Bifurcation from infinity with applications to reaction-diffusion systems

The bifurcation method is one of powerful tools to study the existence of a continuous branch of solutions. However without further analysis, the local theory only ensures the existence of solutions within a small neighborhood of bifurcation point. In this paper we extend the theory of bifurcation from infinity, initiated by Rabinowitz [ 11 ] and Stuart [ 13 ], to find solutions of elliptic partial differential equations with large amplitude. For the applications to the reaction-diffusion systems, we are able to relax the conditions to obtain the bifurcation from infinity for the following nonlinear terms; (ⅰ) nonlinear terms satisfying conditions similar to [ 11 ] (all directions), (ⅱ) nonlinear terms satisfying similar conditions only on the strip domain along the direction determined by the eigenfunction, (ⅲ) \begin{document}$ p $\end{document} -homogeneous nonlinear terms with degenerate conditions.

1. Introduction. Since in 1952 Turing [14] discovered a diffusion mechanism to generate periodic patterns, many interesting patterns have successfully been found out following from this ingenious idea [7]. This phenomenon is often referred to as diffusion-induced instability; that is, stable homogeneous states are destabilized by changing parameters and inhomogeneous periodic patterns emerge. In many situations, local bifurcation theory only ensures the existence of solutions near the bifurcation point, since it is difficult to trace the bifurcation branch globally unless further analysis can be made [1,2,3,10] to obtain enough information on the eigenvalues for the linearization. To construct the solution with large amplitude, the problem-based-studies are often required.
The linear Strum-Liouville eigenvalue problem −(pu ) + qu = µa(x)u together with suitable boundary conditions possesses an increasing sequence of eigenvalues µ n (n ∈ N) with µ n → ∞. They showed that a branch of solutions bifurcate from (λ, u) = (µ n , ∞), provided that µ n is a simple eigenvalue. The first purpose of this paper is to extend the above result to the following class of reaction-diffusion systems with two components: in a bounded domain Ω ⊂ R N with homogeneous Dirichlet or Neumann boundary conditions on ∂Ω, where d i > 0 and h ij ∈ R are constants (i, j = 1, 2) and nonlinear terms f 1 (u, v) and f 2 (u, v) are assumed to satisfy In Section 2.1, seeking the bifurcation from infinity, we give an extension of the above mentioned results, ( [11], [13]) for a single equation like (1.1), to two-component reaction-diffusion system (1.2). The second purpose of this paper is to relax the sufficient condition for the bifurcation from infinity, especially in dealing with Dirichlet boundary conditions. We weaken assumption (1.3) to certain strip domains which are depending on the eigenspaces, and call it a bifurcation directionally from infinity. More precisely, the following result will be established. Theorem 1.1. Let σ be the simple eigenvalue of −∆ with homogeneous Dirichlet boundary condition on ∂Ω and let φ(x) be the corresponding eigenfunction with max x∈Ω φ(x) = 1. Assume that there exist d * 2 > 0 and (a 1 , a 2 ) ∈ S 1 such that (U, V ) = (a 1 , a 2 )φ is a solution of d 1 ∆U + h 11 U + h 12 V = 0, d * 2 ∆V + h 21 U + h 22 V = 0 in Ω and U = V = 0 on ∂Ω. Furthermore, assume that If there exist a positive constant k f and a non-negative continuous function k(t) such that lim t→0 k(t) = 0 and for any i = 1, 2, t = 0 and z 1 , z 2 ∈ R, then, emanated from (d * 2 , ∞), there exists a bifurcation branch of {(d 2 , u) | u(x) = (u(x; d 2 ), v(x; d 2 ))} of (1.2) subject to the homogeneous Dirichlet boundary conditions. Theorem 1.1 enables us to employ the method of bifurcation from infinity to study the stationary solutions in wider classes of reaction-diffusion systems. For examples, in some prey-predator models of Holling type II or III, when f i (u, v) do not satisfy (1.3) but the hypotheses of Theorem 1.1. See Sections 2.2 and 2.3 for more details. We remark that different types of bifurcation from infinity can be found in [12].
