Large Solutions for a System of Elliptic Equations Arising from Fluid Dynamics

Transcription

1 Lage Solutions fo a System of Elliptic Equations Aising fom Fluid Dynamics J. I. Díaz, M. Lazzo and P. G. Schmidt Abstact. This pape is concened with the elliptic system v = φ, φ = v 2, (.1) posed in a bounded domain Ω R N, N N. Specifically, we ae inteested in the existence and uniqueness o multiplicity of lage solutions, that is, classical solutions of (.1) that appoach infinity at the bounday of Ω. Assuming that Ω is a ball, we pove that the system (.1) has a unique adially symmetic and nonnegative lage solution with v() = (obviously, v is detemined only up to an additive constant). Moeove, if the space dimension N is sufficiently small, thee exists exactly one additional adially symmetic lage solution with v() = (which, of couse, fails to be nonnegative). We also study the asymptotic behavio of these solutions nea the bounday of Ω and detemine the exact blow-up ates; those ae the same fo all adial lage solutions and independent of the space dimension. Ou investigation is motivated by a poblem in fluid dynamics. Unde cetain assumptions, the unidiectional flow of a viscous, heat-conducting fluid is govened by a pai of paabolic equations of the fom v t v = θ, θ t θ = v 2, (.2) supplemented by suitable initial and bounday conditions; v and θ epesent the fluid velocity and tempeatue, espectively. We conjectue that the solutions of (.2) may blow up in finite time and that this behavio is elated to the existence of lage solutions of the associated elliptic system (that is, the system (.1) with φ = θ ). Key wods. Elliptic system, bounday blow-up, lage solutions, adial solutions, existence and multiplicity, asymptotic behavio. AMS subject classifications. 35J6, 35J55, 35Q35. Abbeviated title: Lage Solutions fo an Elliptic System. Depatamento de Matemática Aplicada, Univesidad Complutense de Madid, Avenida Complutense s/n, 284 Madid, Spain; ji diaz@mat.ucm.es. Dipatimento di Matematica, Univesità di Bai, via Oabona 4, 7125 Bai, Italy; lazzo@dm.uniba.it. Depatment of Mathematics, Pake Hall, Aubun Univesity, AL , USA; pgs@cam.aubun.edu. 1

2 1 Intoduction and Main Results This pape is a contibution to the study of explosive behavio in cetain systems of elliptic and paabolic PDEs. Ou investigation is motivated by a long-standing question egading the dynamics of a viscous, heat-conducting fluid. In geneal, the flow of a viscous, heat-conducting fluid is govened by a system of balance equations fo momentum, mass, and enegy. In the socalled Boussinesq appoximation, this system educes to the Navie-Stokes equations fo an incompessible fluid, along with a heat equation; these equations ae nonlinealy coupled though the buoyancy foce and viscous heating. If viscous heating (that is, the poduction of heat due to intenal fiction) is neglected, the esulting initial and bounday-value poblems ae well posed in the same sense as fo the classical Navie-Stokes equations without themal coupling; but if viscous heating is taken into account, wellposedness is an open question. In fact, we conjectue that the solutions, in this case, may exhibit explosive behavio. Such behavio would have seious implications fo the viability of the Boussinesq appoximation in situations whee viscous heating cannot be neglected. To addess this issue, we study a simple pototype poblem, which can be physically justified by consideing a unidiectional flow, independent of distance in the flow diection: v t v = θ, θ t θ = v 2. (1.1) Hee, v (the velocity) and θ (the tempeatue) ae scala functions of time t and position x; the spatial vaiable x vaies ove a bounded domain Ω R N with N N (N = 2 in the physically elevant case, whee Ω is the cosssection of the flow channel). The souce tems θ and v 2 epesent the buoyancy foce and viscous heating, espectively. The system (1.1) must be supplemented by suitable initial conditions at time t = and bounday conditions on the bounday Ω of the domain Ω (fo example, a homogeneous Diichlet condition fo v and a homogeneous Neumann condition fo θ if the walls of the flow channel ae impemeable and themally insulated). Note that we cannot hope to find weak solutions of the esulting initialbounday value poblem in the usual Hilbet-space setting: if v takes values in H 1 (Ω), then the ight-hand side of the second equation in (1.1) maps, a pioi, only into L 1 (Ω). Howeve, local-in-time existence and uniqueness of a stong solution can be established by means of semigoup theoy in a suitable L p -space setting (details will appea in a fothcoming publication). We conjectue that the solution may blow up in finite time (in the sense that 2

