Bifurcation from zero of a complete trajectory for non-autonomous logistic PDEs
Abstract
In this paper we extend the well-known bifurcation theory for autonomous logistic equations to the non-autonomous equation ut − ∆u = λu − b(t)u 2 with b(t) ∈ [b0, B0], 0 < b0 < B0 < 2b0. In particular, we prove the existence of a unique uniformly bounded trajectory that bifurcates from zero as λ passes through the first eigenvalue of the Laplacian, which attracts all other trajectories. Although it is this relatively simple equation that we analyse in detail, other more involved models can be treated using similar techniques.
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Bifurcation from zero of a complete trajectory for non-autonomous logistic PDEs Jos´e A. Langa∗ James C. Robinson† Antonio Su´arez∗ August 5, 2004 Abstract In this paper we extend the well-known bifurcation theory for autonomous logistic equations to the non-autonomous equation ut−∆u=λu −b(t)u2with b(t)∈[b0, B0], 0< b0< B0<2b0. In particular, we prove the existence of a unique uniformly bounded trajectory that bifurcates from zero as λpasses through the first eigenvalue of the Laplacian, which attracts all other trajectories. Although it is this relatively simple equation that we analyse in detail, other more involved models can be treated using similar techniques. Keywords: Non-autonomous differential equations; bifurcation from zero; comparison techniques; complete trajectories Mathematics Subject Classification 2000: 35B05, 35B22, 35B41, 37L05. 1 Introduction The bifurcation behaviour of positive solutions of the autonomous logistic PDE ut−∆u=λu −b(x)u2x∈Ω,with u= 0 on ∂Ω under various conditions on the coefficient b(x) is well-known (see for example Smoller (1983) and references therein). The simplest such result, when b(x)≡ b > 0, guarantees that while λ < λ1(the first eigenvalue of the Laplacian on ∗Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain †Mathematics Institute, University of Warwick, Coventry CV4 7AL. U.K. 1
Ω) all such solutions tend to zero, while for λ > λ1there is a unique positive solution of the equation −∆u=λu −bu2 which attracts all solutions as t→+∞(see also Section 2, below). In this paper we extend this result to the non-autonomous equation ut−∆u=λu −b(t)u2in Ω ×(s, ∞), u= 0 ∂Ω, u(s) = u0in Ω, (1) where Ω is a bounded and regular domain of IRN,λis a real parameter, u0∈ C(Ω) with u0>0 and b∈C(IR) satisfies 0< b0≤b(t)≤B0,for all t∈IR, with B0<2b0. We note here that Hess (1991, Theorem 28.1) proved a similar result (without the condition relating b0and B0) when b(t) is a periodic function. Two previous papers by the current authors (Langa & Su´arez, 2000; Langa et al., 2003) treated related examples in which information could only be obtained ‘in the pullback sense’ (see Section 3); although here we still use a construction based on pullback ideas we focus on attraction forwards in time: to our knowledge this is the first result proving the existence of trajectories of (1) that are attracting in the conventional sense (as t→+∞) that allows for a more general non-autonomous term. Whether the condition B0<2b0is in fact necessary in general, or merely a technical artefact of our proof is unclear. From one point of view this condition is natural, in that it constrains b(t) to be ‘close to autonomous’ in that it cannot fluctuate too wildly. On the other hand the non-autonomous logistic ODE ˙x=λx −b(t)x2with 0 < b0≤b(t)≤B0 has a positive attracting trajectory when λ > 0 for any choice of b0and B0(see also Section 3). 2 Notation and preliminaries 2.1 Results for autonomous equations Before studying equation (1), we first recall various results for the autonomous equation ut−∆u=λu −Au2x∈Ω,with u= 0 on ∂Ω.(2) Given an initial condition u(s) = u0this equation has a unique positive solution uA(t, s;u0) for all t≥s. 2