The third purpose of this paper is to give a plausible reason for the blow-up induced by diffusion, as demonstrated by Mizoguchi, Ninomiya and Yanagida [5]. In such a reaction-diffusion system there exist solutions to blow up, while in the absence of diffusion all the solutions for the system of ordinary differential equations actually converge to the origin as t tends to infinity (see also [4,6]). A natural question we encountered is: What kind of equations do not inherit the global existence of solutions by adding diffusion ? To clarify this question, Ninomiya and Weinberger [9] studied the influence of the linear perturbation on the global existence of solutions to the homogeneous nonlinearity (see also [8]). The authors shown that the linear term can result in blow-up phenomenon when the solutions, induced from the homogeneous nonlinearity, go to infinity as t tends to infinity. They also pointed out that in this case there are stationary solutions which bifurcate from infinity. We extend this investigation to the reaction-diffusion system with p-homogeneous nonlinearity. Under a highly degenerate condition, we show that the bifurcation from infinity leads to the existence of stationary solutions (see Sections 3.1 and 3.2). Along the bifurcation branch, if the stationary solutions are unstable, it is plausible that the unstable manifold of such stationary solutions will reach to infinity which results in the time dependent solution to blow up in finite time. Therefore if a system of ordinary differential equations possesses a global attractor, it seems plausible to create blow-up phenomenon induced by diffusion when the bifurcation from infinity takes place.
2.1. Bifurcation from infinity. In this subsection we apply the theorem for the bifurcation from infinity as in [11,13] to the reaction diffusion system. Let us consider the semilinear system in a bounded domain Ω ⊂ R N with homogeneous Neumann boundary conditions ∂u ∂n = ∂v ∂n = 0 on ∂Ω or homogeneous Dirichlet boundary conditions In this section, we assume that the nonlinear terms f 1 (u, v) and f 2 (u, v) satisfy (1.3), whereas h 1 (u, v) and h 2 (u, v) are linear terms defined by In what follows, we often regard d 2 as a bifurcation parameter. For use of the bifurcation theorem, we assume that has a zero eigenvalue of odd multiplicity at d 2 = d * 2 . More precisely, we assume that, if d 2 lies in a neighborhood of λ * 2 , then there exists an eigenvalue λ(d 2 ) of odd multiplicity such that 1 , ϕ d2 2 ) T represents the corresponding eigenfunction. In order to apply the bifurcation theorems by Rabinowitz [10,11] (2. 2) The linear and nonlinear parts of the right-hand side of (2.2) will be denoted by respectively. Then (2.2) is expressed as Here we remark that L = L d2 and K = K d2 can be regarded as compact mappings from equipped with the norm to E, For simplicity of notation we may also use f 1 (u), h 1 (u) and so on. Then we can obtain the following result similar to [11,Theorem 1.6] or [13] concerning the bifurcation from infinity.
Theorem 2.1. If L d2 has a zero eigenvalue of odd multiplicity when Proof. To use [10, Lemma 1.2] or [11,Lemma 1.2], it suffices to confirm (i) L is compact, (ii) H(w) = o( w ) as w → 0, (iii) H is compact, (iv) 1 is an eigenvalue of L of odd multiplicity when d 2 = d * 2 . The first statement (i) is obvious. The statement (ii) follows from (1.3). The statement (iii) is a direct consequence of (i), (ii) and the definition of H. The last statement follows from the assumption of the eigenvalue of L.
As a consequence of [10, Lemma 1.2], if 1 is an eigenvalue of L of odd multiplicity, then the unbound component of the bifurcation branch meets (d * 2 , ∞). Remark 1. We can use the other parameters such as h ij as a bifurcation parameter. We note that K(0) = 0 is not needed.