3 a suitable nom of (v, θ) appoaches infinity as t T, fo some T > ) and that this behavio is linked to the existence of lage solutions of the associated elliptic system v = θ, θ = v 2, (1.2) posed in the domain Ω. By a lage solution of (1.2) we mean a classical solution that blows up at the bounday of Ω, that is, (v(x), θ(x)) as dist(x, Ω). Note that the θ-component of any solution of (1.2) is supehamonic in Ω and thus, cannot appoach at the bounday (by the maximum pinciple); fo a simila eason, v and θ cannot simultaneously appoach at the bounday. We theefoe expect any lage solution (v, θ) of (1.2) to satisfy v(x) and θ(x) as dist(x, Ω). Hencefoth, we assume that Ω is a ball in R N, centeed at the oigin; that is, Ω = BR N () fo some R >. Fo convenience, we intoduce the function φ = θ and seek adially symmetic lage solutions of the system { v = φ φ = v 2 in BR N (), (1.3) that is, adial solutions with (v(x), φ(x)) as x R. Remak 1.1 The system (1.3) has a scaling popety that we will exploit epeatedly. Suppose (v 1, φ 1 ) is a (lage) solution of (1.3) in B N R 1 (). Fo λ (, ), let R λ := λ 1 R 1. Fo x B N R λ (), define v λ (x) := λ 2 v 1 (λx), φ λ (x) := λ 4 φ 1 (λx). Then (v λ, φ λ ) is a (lage) solution of (1.3) in B N R λ (). Remak 1.2 If (v, φ) is a (lage) solution of (1.3), then so is (v + c, φ), fo any constant c R. Thus, we may estict attention to solutions with v() =. We will now state ou main esults, the fist of which guaantees the existence of a unique (up to a shift in v) adially symmetic and nonnegative lage solution fo any space dimension. Theoem 1.3 Fo evey N N and R >, the system (1.3) has a unique adially symmetic lage solution (v, φ) with v() = and φ() >. Both components of this solution ae inceasing functions of the adial vaiable. 3

4 If the space dimension is sufficiently small, the system (1.3) has exactly one additional adially symmetic lage solution with v() =, which, of couse, fails to be nonnegative. Theoem 1.4 Fo evey N N with N 1 and evey R >, the system (1.3) has a unique adially symmetic lage solution (v, φ) with v() = and φ() <. The φ-component of this solution is an inceasing function of the adial vaiable, while the v -component is deceasing to a negative minimum and inceasing theeafte. Let us note that the bound on N in the above esult is not shap. In fact, based on numeical evidence (see Remaks 3.4 and 4.4), we conjectue that the solution of Theoem 1.4 exists if, and only if, N 14. Regading the asymptotic behavio of the solutions, we find that, as expected, both components appoach infinity, and we detemine the exact blow-up ates; those ae the same fo all adially symmetic lage solutions and independent of the space dimension. Hee and in the sequel, we wite f(x) g(x) if f, g : B N R () R satisfy f(x)/g(x) 1 as x R. Theoem 1.5 Let (v, φ) be a adially symmetic lage solution of (1.3), fo a given N N and R >. Then, as x R, v(x) 3 (R x ) 2 and φ(x) 18 (R x ) 4. The study of explosive behavio, be it finite-time blow-up in evolutionay poblems o bounday blow-up in stationay poblems, has a long histoy, going back to seminal wok by Kelle [1] and Osseman [15] in the 195 s; we efe to the papes [2, 3, 5, 18] and the extensive efeences theein. Howeve, vitually all of the existing liteatue is concened with scala equations. Coupled systems of equations have been attacked only ecently; see fo example [4, 6, 7, 11]. Due to the lack of vaiational stuctue and compaison pinciples, methods that have poven successful fo scala equations will, in geneal, fail to be useful fo systems, even if the expected esults ae analogous. Fo example, ou existence and multiplicity esult fo the elliptic system (1.3) (existence of one lage nonnegative solution fo any space dimension, existence of a second lage solution fo sufficiently small space dimension) is analogous to a esult by McKenna, Reichel and Walte [12] fo a class of scala equations with vaiational stuctue. Howeve, ou method of poof is entiely diffeent, and ou esult appeas to be the fist of its kind fo an elliptic system. We expect that ou wok, while cuently 4

5 focussed on a vey specific poblem, will lead to geneal insights and new methods with potential applications to a much wide class of elliptic and paabolic systems. The pape is oganized as follows. In Section 2 we educe the poblem to the study of a system of fist-ode ODEs, establish some basic popeties of its solutions, and pove the existence and uniqueness of nonnegative lage adial solutions fo the system (1.3); Theoem 1.3 is an immediate consequence of Poposition 2.5. Section 3 is devoted to the poof of Theoem 1.4 (existence of a second lage adial solution fo sufficiently small space dimension), which follows fom Lemma 3.1(a) and Poposition 3.3. This section also includes a discussion of numeical expeiments, suggesting a shape vesion of Theoem 1.4, and emaks on a elated Diichlet poblem. In Section 4 we analyze the asymptotic behavio of lage adial solutions; Theoem 1.5 follows fom Poposition 4.1, whose poof elies on dynamical-systems theoy, applied to an asymptotically autonomous and coopeative ODE system in R 3. Acknowledgements. This poject was stated while M. Lazzo and P. G. Schmidt wee visiting the Univesidad Politécnica de Madid and the Univesidad Complutense de Madid; both wish to expess thei gatitude to J. F. Padial and J. I. Díaz fo thei geneous suppot. Most of the wok was done while M. Lazzo was visiting Aubun Univesity; she would like to thank the faculty and staff of the Depatment of Mathematics fo thei kind hospitality. Both M. Lazzo and P. G. Schmidt ae gateful fo stimulating discussions with W. Shen and G. Hetze at Aubun. The eseach of J. I. Díaz was patially suppoted by the pojects REN C3 of the DGISGPI (Spain) and RTN HPRN-CT of the Euopean Union. 2 Peliminaies and Nonnegative Lage Solutions Given N N and R >, adially symmetic solutions of the system (1.3) coespond to solutions of the ODE system v + N 1 φ + N 1 v = φ φ = v 2 in (, R) (2.1) with v () = φ () = ; lage solutions ae those with (v(), φ()) as R. In view of Remak 1.2, we may impose the initial condition v() =. Finding adially symmetic lage solutions of the system (1.3) is theefoe equivalent to finding initial conditions φ() = p such that the 5