Denote by λ1the first eigenvalue of the Laplacian on Ω with Dirichlet boundary conditions. Then if λ < λ1we have lim t→+∞uA(t, s;u0) = lim s→−∞ uA(t, s;u0) = 0, while if λ > λ1then lim t→+∞uA(t, s;u0) = lim s→−∞ uA(t, s;u0) = θ[λ,A], where θ[λ,A]is the unique positive solution of ½−∆u=λu −Au2in Ω, u= 0 on ∂Ω. (3) We will need a condition to guarantee that solutions of the linear equation wt−∆w=a(x)win Ω ×(s, ∞), w= 0 ∂Ω, w(s) = u0in Ω. (4) (with a∈L∞(Ω)) tend to zero. To this end, given q∈L∞(Ω) we will denote by λ1(q) the principal eigenvalue of the problem ½−∆w+q(x)w=σw in Ω, w= 0 on ∂Ω (note that λ1=λ1(0)). If λ1(−a)<0 then w→ ∞ as t→+∞, while if λ1(−a)>0 then w→0 as t→+∞. It is shown by L´opez-G´omez (1996, Theorem 2.5) that if there exists a function ψthat is positive within the interior of Ω and satisfies −∆ψ+qψ > 0 (5) then λ1(q)>0. 2.2 Existence, uniqueness, and regularity properties Following arguments due to Mora (1983) it is possible to show that solutions of (1) with continuous initial conditions exist and are unique, and enjoy parabolic smoothing. We denote by Xthe space C0 0(Ω) of all continuous functions on Ω that are zero on ∂Ω, equipped with its natural norm k·k∞. Theorem 2.1 Given an initial condition u(s) = us∈X, then there exists a unique solution u(t, s;us)of (1), which satisfies u(t, s;us)∈C0([s, T]; X) for each T > s, and depends continuously on us. Furthermore for each t > s and k > 0the solution operator S(t, s)is a bounded map from Xinto Ck(Ω). 3
Note that in particular the solution operator S(t, s) defined by S(t, s)us=u(t, s;us) is a continuous and compact operator from Xinto itself. 2.3 Order-preserving properties Many of our arguments rely on the order-preserving nature of equation (1), and the resulting possibility of comparing solutions to those of suitable autonomous equations. The following comparison principle, which in particular shows that solutions of (1) preserve the order of their initial conditions, is key to all that follows. We say that u≥vif u(x)≥v(x) everywhere in Ω. Lemma 2.2 Denote by u(t, s;u0)the solution of ut−∆u=λu −b(t)u2u(s) = u0 and by v(t, s;v0)the solution of vt−∆v=λv −β(t)v2v(s) = v0. Then u0≥v0and b(t)≤β(t)⇒u(t, s;u0)≥v(t, s;v0)for all t≥s. In particular if u(t, s;us)denotes the solution of (1) with u(s) = usthen (i) if us≥0then u(t, s;us)≥0for all t≥s, and (ii) when b0≤b(t)≤B0we have uB0(t, s;u0)≤u(t, s;u0)≤ub0(t, s;u0),(6) where uβis the solution of the autonomous equation that has b(t)≡β. We will deduce this as a corollary of a more general nonlinear comparison principle. Proposition 2.3 (Comparison principle) Suppose that uand vare C2in space and C1in time, (i) f(u, t)≥g(u, t), (ii) f(u+s, t)−f(u, t)≤Ls for all 0< s < δ for some δ > 0, (ii) u|∂Ω≥v|∂Ω, 4