Let λ and µ be any eigenvalue of L and L, respectively. Furthermore, let σ be any eigenvalue of −∆ with homogeneous Neumann boundary condition or Dirichlet boundary condition on ∂Ω. The following transversality condition for the simple zero eigenvalue will be used in the next section.
Then the proof of Lemma 2.2 is complete.

2.2.
Bifurcation directionally from infinity. The condition (1.3) requires that f i (u, v) (i = 1, 2) remain bounded along any directions when |(u, v)| → ∞. Such a uniform boundedness on (f 1 , f 2 ) results in the restriction on the application of Theorem 2.1 in considering certain realistic models. To relax the assumption (1.3), we propose a new mathematical framework for the bifurcation from infinity in the next theorem.
Theorem 2.3. Assume that L = L d2 has a simple eigenvalue µ d2 which passes 1 at d 2 = d * 2 transversally. Let ϕ d2 be the eigenfunction of L corresponding to µ d2 and P = P d2 be a projection on the one-dimensional space spanned by ϕ d2 , and set Furthermore, assume that, for any small positive constant α, there exist positive constant δ, M , C K ∈ (0, M/2] and non-negative continuous function k 0 = k 0 (s) satisfying lim s→0 k 0 (s) = 0 such that , a prey-predator model with Holling-type III nonlinearity (2.21) will be introduced in the next sub-section.
Proof. In order to construct a branch of solutions for (2.3), we decompose the unknown function u into the P E component and the QE component as follows: Then we substitute this orthogonal decomposition into equation (2.3), and decompose the equation itself to obtain As the Lyapunov-Schmidt reduction procedure, we first look for the solution w of (2.10) for any fixed s = 0. For the use of the contraction mapping theorem, we introduce the closed subset M = M d2 of E as Here C K is a positive constant in (2.8). Then it can be shown that Φ(s) is the mapping from M to itself if |s| > 0 is sufficiently small. Actually, (2.7) ensures for w E ≤ M . Then it follows from the above inequality and (2.8) that By the assumption lim s→0 k 0 (s) = 0, we may take a constant α 1 ∈ (0, α] to satisfy It follows from (2.12) and (2.13) that and thereby Φ(s) is a contraction mapping from M to itself if 0 < |s| ≤ α 1 . The contraction mapping theorem gives a unique fixed point w * (s) ∈ M(s) which solves (2.10) as follows for any 0 < |s| ≤ α 1 . Since (2.10) implies Hereafter, in the proof, we regard d 2 as a positive unknown number and construct a curve d 2 = d 2 (s) which solves the equation (2.9). For simplicity of notation, we introduce the projection P d2 onto R instead of P d2 by Then (2.9) is rewritten as The function H(s, d 2 ) is continuous in |s| < α 1 and d 2 close to d * 2 . Indeed, it follows from (2.8) and (2.11) that Thus H(s, d 2 ) is continuous in s for |s| < α 1 . For use of the contraction mapping theorem, we set By the assumption that µ d2 passes 1 at d * 2 transversally, it can be verified that for any d 2 , d 2 in the neighborhood of d * 2 . Next we show the Lipschitz continuity of w * with respect to d 2 . To clarify the dependency of d 2 , we write w * (s; d 2 ) instead of w * (s). It follows from (2.7) and (2.14) that for any 0 < |s| ≤ α 1 . This immediately implies that where C 1 , C 2 and C 3 are positive constants independent of d 2 , d 2 and s. Namely, the Lipschitz constant of H with respect to d 2 is sufficiently small near (s, for d ∈ C[−α 1 , α 1 ] . By the transversality of the eigenvalue µ d2 at d 2 * , (2.16) and (2.17) imply that there exists a constant θ ∈ (0, 1) satisfying for d, d ∈ C[−α 1 , α 1 ] and |s| ≤ α 1 , which is retaken smaller if necessary. Since we can choose δ and α 1 so small that η is a contraction mapping onto for any x ∈ Ω. Applying the contraction mapping theorem, we obtain a unique curve d 2 = d 2 (s) as the fixed point of η such that for any x ∈ Ω and |s| < α 1 . This implies the existence of a solution u = ϕ d2(s) /s + w(s) to (2.3) when d 2 is close to d * 2 . Namely, the solution bifurcates from (d * 2 , ∞).