6 solution of the Cauchy poblem v + N 1 φ + N 1 v = φ, v() =, v () = φ = v 2, φ() = p, φ () = (2.2) exists on the inteval [, R) and blows up at R. Despite the singulaity at = fo N > 1, the Cauchy poblem (2.2) is well posed. Indeed, fo evey p R, thee exists a unique maximal solution, which depends continuously on p (in the usual sense); see Lemma 2.3 below fo details. Remak 2.1 The scaling popety of the system (1.3), as descibed in Remak 1.1, and the well-posedness of (2.2) imply that all solutions of the Cauchy poblem with p > (p < ) ae escalings of the solution with p = 1 (p = 1). Indeed, if (v 1, φ 1 ) is the maximal solution with initial value p = 1 (p = 1), then the maximal solution with initial value p > (p < ) is given by (v λ, φ λ ) with λ = p 1/4. Consequently, if the maximal solution with initial value p = 1 (p = 1) blows up at R 1, then the maximal solution with initial value p > (p < ) blows up at R p = p 1/4 R 1. Remak 2.2 In light of the peceding emak, it is clea that the elliptic system (1.3) has lage adial solutions, fo any given R >, if and only if the solutions of the Cauchy poblem (2.2) with p = ±1 exhibit finite-time blow-up. Moe pecisely, (1.3) has exactly one lage adial solution with v() = and φ() > if and only if the solution of (2.2) with p = 1 blows up in finite time; (1.3) has exactly one lage adial solution with v() = and φ() < if and only if the solution of (2.2) with p = 1 blows up in finite time. In paticula, (1.3) cannot have moe than two lage adial solutions. The Cauchy poblem (2.2) is equivalent to the fist-ode system v = w, v() =, w + N 1 w = φ, w() =, φ = ψ, φ() = p, ψ + N 1 ψ = w 2, ψ() =. 6

7 Obviously, we can eliminate v and dop the fist equation and initial condition; v is ecoveed fom w via anti-diffeentiation. Futhemoe, we may eplace N 1 (N N) with a continuous paamete µ R +. Thus, we ae led to the Cauchy poblem w + µ w = φ, w() =, φ = ψ, φ() = p, (2.3) ψ + µ ψ = w2, ψ() =. Lemma 2.3 Fo evey µ R + and p R, the Cauchy poblem (2.3) has a unique maximal, that is, noncontinuable, solution (w, φ, ψ) C 1 ([, R), R 3 ), fo some R (, ]. If R <, then (w(), φ(), ψ()) as R. Moeove, (w, φ, ψ) depends continuously on µ and p. Poof. What we claim is that, despite the singulaity at = in the case µ >, the Cauchy poblem (2.3) has the usual, well-known popeties of a egula initial-value poblem in R 3. Since we could not find a geneal esult in the liteatue that would cove ou poblem, we povide a few emaks on the poof. Note that the fist equation in (2.3) can be witten as ( µ w) = µ φ. Togethe with the initial condition w() =, this is equivalent to the integal equation ( s ) µ w() = φ(s) ds. (2.4) Similaly, the emaining diffeential equations and initial conditions in (2.3) ae equivalent to the integal equations and φ() = p + ψ() = ( s ψ(s) ds (2.5) ) µw 2 (s) ds. (2.6) Since we have < s/ < 1 fo < s <, the singula tem (s/) µ does not cause any difficulties in poving the existence and uniqueness of a solution (w, φ, ψ) C([, ε], R 3 ) of Equations (2.4) (2.6), fo some ε >, by means of the contaction mapping pinciple. Clealy, w, φ and ψ ae continuously diffeentiable on (, ε] and satisfy the diffeential equations and initial conditions in (2.3). In fact, all thee components ae continuously diffeentiable on the closed inteval [, ε]. This is obvious fo φ, but less so 7