(iii) u0≥v0, and (iv) ut−∆u−f(u, t)≥vt−∆v−g(v, t). The u(x, t)≥v(x, t)for all t≥0. (Proposition 2.3) Following Walter (2002) we first prove Nagumo’s Lemma, in which most of the inequalities are strict: suppose that (i) f(u, t)≥g(u, t), (ii) u|∂Ω> v|∂Ω, (iii) u0> v0, and (iv) ut−∆u−f(u, t)> vt−∆v−g(v, t). Then u(x, t)> v(x, t) for all t≥0. Indeed, consider the difference w(x, t) = u(x, t)−v(x, t). Then if the result does not hold there exists a pair (x0, t0) with x0∈int Ω such that w(x0, t0) = 0,∆w(x0, t0)≥0,and wt(x0, t0)<0. Equivalently, at (x0, t0) we have u(x0, t0) = v(x0, t0),∆u(x0, t0)≥∆v(x0, t0),and ut(x0, t0)≤vt(x0, t0). It follows that [ut−∆u−f(u, t)](x0, t0)] ≤[vt−∆v−f(v, t)](x0, t0), contradicting (iv). In order to prove the result as stated we now consider ˜u(x, t) = u(x, t)+²eLt. Then ˜u0> u0≥v0,˜u|∂Ω> u|∂Ω≥v|∂Ω, and ˜ut−∆˜u−f(˜u, t) = ut+L²eLt −∆u−f(u+²eLt, t) > ut−∆u−f(u, t), since f(u+²eLt, t)< f(u, t) + L²eLt. Nagumo’s lemma implies that ˜u(x, t)> v(x, t) for all t≥0, and the result holds on taking the limit as ²→0. (Lemma 2.2). We note from the regularity results in Theorem 2.1 that uand vsatisfy the smoothness requirements for all s > t. Set f(u, t) = λu −b(t)u2 and g(u, t) = λv −β(t)v2; clearly f(u, t)≥g(u, t) if b(t)≤β(t), and for u, s ≥0 f(u+s, t)−f(u, t) = λs −b(t)[s2+ 2us]< λs. Finally we introduce sub- and supertrajectories as a generalisation of sub- and super- and equilibria in Hess (1991); see also Chueshov (2001) or Langa and Su´arez (2002). 5
Definition 2.4 We call u(u) : R→Xa sub-trajectory (super-trajectory) of (1) if it satisfies u(t, s;u(s)) ≥u(t)for all t≥s(u(t, s;u(s)) ≤u(t)for all t≥s). 3 An attracting trajectory bifurcating from zero In this section we prove our main result, namely that while all solutions tend to zero for λ < λ1, when λ > λ1there exists a unique complete trajectory, i.e. a u:R→Xsuch that u(t, s;u(s)) = u(t) for all t≥s, that is bounded above and below, and that this trajectory is attracting. Note that it is clear that when b(t)≥0 and λ < λ1, the solution of (1) tends to zero as t→+∞. We therefore concentrate on the behaviour of solutions for λ > λ1. First we show that there exists at least one complete trajectory that is bounded above and below. The argument showing the uniqueness of this trajectory also shows that it attracts all other trajectories as t→+∞. It is interesting to remark that although we are seeking a trajectory that attracts as t→+∞, the construction in fact involves the notion of ‘pullback attraction’. This can be clearly illustrated by considering the non-autonomous logistic ODE ˙x=λx −b(t)x2with λ > 0 and 0 < b0≤b(t)≤B0.(7) For the initial condition x(s) = xsthis equation admits the explicit solution x(t, s;xs) = eλt eλsx−1 s+Rt seλrb(r) dr. Fixing xsand letting s→ −∞ we obtain the ‘pullback limit’ x∗(t) = eλt Rt −∞ eλrb(r) dr, which is a complete trajectory of (7). (The lower bound on b(t) ensures that x∗(t) is well-defined, while the upper bound ensures that it is bounded away from zero.) While it is clear by construction that x∗(t) is ‘pullback attracting’, i.e. that lim s→−∞ x(t, s;xs) = x∗(t) 6
