following Hölder inequality:where is a nonnegative function and , , , and are positive constants satisfying .

We also need the following Sobolev inequality [22]:with which is a Sobolev embedding constant depending on and .

And the classical (or elementary) inequality iswhere , , and are positive constants, and satisfies .

2. Lower Bound for the Blow-Up Time

In this part, we define an auxiliary function of the formwhere , , and .

We establish the following theorem:

Theorem 1. Let be the weak solution of problems (1)–(3) in a bounded convex domain . Then, the quantity defined in (8) satisfies the integral inequalitywhich follows that the blow-up time is bounded below. We havewhere , , , , and will be defined later.

Now, we prove Theorem 1. For simplicity, assume that the solution is classical of problems (1)–(3). The general case can be done by approximation. Differentiating , we havewhere .

For the second term on the right side of (11), we apply the divergence theorem, the trace embedding, and (3) to getwhere , is the outward normal vector of and .

For the second term on the right side of (12), using (11), we obtainwhere is a positive constant which will be defined later. Using (4), we havewhere .

Choosing and and using (4), (5), and (7), we havewhere and is a positive constant which will be defined later.

For the first term on the right side of (15), using (4) and Young’s inequality, we havewhere , , and is a positive constant which will be defined later.

Combining (15) and (16), if we choose suitable such that , we havewhere , , and .

Combining (12), (13), (14), and (17), we havewhere and .

Similarly, for the fourth term on the right side of (11), using the divergence theorem and (3), we have

For the second term on the right side of (19), using (4), we obtainwhere is a positive constant which will be defined later. Similarly, we havewhere and .

Choosing and and using (4), (6), and (7), we havewhere and is a positive constant which will be defined later.

For the first term on the right side of (22), using (4), we havewhere , , and is a positive constant which will be defined later.

Combining (22) and (23), if we choose suitable such that , we havewhere .

Combining (19)–(21) and (24), we havewhere and

For the third term on the right side of (11), using Hölder inequality and Young’s inequality, we have

For the first term on the right side of (26), using (4), (5), and (7) and taking care of the given condition , we havewhere , , and .

For the first term on the right side of (27), using (4) and Young’s inequality, we getwhere , , and is a positive constant which will be defined later.

Combining (27) and (28), if we choose suitable such that , we havewhere , , and .

Combining (26) and (29), we obtain

By the same way, for the fifth term on the right side of (11), using (4), we have

For the first term on the right side of (31), using (4), (6), and (7) and taking care of the given condition , we havewhere , , , and is a positive constant which will be defined later.

For the first term on the right side of (32), using (4), we getwhere , , and is a positive constant which will be defined later.

Combining (32) and (33), if we choose suitable such that , we havewhere , , and .

Combining (31) and (34), we obtain

Combining (11), (18), (25), (30), and (35), we havewhere , , , and

If we choose suitable , , , and such that , we can rewrite (36) aswhere , , , and .

Letwhere .

Integrating (37) from 0 to , we have

Considering , the integration of the right side of (39) exists. It is clear that is an increasing function. So, we can getwhere is the inverse function of .

The proof of Theorem 1 is complete.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 61907010), the Foundation for Natural Science in Higher Education of Guangdong, China (Grant no. 2018KZDXM048), the General Project of Science Research of Guangzhou (Grant no. 201707010126), and Huashang College Guangdong University of Finance & Economics (Grant no. 2019HSDS26).