8 fo w and ψ. Note, howeve, that ( lim w () = lim φ() µ ) + + w() = p µ lim 1 = p µ lim + µ ( s ) µ φ(s)ds s µ µ φ() φ(s)ds = p µ lim + (µ + 1) µ φ() = p µ lim + µ + 1 = p µ p µ + 1 = p µ + 1, whee we used l Hospital s ule to get the fouth equality. This implies that w C 1 ([, ε], R) with w () = p/(µ + 1). Similaly, one shows that ψ C 1 ([, ε], R) with ψ () =. Once existence and uniqueness of a local C 1 -solution ae established, the emaining claims about maximal continuation and continuous dependence on paametes and initial data can be poved in the same way as fo egula initial-value poblems. Lemma 2.4 Let (w, φ, ψ) C 1 ([, R), R 3 ) be the maximal solution of (2.3), fo some µ R + and p R, p. Then the function φ is stictly inceasing on [, R), and L := lim R φ() is eithe o. In fact, (a) if L <, then R = and L = ; (b) if L =, then R < and w(r ) = φ(r ) = ψ(r ) =. Poof. Taking into account the equations and initial conditions in (2.3), it is easy to see that the function ψ µ () := µ ψ() is stictly inceasing on [, R). As a consequence, ψ µ (and thus, ψ ) is positive on (, R), and this implies that φ is stictly inceasing on [, R), with L := lim R φ() (p, ]. (a) Assume L <, that is, φ is bounded. By (2.4), w() gows at most linealy with, and by (2.6), ψ() gows no faste than 3. In paticula, (w(), φ(), ψ()) cannot go to infinity in finite time. Thus, R =. Now suppose that L. If L >, choose a numbe > such that φ() L/2 fo evey. It follows that ( s ) µ L ( s ) µ w() φ(s) ds + ds 2 fo evey, and we conclude that lim w() = (note that the last integal is of ode ). If L <, we infe in a simila way that lim w() =. In any case, we can choose a numbe 1 > such that w 2 () 1 fo evey 1. As a consequence, 1 ( s ) µ ( s ) µ ψ() w 2 (s) ds + ds 8 1

9 fo evey 1, and so, lim ψ() =. But this implies L = lim φ() = p + lim ψ(s) ds =, a contadiction. It follows that L =. (b) Assume L = and, by way of contadiction, suppose that R =. Then thee exist c, > such that φ() c fo evey, and as in the poof of Pat (a), it follows that w() and ψ() as. In paticula, we can choose > such that w(), φ(), ψ() > fo evey. Define η := w φ ψ. Then we have whee Q is defined by η = φ 2 ψ + wψ 2 + w 3 φ 2µ w φ ψ = Q ( w, φ, ψ ) η 13/12 2µ η in [, ), (2.7) Q(x, y, z) := y2 z + xz 2 + x 3 y (xyz) 13/12, fo x, y, z >. Note that Q = Q 1 + Q 2 + Q 3, with Q 1 := y 2 z (xyz) 13/12, Q 2 := xz 2 (xyz) 13/12, Q 3 := x 3 y (xyz) 13/12. It is easy to see that Q 5 1 Q4 2 Q3 3 1, which implies that max(q 1, Q 2, Q 3 ) 1. Hence, we have Q(x, y, z) 1 fo all x, y, z >, and (2.7) yields ( η η η 1/12 2µ ) in [, ). (2.8) Recall that w(), φ(), ψ() as and choose such that η( ) > (2µ/ ) 12. Then the maximal solution ζ of the initial-value poblem ( ζ = ζ ζ 1/12 2µ ), ζ( ) = η( ) appoaches infinity in finite time. But due to (2.8), ζ is bounded fom above by η on [, ). This is a contadiction, and it follows that R is finite. In ode to pove ou last claim, we fist note that both w and ψ have (pope o impope) limits as R. Indeed, since φ is eventually positive, the function w µ () := µ w() is eventually inceasing and so, has a 9

10 limit as R. As we obseved ealie, the same holds fo the function ψ µ := µ ψ(). Since R is finite, it follows that w() and ψ(), too, have limits as R. Moeove, since R is finite, all thee of the functions w, φ, ψ would be bounded if one of them wee. But φ is unbounded (by assumption) and so, w and ψ ae unbounded as well. Clealy, this implies that w(r ) = φ(r ) = ψ(r ) =. Poposition 2.5 Fo evey µ R +, the maximal solution of (2.3) with p = 1 blows up in finite time. Poof. Fix µ R + and let (w, φ, ψ) C 1 ([, R), R 3 ) be the maximal solution of (2.3) with p = 1. Accoding to Lemma 2.4, φ is inceasing with L := lim R φ() {, }. Since φ() >, we have L =, and Pat (b) of the lemma implies that R is finite. Poof of Theoem 1.3. Thanks to Remak 2.2, the peceding poposition guaantees that the system (1.3), fo abitay N N and R >, has exactly one lage adial solution (v, φ) with v() = and φ() >. By Lemma 2.4, φ is a stictly inceasing function of the adial vaiable, and the same then holds fo v. (Note that, by Lemma 2.4, φ() appoaches infinity as R, and so do v () and φ (). That the same holds fo v() is not clea yet, but will be shown in Section 4.) The figues below show computed pofiles of the nonnegative lage adial solutions (v, φ) of the system (1.3), with R = 1, fo two values of the space dimension N (see Remak 4.4 fo comments on the numeical method) v phi v phi Fig. 1: N=1 Fig. 2: N=6 1