for any xs, it can also be shown by direct calculation that |x(t, s;xs)−x∗(t)| → 0 as t→+∞ for any choice of sand xs, i.e. that x∗(t) attracts all trajectories forwards in time. Our proof also uses this pullback technique to find an appropriate candidate solution that will attract forwards in time. Theorem 3.1 Suppose that λ>λ1and B0<2b0. Then there exists a unique complete trajectory u(t)for (1) that satisfies θ[λ,B0]≤u(t)≤θ[λ,b0]for all t∈R. (8) This trajectory attracts all other positive solutions as t→+∞, i.e. lim t→+∞ku(t, s;us)−u(t)k∞= 0. We note here that in fact the proof shows more, namely that the trajectory u(t) is uniformly attracting forwards in time lim t→+∞sup s∈R ku(t+s, s;us)−u(t+s)k∞= 0. This implies uniform ‘pullback convergence’ (cf. Cheban et al., 2002), lim s→+∞sup t∈R ku(t, t −s;us)−u(t)k∞= 0. Observe that u(t) := θ[λ,B0]and u(t) := θ[λ,b0], are sub and super-trajectories of (1). Indeed, for uwe have to prove that u(t, s;u(s)) ≥u(t), or equivalently θ[λ,B0]≤u(t, s;θ[λ,B0]), which holds since θ[λ,B0]is a sub-solution of (1) with u0=θ[λ,B0]. We now consider the ‘pullback limit’ u(t) = lim s→+∞u(t, t −s;θ[λ,b0]),(9) following the argument in Langa & Su´arez (2002). To show that this limit exists, observe that u(t, s;θ[λ,b0]) = S(t, s)θ[λ,b0] =S(t, s +²)S(s+², s)θ[λ,b0] ≤S(t, s +²)θ[λ,b0], 7
since θ[λ,b0]is a super-solution. It follows that u(t, t−s;θ[λ,b0]) is pointwise nonincreasing in s. Since it is bounded below by θ[λ,B0]this sequence converges pointwise (in x) to a limit u∗(x, t). Now we also have S(t, s)θ[λ,b0]=S(t, t −1)[S(t−1, s)θ[λ,b0]], and we know that θ[λ,B0]≤S(t−1, s)θ[λ,b0]≤θ[λ,b0], Therefore S(t, s)θ[λ,b0]∈S(t, t −1)I, where Iis the bounded set {u∈X:θ[λ,B0]≤u≤θ[λ,b0]}. Since S(t, t −1) is a compact operator from Xinto itself (see Theorem 2.1), for each fixed twe know that S(t, s)θ[λ,b0]lies in a compact subset of X. It follows that S(t, s)θ[λ,b0]converges uniformly (in x) to u∗(x, t). Suppose not; then there exists an ² > 0 and a sequence sn→ −∞ such that kS(t, sn)θ[λ,b0]−u∗(x, t)k∞> ². (10) But by compactness of S(t, t −1)Ithere must be a subsequence snjthat converges uniformly to some v∗(x). Since S(t, snj)θ[λ,b0]→u∗(x, t) pointwise, we must have v∗(x) = u∗(x, t), contradicting (10). This trajectory u(t) is a complete trajectory, since S(t, s) is continuous from Xinto itself: S(t, τ)u(τ) = S(t, τ) lim s→−∞ S(τ, s)θ[λ,b0] = lim s→−∞ S(t, τ)S(τ, s)θ[λ,b0] = lim s→−∞ S(t, s)θ[λ,b0] =u(t), and it is clear that u(t) satisfies θ[λ,B0]≤u(t)≤θ[λ,b0]for all t∈R. In order to prove both the uniqueness of this bounded trajectory and its forwards attraction property we consider two solutions of (1), u(t, s;us) and v(t, s;vs) with us, vs∈ V+and us, vs6= 0, and show that ku(t, s;us)−u(t, s;vs)k∞→0,as t→+∞or s→ −∞.(11) 8
Since (1) is order-preserving it follows that p(t, s;ps)≤u(t, s;us), v(t, s;vs)≤q(t, s;qs), where ps:= min{us, vs}and qs:= max{us, vs} (p(·) and q(·) are also solutions of (1)). Without loss of generality we will assume that us> vsand hence that w:= u−v > 0 for all t≥s. The function wsatisfies the equation wt−∆w=λw −b(t)(u+v)win Ω ×(s, ∞), w= 0 on ∂Ω, w(s) = us−vs>0 in Ω. (12) Now note that for any ε, it follows from (6) and the forward behaviour of the autonomous logistic equation, that there exists Tεsuch that for t−s≥Tε we have θ[λ,B0]−ε≤u(t, s;us), v(t, s;vs)≤θ[λ,b0]+ε, (13) and therefore for such tand s wt−∆w≤w(λ−2b0(θ[λ,B0]−ε)). We now show that λ1(2b0θ[λ,B0])> λ, (14) for which it suffices to show that λ1(2b0θ[λ,B0]−λ)>0.(15) But it is not hard to prove that θ[λ,B0]>0 is a supersolution of −∆u+ (2b0θ[λ,B0]−λ)u= 0 as required by (5). Indeed, −∆θ[λ,B0]+ (2b0θ[λ,B0]−λ)θ[λ,B0]= (2b0−B0)θ2 [λ,B0]>0 since 2b0> B0. Choosing εsufficiently small that λ1(2b0θ[λ,B0])> λ + 2εb0(16) and using the corresponding value of Tεin (13), the convergence property (11) follows. 9