By taking α 1 smaller if necessary, u = ϕ d2 /s + w(s) is positive in Ω when both components of ϕ d2 are positive. If one of the components of ϕ d2 vanishes, then we need to check case by case.
Concerning (2.1) with homogeneous Dirichlet boundary conditions, the following corollary gives a sufficient condition on f 1 and f 2 for the bifurcation from infinity to occur. In what follows in this section, σ represents the simple eigenvalue of −∆ with the Dirichlet boundary condition and φ(x) denotes the corresponding positive eigenfunction with L ∞ normalization, that is, Corollary 1. Assume that L d2 has a simple eigenvalue λ(d 2 ) satisfying λ(d * 2 ) = 0 when d 2 is near d * 2 , and Suppose there exist a positive constant k f and a non-negative continuous function k(t) such that lim t→0 k(t) = 0 and for t = 0, z 1 , z 2 ∈ R, i = 1, 2. Then for (2.1) with homogeneous Dirichlet boundary conditions, there exists a bifurcation branch of solutions Proof. It suffices to show (2.7) and (2.8). Set ϕ(x) = (a 1 , a 2 )φ(x) and w = (w 1 , w 2 ). From (2.19), it follows that for any x ∈ Ω. By the definition of K, we use the green function g corresponding to (−d 1 ∆ + 1) −1 with Dirichlet boundary condition. Then it is sufficient to consider where we simply write f i (a 1 φ/s + w 1 , a 2 φ/s + w 2 ) as f i (ϕ/s + w) for j = 1, 2. The Hopf Lemma implies that ∂φ/∂n = 0 at ∂Ω. The first inequality of (2.19) and the above fact imply that for all s ≥ 0. Using the second inequality of (2.20) yields a similar estimate for (−d 1 ∆ + 1) −1 ∂f i /∂v(ϕ/s + w). Thus (2.7) is verified.
From the third inequality of (2.19), we see that Thus (2.8) holds and the proof is complete.

Examples.
In this section, we use Corollary 1 to show the existence of bifurcation branches from infinity in a couple of the prey-predator models.
2.3.1. Prey-predator model with Holling Type III nonlinearity. The first example is one of the prey-predator models, so called Holling Type III: with homogeneous Dirichlet boundary conditions. Usually this system includes the saturation term, but it is neglected here. To express (2.21) in the form of (2.1), we take Hence lim u→∞ f 1 (u, v) = lim u→∞ f 2 (u, v) = 0 for every v ∈ R, but (1.3) does not hold. When d * 2 σ = a − 1 and d 1 σ = 1, the operator of L has a zero eigenvalue at d 2 = d * 2 . In this situation, (2.18) is also satisfied since h 11 = 1 and h 22 = a − 1. The eigenfunction corresponding to a zero eigenvalue of L is ϕ d2 := (a 1 , a 2 ) T φ where φ is a eigenfunction of a Laplace operator and Here we note that a 2 = 0 since d 1 σ = 1. We now verify (2.19) to confirm the criteria for the bifurcation from infinity as stated in Theorem 2.3 and Corollary 1. For any t = 0, → 0 as t → 0 since a 1 = 0. Since f 2 = −af 1 , the first condition of (2.19) is fulfilled.

2.3.2.
Prey-predator model with Holling Type II nonlinearity. Let us consider the prey-predator model with Holling Type II nonlinearity: with homogeneous Dirichlet boundary conditions. Namely, we take and ϕ = (a 1 , a 2 ) T φ as above. To confirm (2.19) in Corollary 1, we see that Since f 2 = −af 1 , the first condition of (2.19) is fulfilled.