11 3 Existence of a Second Lage Solution To pove Theoem 1.4, we need to know fo which values of µ (if any) the maximal solution of (2.3) with p = 1 blows up in finite time. Lemma 3.1 Fo µ R +, let R µ denote the exit time of the maximal solution of (2.3) with p = 1. Define M := {µ R + : R µ < } and µ := sup{µ R + : [, µ] M} (with the undestanding that sup = and sup A = if A R + is unbounded). (a) The set M contains the inteval [, µ ). That is, fo evey µ [, µ ), the maximal solution of (2.3) with p = 1 blows up in finite time. (b) The set M is open in R +, and µ is not an element of M. Poof. Pat (a) is clea by the definition of M and µ. As to Pat (b), ecall fom Lemma 2.4 that R µ < if and only if the φ-component of the coesponding maximal solution of (2.3) is eventually positive; since the solution depends continuously on µ, this popety is stable unde small vaiations of µ. Thus, the set M is open in R +, and it follows that µ is not an element of M. Remak 3.2 It is easy to see that the set M contains the inteval [, 1]; that is, we have µ > 1. Indeed, suppose that R µ = fo some µ [, 1] and let (w, φ, ψ) C 1 ([, ), R 3 ) be the coesponding maximal solution of (2.3) with p = 1. Lemma 2.4 implies that φ() as ; thus, ψ(s) ds = 1, due to (2.5). On the othe hand, since ψ µ () := µ ψ() is stictly inceasing fo, we have c := ψ µ (1) > and ψ() = ψ µ () µ c µ fo all 1; thus, ψ(s) ds c 1 s µ ds =. The contadiction shows that R µ < fo evey µ [, 1]. Note that Remak 3.2 guaantees the existence of a second lage adial solution of the system (1.3) at least fo space dimensions 1 and 2. In the following, we will establish a-pioi estimates to impove the lowe bound on the numbe µ. In view of Lemma 2.4, the maximal solution (w, φ, ψ) of the Cauchy poblem (2.3) with p = 1 blows up in finite time if and only if φ cosses zeo at some point >. Due to the scaling popety of the system (see Remak 2.1), this happens if and only if thee exists a (necessaily negative and unique) initial value p such that the φ-component of the coesponding maximal solution of (2.3) cosses zeo at = 1. In othe wods, the maximal 11

12 solution of (2.3) with p = 1 blows up in finite time if and only if the bounday-value poblem w + µ w = φ, w() =, φ = ψ, φ(1) =, (3.1) ψ + µ ψ = w2, ψ() =, has a (necessaily unique) nontivial solution. Clealy, (3.1) can be witten as a paamete-dependent fixed-point equation of the fom u = T (µ, u) (3.2) in X := C([, 1], R 3 ), whee T : R + X X is a completely continuous opeato, defined by ( ( T (µ, u)() := s ) µ 1 φ(s) ds, ψ(s) ds, ( s ) µw 2 (s) ds), fo µ R +, u = (w, φ, ψ) X, and [, 1]. Poposition 3.3 Fo evey µ [, µ ), with µ as in Lemma 3.1, Equation (3.2) has a unique nontivial solution u µ X. The function µ u µ is continuous; its gaph C := {(µ, u µ ) : µ [, µ )} is unbounded in R + X, and the numbe µ is geate than 9. Poof. The peceding discussion and Lemma 3.1(a) imply that fo evey µ [, µ ), Equation (3.2) has a unique nontivial solution u µ X. The continuity of the function µ u µ (as a mapping fom [, µ ) into X ) is easily veified, using the fact that the solution of the Cauchy poblem (2.3) depends continuously on µ, along with the scaling popety. Hence, the gaph C := {(µ, u µ ) : µ [, µ )} is a continuous cuve in R + X. To pove the emaining assetions, we need to establish lowe and uppe a-pioi bounds fo the nontivial solutions u µ = (w µ, φ µ, ψ µ ) of (3.2). Stating with the tivial estimate φ µ () φ µ () fo 1, we obtain a lowe bound fo w µ () = ( s ) µ φµ (s) ds, then an uppe bound fo ψ µ () = ( s ) µw 2 µ (s) ds; indeed, w µ () φ µ() µ + 1 and ψ µ() φ µ () 2 (µ + 1) 2 (µ + 3) 3 fo 1. 12

