Table of Contents Author Guidelines Submit a Manuscript
International Journal of Differential Equations
Volume 2018, Article ID 5053415, 7 pages
https://doi.org/10.1155/2018/5053415
Research Article

Global and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusion

Laboratory of Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

Correspondence should be addressed to Tetsutaro Shibata; pj.ca.u-amihsorih.htama@atabihs

Received 29 March 2018; Accepted 7 August 2018; Published 2 September 2018

Academic Editor: Patricia J. Y. Wong

Copyright © 2018 Tetsutaro Shibata. 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.

Abstract

We consider the nonlinear eigenvalue problem , , , , where , , and is a bifurcation parameter. Here, and () are constants. This equation is related to the mathematical model of animal dispersal and invasion, and is parameterized by the maximum norm of the solution associated with and is written as . Since contains both power nonlinear term and oscillatory term , it seems interesting to investigate how the shape of is affected by . The purpose of this paper is to characterize the total shape of by and . Precisely, we establish three types of shape of , which seem to be new.

1. Introduction

This paper is concerned with the following nonlinear eigenvalue problems: where , , and is a bifurcation parameter. Here, and () are constants. Bifurcation problems have a long history and there are so many results concerning the asymptotic properties of bifurcation diagrams. We refer to [18] and the references therein. Moreover, bifurcation problems with nonlinear diffusion have been proposed in the field of population biology, and several model equations of logistic type have been considered. We refer to [9] and the references therein. In particular, the case () has been derived from a model equation of animal dispersal and invasion in [10, 11]. In this situation, is a parameter which represents the habitat size and diffusion rate. On the other hand, there are several papers which treat the asymptotic behavior of oscillatory bifurcation curves. We refer to [7, 1219] and the references therein. Our equation (1) contains both nonlinear diffusion term and oscillatory nonlinear terms. The purpose of this paper is to find the difference between the structures of bifurcation curves of the equations with only oscillatory term and those with both nonlinear diffusion term and the oscillatory term in (1). To clarify our intention, let and . Then (1) is given as The corresponding equation without nonlinear diffusion is the case and , namely, It should be mentioned that, by using a generalized time-map argument in [9], for any given , there exists a unique classical solution pair of (1)–(3) satisfying . Furthermore, is parameterized by as and is continuous in . For (5), the following asymptotic formula for as has been obtained.

Theorem 1 . (see [12]). Consider (5) with (2)–(3). Then as ,

For (5) with (2)–(3), the asymptotic behavior of as is as follows. For a solution pair satisfying , put and let . Then we easily obtain the function which satisfies , for with . This implies . By this fact and Theorem 1, the bifurcation curve starts from and tends to with oscillation and intersects the line infinitely many times for .

Since (4) includes both the nonlinear diffusion function and oscillatory term, it seems interesting how the nonlinear diffusion functions give effect to the structures of bifurcation curves.

Now we state our main results.

Theorem 2. Consider (1) with (2)–(3). Then as ,where

By Theorem 2, we obtain the global behavior of as for and see that the asymptotic behavior of is completely different from that for by comparing Figures 1 and 2.

Figure 1: The graph of for (5) ().
Figure 2: The graph of for .

Now we establish the asymptotic behavior of as to obtain a complete understanding of the structure of . Let

Theorem 3. Consider (1) with (2)–(3). Then the following asymptotic formulas hold as .(i)Assume that . Then(ii)Assume that . Then(iii)Assume that . Then(iv)Assume that . Then(v)Assume that . Then

The rough images of the graphs of for , , and are given in Figures 3, 4, and 5.

Figure 3: The graph of for , .
Figure 4: The graph of for .
Figure 5: The graph of for , .

The proofs depend on the generalized time-map argument in [9] and stationary phase method (cf. Lemma 4). It should be mentioned that if we apply Lemma 4 to our situation, careful consideration about the regularity of the functions is necessary.

2. Proof of Theorem 2

We putIt was shown in [9, ] that if , then is well defined. In our situation, it is clear that, for , , , so . Therefore, . By this and the generalized time-map obtained in [9] (cf. (24)) and the time-map argument in [8, Theorem 2.1], we see that, for any given , there exists a unique classical solution pair of (1)–(3) satisfying . Furthermore, is parameterized by as and is continuous in . For , we put It is known from [9] that if satisfies (1)–(3), then In what follows, we denote by various positive constants independent of . For and , we have By this, (19), and Taylor expansion, we have from [9, ] that where We see from (24) and (25) that if we obtain the precise asymptotic formula for as , then we obtain Theorem 2. To do this, we apply the stationary phase method to our situation. By combining [13, Lemma 2] and [7, Lemmas 2.24], we have the following equality.

Lemma 4 (see [13, Lemma 2 and 10, Lemma 2.24]). Assume that the function , , andThen as In particular, by taking the imaginary part of (27), as ,

We note that, to obtain (27), we have to be careful about the regularity of and .

Lemma 5. As ,

Proof. We put and By integration by parts, (25) and (30), we haveBy l’Hôpital’s rule, we obtain This implies that . Next, We first calculate . Assume that . ThenThis implies that, for ,By direct calculation, we also obtain (35) for the case where . By integration by parts, we obtain By (35) and (36), for , we obtain Since by putting , we obtainLetCase 1. Assume that or . Then clearly , and we are able to apply Lemma 4 to (39). Then we obtain By this, (33), and (37), we obtain (29).
Case 2. Assume that . Then with . Therefore, does not satisfy the condition in Lemma 4. However, we found in [14] that we can still apply Lemma 4 to (39) in this situation and obtain (41). For completeness, the reason will be explained in the Appendix. By this, (33), and (41), we obtain (29). Thus the proof is complete.