Since ∂f 1 /∂u = −v(1 + u) −2 , then for any t = 0 and z i ∈ R, ∂f 1 ∂u Thus the second condition of (2.19) is fulfilled. The verification of the third condition is similar. Then Corollary 1 deduces that the stationary solutions for (2.21) with Dirichlet boundary conditions bifurcate from infinity at d 2 = d * 2 = (a − 1)/σ. Next we consider the positivity of the solution. By the property of the Green function, we have |w| ≤ Cφ.
3. The case of p-homogeneous nonlinearities. In the previous section, we consider the case where nonlinear term converges to 0 near infinity; there it is assumed that f 1 (u, v) and f 2 (u, v) converge to 0 as u → ∞ (or as u → ∞ as in (2.21)). Instead if we assume that both f 1 (u, v) and f 2 (u, v) vanish in one direction, then nonlinear terms are degenerate and in this case we expect that the lower order terms such as linear ones dominate the dynamics.
First we define p-homogeneous nonlinearity f by requiring for any u ∈ R 2 and t ∈ R where p is a positive integer. Let f 1 and f 2 be phomogeneous nonlinearities. Consider the following reaction-diffusion system with p-homogeneous nonlinear terms: in Ω = (−1, 1) with Dirichlet or Neumann boundary conditions. We assume that there is a constant k * satisfying Then there exist a family of stationary solutions {(s, sk * ) | s ∈ R}. To find stationary solutions, we introduce a new variable k defined by v(x) = ku(x).
Next we consider the stationary solution depending on x. Substituting v(x) = ku(x) into the second equation of (3.1) and removing u from the first equation, we derive the following condition: where k is independent of x. 3.1. p-homogeneous systems. In this subsection we show the existence of stationary solutions of (3.1) that bifurcate from infinity. > 0 for all δ ∈ (0, δ 0 ) or for all δ ∈ (−δ 0 , 0). Then there exist an interval V , a constantk =k(δ) and a functionū =ū(x; δ) such that Note that assumption (iv) implies that k * = 0.
Note that U is not necessarily positive. We also see that infinitely many nodal solutions of (3.5) bifurcate from infinity at the same time.

Perturbed systems.
Hereafter we consider the following reaction-diffusion system: in a domain Ω = (0, 1) with the homogeneous Dirichlet or Neumann boundary condition, where f 1 , f 2 are p-homogeneous and Suppose the stationary solution of (3.6) is of the form (u, ku) then Since ∆(ku) = u∆k + 2∇k · ∇u + k∆u = (∇(u 2 ∇k))/u + k∆u, the second equation is transformed into Assume that k is a constant as in the previous section. If (u, ku) is a solution, the following conditions hold: Next consider an auxiliary equation with Dirichlet or Neumann boundary condition. It is known that the Dirichlet problem has a positive solution for any d 1 > 0 , while the Neumann problem has a positive solution for any small d 1 > 0.
Theorem 3.2. Assume that the hypotheses of Theorem 3.1 are satisfied with the same notation V andk(δ). Then there are parameters h ij = h ij (δ) and a bifurcation branch and U is given by (3.10). (ii) the following condition holds: Proof. As in the proof of Theorem 3.1, there is a k =k(δ) satisfying (3.9). Substituting u = ηU (x) into (3.7) yields Thus the above equation holds by (3.11) and (3.10); that is, is a solution of (3.7) and (3.8). Since η(δ) → ∞ ask(δ) → k * , this family of solutions forms a bifurcation branch emanating from infinity. When (3.12) holds. Namely, h 12 = h 21 = 0, h 11 > 0, h 22 = d 2 h 11 /d 1 , which means that h 1 , h 2 are independent ofk. The condition (3.12) can be relaxed to Inhomogeneous case. In this subsection, we put ε in front of the linear term: in an interval Ω = (−1, 1) with the Neumann boundary condition, where f 1 , f 2 are p-homogeneous and h 1 , h 2 are linear. By assuming the odd symmetry of u and v, we impose the Dirichlet boundary condition Thus we consider the problem (3.13) on the interval Ω = (0, 1).