13 Using the last estimate and the fact that φ µ () = 1 ψ µ(s) ds, we infe that φ µ () 4(µ + 1) 2 (µ + 3) 12 fo all µ. (3.3) Moeove, the uppe bound fo ψ µ () can be used to deive an uppe bound fo φ µ () = φ µ () + ψ µ(s) ds; indeed, (( ) 4 ) φ µ () φ µ () 1 fo 1, µ whee µ := ( 4(µ + 1) 2 (µ + 3)/ φ µ () ) 1/4 ; due to (3.3), we have µ 1. Using this uppe bound fo φ µ on the inteval [, µ ] and the tivial, yet bette, estimate φ µ on ( µ, 1], we obtain an uppe bound fo w µ, then a lowe bound fo ψ µ. Recalling, once again, that φ µ () = 1 ψ µ(s) ds, we aive at an inequality fo φ µ () of the fom a(µ) φ µ () (1 µ)/2 b(µ) φ µ () (1 µ)/4 + c(µ), (3.4) with coefficients a(µ), b(µ), c(µ) defined fo µ R + \ {1} and depending only on µ; moeove, a(µ) >, b(µ) >, and b 2 (µ) 4 a(µ) c(µ) >, while c(µ) has the same sign as µ + µ 2 µ 3 (we used a compute algeba system to compute the coefficients and veify the elevant popeties). Hence, thee exists a numbe µ such that c(µ) > fo µ [, µ) \ {1} and c(µ) < fo µ ( µ, ). It follows that the quadatic polynomial a(µ) t 2 b(µ) t + c(µ) has a pai of eal oots t 1 (µ), t 2 (µ) with t 1 (µ) < t 2 (µ); while t 2 (µ) is always positive, t 1 (µ) is positive if and only if µ < µ. Thus, the inequality (3.4) implies t 2 (µ) 4/(1 µ) if µ < 1, φ µ () f(µ) := t 1 (µ) 4/(1 µ) if 1 < µ < µ. The function f can be extended to a continuous function f on [, µ), and by continuity, φ µ () f(µ) fo all µ [, µ). Note that f(µ) as µ µ. Of couse, the inequality (3.4) yields also a lowe bound fo φ µ (), which significantly impoves (3.3), but this will not be needed. Moe impotantly, with a efinement of the above agument and plenty of compute algeba, we ae able to explicitly constuct a continuous function ˆf, defined on an inteval [, ˆµ) with ˆµ 9.73, such that φ µ () ˆf(µ) fo all µ [, ˆµ). (3.5) We omit the vey technical details of this constuction. 13

14 and Now obseve that φ µ = φ µ (), w µ φ µ() µ + 1 φ µ(), ψ µ φ µ () 2 (µ + 1) 2 (µ + 3) 1 3 φ µ() 2, fom which we infe that (w µ, φ µ, ψ µ ) φ µ () 2 (note that φ µ () 12, due to (3.3)). Consequently, we have φ µ () u µ φ µ () 2, and ecalling (3.3) and (3.5), we conclude that 12 4(µ + 1) 2 (µ + 3) u µ ˆf 2 (µ) fo all µ [, ˆµ). (3.6) Next we will compute the (Leay-Schaude) fixed-point index of T (, ) in u, the nontivial solution of (3.2) fo µ =. (Note that the existence of u is aleady clea by Remak 3.2; but the following agument will pove it once again.) Inspied by simila easoning in [1], we define a completely continuous opeato S : R + X X by ( S(λ, u)() := φ(s) ds, 1 ψ(s) ds, ( w 2 (s) + λ ) ) ds, fo λ R +, u = (w, φ, ψ) X, and [, 1], and conside the paametedependent fixed-point poblem in X, u = S(λ, u). (3.7) Note that S(, ) = T (, ); that is, if λ =, then Equation (3.7) coincides with Equation (3.2) with µ =. Now let λ R + and suppose that u λ = (w λ, φ λ, ψ λ ) X is a solution of (3.7). Since φ λ = wλ 2 + λ, the function φ λ is convex; thus, φ λ () φ λ ()(1 ) fo all [, 1]. Aguing as in ou ealie estimates, we deive an uppe bound fo w λ, then a lowe bound fo ψ λ. Again, we have φ λ () = 1 ψ λ(s) ds; this and the lowe bound fo ψ λ yield a quadatic inequality fo φ λ (), namely, φ λ () 2 24 φ λ () + 12λ. It follows that λ 12 and φ λ () 24. This shows that (3.7) does not have any solutions if λ > 12; moeove, the unifom bound on φ λ () implies a unifom bound on u λ. Choosing a sufficiently lage constant ρ >, we infe that the numbe deg ( Id X S(λ, ), Bρ X (), ) is well defined fo λ R +, independent of λ 14