By (24) and Lemma 5, we obtain Theorem 2 immediately. Thus the proof is complete.

3. Proof of Theorem 3

In this section, let . The proofs of Theorem 3 (i)-(v) are similar. Therefore, we only prove (i) and (iv).

Proof of Theorem 3 (i). We assume that . Then by Taylor expansion, for , we have By this, (24), Taylor expansion, and putting , we obtain This implies (13). Thus the proof is complete.

Proof of Theorem 3 (iv). We assume that . Then by (42), for , we have By this, (24), and putting , we obtainThis impliesThis implies (16). Thus the proof is complete.

Appendix

In this section, by following the argument in [14], we show that Case 2 in Lemma 5 holds for completeness. We put Note that . We see that . The essential point of the proof of (27) in this case is to show Lemma 2.24 in [7] (see also [7, Lemma 2.25]). Namely, as , We put . Then we have . We know from [7, Lemma 2.24] that, for , By (A.2) and (A.3), we obtain We put Now we prove that , because if it is proved, then by integration by parts, we easily show that and our conclusion (A.2) follows immediately from (A.4) and (A.5). For , we have Then we have . Furthermore, by direct calculation, we can show that . It is reasonable, because by Taylor expansion, for , we have Thus the proof is complete.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This work was supported by JSPS KAKENHI Grant Number JP17K05330.

References

  1. A. Ambrosetti, H. Brézis, and G. Cerami, “Combined effects of concave and convex nonlinearities in some elliptic problems,” Journal of Functional Analysis, vol. 122, no. 2, pp. 519–543, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. S. Cano-Casanova and J. López-Gómez, “Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line,” Journal of Differential Equations, vol. 244, no. 12, pp. 3180–3203, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. Y. J. Cheng, “On an open problem of Ambrosetti, Brezis and Cerami,” Differential and Integral Equations, vol. 15, pp. 1025–1044, 2002. View at Google Scholar
  4. R. Chiappinelli, D. G. De Figueiredo, and P. Hess, “Bifurcation from infinity and multiple solutions for an elliptic system,” Differential and Integral Equations, vol. 6, no. 4, pp. 757–771, 1993. View at Google Scholar · View at Scopus
  5. R. Chiappinelli, “Upper and lower bounds for higher order eigenvalues of some semilinear elliptic equations,” Applied Mathematics and Computation, vol. 216, no. 12, pp. 3772–3777, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. R. Chiappinelli, “Approximation and convergence rate of nonlinear eigenvalues: Lipschitz perturbations of a bounded self-adjoint operator,” Journal of Mathematical Analysis and Applications, vol. 455, no. 2, pp. 1720–1732, 2017. View at Publisher · View at Google Scholar · View at Scopus
  7. P. Korman, Global solution curves for semilinear elliptic equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar
  8. T. Laetsch, “The number of solutions of a nonlinear two point boundary value problem,” Indiana University Mathematics Journal, vol. 20, pp. 1–13, 1971. View at Publisher · View at Google Scholar · View at MathSciNet
  9. Y. H. Lee, L. Sherbakov, J. Taber, and J. Shi, “Bifurcation diagrams of population models with nonlinear, diffusion,” Journal of Computational and Applied Mathematics, vol. 194, no. 2, pp. 357–367, 2006. View at Publisher · View at Google Scholar · View at Scopus
  10. J. D. Murray, “Mathematical biology. I. An introduction,” in Interdisciplinary Applied Mathematics, vol. 17, Springer-Verlag, New York, NY, 3rd edition, 2002. View at Google Scholar
  11. P. Turchin, “Population consequences of aggregative movement,” Journal of Animal Ecology, vol. 58, no. 1, pp. 75–100, 1989. View at Publisher · View at Google Scholar · View at Scopus
  12. T. Shibata, “Asymptotic length of bifurcation curves related to inverse bifurcation problems,” Journal of Mathematical Analysis and Applications, vol. 438, no. 2, pp. 629–642, 2016. View at Publisher · View at Google Scholar · View at Scopus
  13. P. Korman and Y. Li, “Infinitely many solutions at a resonance,” Electronic Journal of Differential Equations, pp. 105–111, 2000, Presented at the Differential Equations Conference 5. View at Google Scholar
  14. T. Shibata, “Global and local structures of oscillatory bifurcation curves,” Journal of Spectral Theory.
  15. A. Galstian, P. Korman, and Y. Li, “On the oscillations of the solution curve for a class of semilinear equations,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 576–588, 2006. View at Publisher · View at Google Scholar · View at Scopus
  16. P. Korman, “An oscillatory bifurcation from infinity, and from zero,” Nonlinear Differential Equations and Applications NoDEA, vol. 15, no. 3, pp. 335–345, 2008. View at Publisher · View at Google Scholar · View at Scopus
  17. T. Shibata, “Oscillatory bifurcation for semilinear ordinary differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, no. 44, pp. 1–13, 2016. View at Publisher · View at Google Scholar
  18. T. Shibata, “Global and local structures of oscillatory bifurcation curves with application to inverse bifurcation problem,” Topological Methods in Nonlinear Analysis, vol. 50, no. 2, pp. 603–622, 2017. View at Google Scholar · View at Scopus
  19. T. Shibata, “Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms,” Communications on Pure & Applied Analysis, vol. 17, no. 5, pp. 2139–2147, 2018. View at Publisher · View at Google Scholar