Without loss of generality, we may assume v(0) ≥ 0. Then v is positive near x = 0.
We omit the proof, since it is easy to check.
Next we consider the perturbed system (3.13) with ε > 0. Here we seek for stationary solutions in the form (u, k(x)u), which is different from the previous two subsections since k is inhomogeneous. It follows that To seek a solution near (ū,kū), we set (u, k) = (ū + ρw,k + z) = (ρ(U + w),k + z).
for simplicity in notation. Since ρ p−1 = d 1 /f 1 (1,k), the system can be rewritten as Therefore where L 1 is given by (3.16) and with z x (0) = z x (1) = 0. Since U (0) = 0, then −(U 2 z x ) x − q z (0; δ)U p+1 z = 0 has a regular singular point. We recall that the Green functions for L 1 and L 2 are denoted by G 1 and G 2 respectively and that the fundamental solutions for L 1 ψ = 0 (resp. L 2 ) are denoted by ψ 11 , ψ 12 (resp. ψ 21 , ψ 22 ). Namely, Lemma 3.7. The followings hold: Note that the norm of the left hand side in the second inequality is C 1 , not C 0 .
Proof. Observe that if y < x ≤ 1. Then Similarly, Thus (3.17) follows. By direct calculation can be treated similarly, we omit the detail and thus (3.18) is established.
Consider the following metric space: with some positive constant η. Using the integration by parts yields for (w, z) ∈ M. It is also denoted by K 5 , namely, G 2y (x, y)(2U (y)w(y) + w(y) 2 )z x (y)dy.

This leads to
Now the proof is complete.
Since K 1 (0, 0) = 0, we see that T is a contraction mapping on M with appropriate constants M 1 , M 2 , ε, for any δ > 0. Thus the existence of non-constant stationary solutions through the bifurcation from infinity has been shown.
As an application of Theorem 3.3, we will see that adding an arbitrarily small linear perturbation into a p-homogeneous system could change bounded solutions to be blow-up.
Thus the orbit through (r 0 cos θ 0 , r 0 sin θ 0 ) is a circle centred on the line u + v = 0 and tangent to the line u = v = 0 at the origin. Since the points of the line u−v = 0 are all equilibria, we see that all solutions of the 3-homogeneous system which is obtained by setting α = 0 in (3.26) are bounded.
On the other hand, we observe that h(π/4) = (π/4) = 0, that h and are positive in the interval (π/4, 3π/4), that Q(π/4) = −α , and that h = 2 , so that ν * = 1/2. Thus Theorem 6.1 of [9] shows that if α < 0, the system (3.26) has solutions which to be blow-up in a finite time. In particular, we have shown that for any fixed ε > 0, all solutions of the system (3.26) with α = 0 are bounded, while if α < 0, some solutions turn out to blow up in finite time. Thus even adding an arbitrarily small linear perturbation into a p-homogeneous system results in blow-up phenomenon.
Mizoguchi, Ninomiya, and Yanagida [5] have shown that the Neumann problem on a bounded domain Ω has the property that all solutions converge to the origin when = 0, but that there are solutions which blow up in a finite time when 0 < < 1.
If we think ofū(t) as the value at x = 0 of a functions which is defined to be 0 at all the other integer points x = k = 0, then the the second difference of this grid function at x = 0 is δ 2ū (t, 0) = −2ū(t, 0). Thus the system in the following example may be thought of as a finite difference analog of the system (3.27) of Mizoguchi, Ninomiya, and Yanagida, but with Dirichlet rather than Neumann boundary conditions. Example 3.11. Consider the system (3.28) In this example, we see that k * = 1. Moreover, it is easily confirmed that µ * = lim