15 (due to homotopy invaiance), and in fact equal to zeo (since (3.7) has no solutions fo λ > 12). It follows that deg ( Id X T (, ), B X ρ (), ) = deg ( Id X S(, ), B X ρ (), ) =. Howeve, a outine homotopy agument shows that the index of T (, ) in the tivial fixed point equals 1. This poves, once again, the existence of the nontivial fixed point u and shows, moe impotantly, that the index of T (, ) in u is 1. We ae now in a position to complete the poof of the poposition by applying a Rabinowitz-type agument (see [16]). Suppose, by way of contadiction, that the solution banch C := {(µ, u µ ) : µ [, µ )} is bounded. Then µ <, and thee exists a constant ρ > such that u µ < ρ fo all µ [, µ ). Also, due to (3.3), u µ 12 fo all µ [, µ ), and accoding to Lemma 3.1(b), the equation (3.2) has no nontivial solution fo µ = µ. It follows that deg ( Id X T (µ, ), Bρ X () \ B X 1 (), ) is well defined fo µ [, µ ], independent of µ, and in fact equal to zeo. Clealy, this contadicts the fact that T (, ) has index 1 in u. It follows that C is unbounded in R + X. Due to (3.6), the pojection of C onto R + {} contains the inteval [, ˆµ), which implies µ ˆµ and thus, µ > 9. Poof of Theoem 1.4. In view of Remak 2.2, Lemma 3.1(a) implies that the elliptic system (1.3), fo abitay R >, has exactly one lage adial solution (v, φ) with v() = and φ() <, povided that N 1 < µ ; since µ > 9, this is the case if N 1. Lemma 2.4 shows that φ is a stictly inceasing function of the adial vaiable and cosses zeo at a point (, R). Hence, the function w µ () := µ w(), with w = v, is stictly deceasing fo < and stictly inceasing fo > ; it cosses zeo at a point 1 (, R). Thus, v is negative on (, 1 ), positive on ( 1, R), and consequently, v is stictly deceasing to a negative minimum at 1, stictly inceasing theeafte. Remak 3.4 Numeical evidence suggests that µ and that the set M of Lemma 3.1 coincides with the inteval [, µ ). In othe wods, the maximal solution of the Cauchy poblem (2.3) with p = 1 blows up in finite time if and only if µ < µ As a consequence, we conjectue that the lage solution of Theoem 1.4 exists if and only if N

16 Figues 3 6 depict computed pofiles of the lage adial solutions (v, φ) with v() = and φ() < of the system (1.3), with R = 1, fo seveal values of the space dimension N (see Remak 4.4 fo comments on the numeical method). Specifically, Figue 5 shows the solution fo N = 1, the lagest space dimension fo which we poved its existence (Theoem 1.4); Figue 6 shows the solution fo N = 14, the lagest space dimension fo which we found it numeically. Of couse, we can compute the lage solution (v, φ) with v() = and φ() < of the system (2.1) (the adial vesion of (1.3)) fo evey µ [, µ ), whee µ It is notewothy that as µ µ, the φ-component of the solution appeas to appoach c R δ R c δ, fo some positive constants c, c R, whee δ, δ R denote the Diac distibutions centeed at and R, espectively v phi v phi Fig. 3: N=1 Fig. 4: N=3 2e+6 4e+7 1.5e+6 3e+7 1e+6 2e+7 5 1e e+7 v phi v phi Fig. 5: N=1 Fig. 6: N=14 16

17 We conclude this section with a comment on the Diichlet poblem fo the elliptic system (1.2) on a ball in R N, v = θ in BR N(), θ = v 2 in BR N(), (3.8) v = θ = on BR N(). Assuming that R = 1 (due to the scaling popety, this entails no loss of geneality), thee is a one-to-one coespondence between the adial solutions of (3.8) and the solutions of the bounday-value poblem (3.1) o the equivalent fixed-point equation (3.2), with µ = N 1, θ = φ, and v() = 1 w(s) ds fo [, 1]. Hence, Poposition 3.3 implies that (3.8) has a unique nontivial adial solution (v, θ) at least fo N 1; it is easily checked that both components of this solution ae stictly deceasing functions of the adial vaiable. Coollay 3.5 Fo evey N N with N 1 and evey R >, the Diichlet poblem (3.8) has a unique nontivial adially symmetic solution (v, θ). Both components of this solution ae deceasing functions of the adial vaiable v theta v theta Fig. 7: N=1 Fig. 8: N=14 The numeical evidence cited in Remak 3.4 suggests that the (necessaily unique) nontivial adial solution of (3.8) exists, in fact, if and only if N 14. Figues 7 and 8 show computed pofiles of this solution fo R = 1 and the two exteme values of the space dimension, N = 1 and N =

18 Of couse, we can compute the nontivial solution (v, θ) of the adial vesion of (3.8) fo any µ [, µ ) in place of the intege N 1. As µ µ , the θ-component of the solution appeas to appoach a multiple of δ (the Diac distibution centeed at ). 4 Asymptotic Behavio Let (w, φ, ψ) C 1 ([, R), R 3 ) be the maximal solution of the Cauchy poblem (2.3) fo a given µ R + and p R. By Lemma 2.4, we know that R is finite if and only if φ() is eventually positive and that in this case, w(), φ(), ψ() as R. In view of the existing liteatue on bounday blow-up in elliptic equations (see, fo example, [3, 18]), it is natual to expect asymptotic behavio of the fom Q/(R ) q, with positive constants Q and q. In fact, we will pove the following esult. Poposition 4.1 Let (w, φ, ψ) C 1 ([, R), R 3 ) denote the maximal solution of (2.3), fo a given µ R + and p R, and suppose that R is finite. Then, as R, w() , φ(), ψ() (R ) 3 (R ) 4 (R ) 5. Let us note that if all thee of the functions w, φ, ψ exhibit asymptotic behavio of the fom Q/(R ) q, it is easy to see that the constants Q and q ae necessaily as above. We will pove Poposition 4.1 unde the assumption that R = 1; thanks to the scaling popety, this entails no loss of geneality. The poof will be achieved by analyzing a system of equations deived fom (2.3) by a suitable change of vaiables. Given any solution (w, φ, ψ) of (2.3), define functions α, β, γ by α() := (1 )3 6 w(), β() := (1 )4 18 φ(), γ() := (1 )5 72 then (α, β, γ) is a solution of (1 ) (α + µ ) α = 3(β α), α() =, (1 ) β = 4(γ β), β() = p/18, (1 ) (γ + µ ) γ = 5(α 2 γ), γ() =. ψ() ; (4.1) 18

