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 [1–8] 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, 12–19] 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.
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.
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.