References

  1. B. Straughan, Explosive Instabilities in Mechnics, Spring, Berlin, Germany, 1998.
  2. W. Deng, “Global existence and finite time blow up for a degenerate reaction-diffusion system,” Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 5, pp. 977–991, 2005. View at: Publisher Site | Google Scholar
  3. L. E. Payne and P. W. Schaefer, “Lower bounds for blow-up time in parabolic problems under Dirichlet conditions,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1196–1205, 2007. View at: Publisher Site | Google Scholar
  4. J. Ding and H. Hu, “Blow-up and global solutions for a class of nonlinear reaction diffusion equations under Dirichlet boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 433, no. 2, pp. 1718–1735, 2016. View at: Publisher Site | Google Scholar
  5. L. E. Payne and P. W. Schaefer, “Lower bounds for blow-up time in parabolic problems under Neumann conditions,” Applicable Analysis, vol. 85, no. 10, pp. 1301–1311, 2006. View at: Publisher Site | Google Scholar
  6. Y. Liu, “Blow-up phenomena for the nonlinear nonlocal porous medium equation under Robin boundary condition,” Computers & Mathematics with Applications, vol. 66, no. 10, pp. 2092–2095, 2013. View at: Publisher Site | Google Scholar
  7. Y. F. Li, “Blow-up and global existence of the solution to some more general nonlinear parabolic problems with Robin boundary conditions,” Acta Mathematica Sinica, English Series, vol. 41, pp. 257–267, 2018. View at: Google Scholar
  8. L. E. Payne and P. W. Schaefer, “Blow-up in parabolic problems under Robin boundary conditions,” Applicable Analysis, vol. 87, no. 6, pp. 699–707, 2008. View at: Publisher Site | Google Scholar
  9. L. E. Payne and J. C. Song, “Lower bounds for the blow-up time in a temperature dependent Navier-Stokes flow,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 371–376, 2007. View at: Publisher Site | Google Scholar
  10. P. W. Schaefer, “Blow up phenomena in some porous medium problems,” Dynamic Systems and Applications, vol. 18, pp. 103–109, 2009. View at: Google Scholar
  11. J. C. Song, “Lower bounds for the blow-up time in a non-local reaction-diffusion problem,” Applied Mathematics Letters, vol. 24, no. 5, pp. 793–796, 2011. View at: Publisher Site | Google Scholar
  12. Y. Li, Y. Liu, and C. Lin, “Blow-up phenomena for some nonlinear parabolic problems under mixed boundary conditions,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3815–3823, 2010. View at: Publisher Site | Google Scholar
  13. Y. Liu, S. Luo, and Y. Ye, “Blow-up phenomena for a parabolic problem with a gradient nonlinearity under nonlinear boundary conditions,” Computers & Mathematics with Applications, vol. 65, no. 8, pp. 1194–1199, 2013. View at: Publisher Site | Google Scholar
  14. L. E. Payne, G. A. Philippin, and S. Vernier Piro, “Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I,” Zeitschrift für angewandte Mathematik und Physik, vol. 61, no. 6, pp. 999–1007, 2010. View at: Publisher Site | Google Scholar
  15. G. Tang, “Blow-up phenomena for a parabolic system with gradient nonlinearity under nonlinear boundary conditions,” Computers & Mathematics with Applications, vol. 74, no. 3, pp. 360–368, 2017. View at: Publisher Site | Google Scholar
  16. A. Bao and X. Song, “Bounds for the blowup time of the solutions to quasi-linear parabolic problems,” Zeitschrift für angewandte Mathematik und Physik, vol. 65, no. 1, pp. 115–123, 2014. View at: Publisher Site | Google Scholar
  17. H. X. Li, W. J. Gao, and Y. Z. Han, “Lower bounds for the blow up time of solutions to a nonlinear parabolic problems,” Electronic Journal of Differential Equations, vol. 20, 2014. View at: Google Scholar
  18. W. H. Chen and Y. Liu, “Lower bound for the blow-up time for some nonlinear parabolic equations,” Boundary Value Problems, vol. 161, no. 1, pp. 1–6, 2016. View at: Publisher Site | Google Scholar
  19. J. Ding and W. Kou, “Blow-up solutions for reaction diffusion equations with nonlocal boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 470, no. 1, pp. 1–15, 2019. View at: Publisher Site | Google Scholar
  20. X. Shen and J. Ding, “Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions,” Computers & Mathematics with Applications, vol. 77, no. 12, pp. 3250–3263, 2019. View at: Publisher Site | Google Scholar
  21. X. Tao and Z. B. Fang, “Blow-up phenomena for a nonlinear reaction-diffusion system with time dependent coefficients,” Computers & Mathematics with Applications, vol. 74, no. 10, pp. 2520–2528, 2017. View at: Publisher Site | Google Scholar
  22. H. Brezis, “Functional analysis, Sobolev spaces and partial differential equations,” in Universitext, Springer, New York, NY, USA, 2011. View at: Google Scholar

Copyright © 2020 Baiping Ouyang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views71
Downloads119
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.