19 Just like (2.3), the system (4.1) is singula at =, but this does not affect the well-posedness of the initial value poblem; in addition, (4.1) is singula at = 1. Remak 4.2 Suppose that (w, φ, ψ) is the maximal solution of (2.3), fo a given µ R + and p R, with inteval of existence [, R p ); let (α, β, γ) be the coesponding solution of (4.1), as defined above. Clealy, if R p < 1, then (α, β, γ) ceases to exist befoe eaching the singulaity at = 1; in fact, α(), β(), γ() as R p. Also, if R p > 1, then (α, β, γ) can be continued beyond the singulaity at = 1, and α(), β(), γ() as 1. Finally, if R p = 1, then (α, β, γ) exists up to the singulaity at = 1, but the behavio nea the singulaity is not obvious. The assetion of Poposition 4.1 (with R = 1) is that, in this case, α(), β(), γ() 1 as 1. Remak 4.3 Recall that fo evey µ R +, thee is exactly one initial value p + µ > and at most one initial value p µ < such that the maximal solution of (2.3) with p = p ± µ blows up at = 1. Let p µ denote one such value. Due to the scaling popety of the system (2.3), all solutions with sign(p) = sign(p µ) blow up in finite time. Moeove, thee is a one-to-one coespondence between the initial value p and the exit time R p of the solution; in fact, Rp 4 = p µ/p (see Remak 2.1). It follows that if p > p µ > o p < p µ <, then R p < 1, and consequently, α(), β(), γ() as Rp. Also, if < p < p µ o p µ < p <, then R p > 1, and consequently, α(), β(), γ() as 1. Remak 4.4 The obsevations in the peceding emak allow us to use a shooting method to numeically appoximate p + µ and, if it exists, p µ, that is, the citical initial values p fo which the maximal solution of (2.3) blows up at = 1. Solving the Cauchy poblem (4.1) with p = p ± µ, we can then constuct the solutions of (2.3) that blow up at = 1, and theeby, the lage adial solutions of the elliptic system (1.3) on the unit ball. All the gaphs in the peceding sections wee geneated in this way (with a suitable escaling in the case of Figues 7 and 8). Ou expeiments suggest that p µ exists if and only if µ is smalle than a cetain numbe, appoximately equal to , which, due to the scaling popety, must coincide with the numbe µ in Lemma 3.1. In fact, we find that p µ is a stictly deceasing function of µ that appoaches as µ appoaches the above numbe. 19

20 alpha beta gamma alpha beta gamma Fig. 9: µ=, p/18= Fig. 1: µ=2, p/18= alpha beta gamma alpha beta gamma Fig. 11: µ=, p/18= Fig. 12: µ=2, p/18= In ode to pove Poposition 4.1 fo R = 1, we must show that all thee components of the maximal solution (α, β, γ) of the Cauchy poblem (4.1) with µ R + and p = p ± µ convege to 1 as 1 (ecall Remaks 4.2 and 4.3). While ou numeical expeiments leave no doubt about this (see Figues 9 12 fo examples of computed solutions), the poof equies a small detou in dynamical systems; we efe to [17] fo teminology and basic popeties. 2

21 It is vey convenient to pefom a change of vaiables in the system (4.1), letting = 1 e t and a(t) = α(), b(t) = β(), c(t) = γ(). With this escaling of the independent vaiable, (4.1) is equivalent to a + µ e t a = 3(b a), 1 a() =, b = 4(c b), b() = p/18, (4.2) c + µ e t 1 c = 5(a2 c), c() =. Note that the singulaity of (4.1) at = 1 has been moved to t = ; moeove, the system (4.2) is autonomous fo µ = and asymptotically autonomous fo µ >. Fo notational convenience, we wite the system of diffeential equations in (4.2) as x + µ e t E(x) = F (x), (4.3) 1 whee x = (a, b, c), E : R 3 R 3 is the linea mapping defined by E(a, b, c) := (a,, c), and F : R 3 R 3 is the vecto field defined by F (a, b, c) := ( 3(b a), 4(c b), 5(a 2 c) ). Remak 4.5 Fo t > and a, the system (4.3), with abitay µ R +, satisfies the Kamke condition and thus, a compaison pinciple (see, fo example, [19]). In fact, let t, t 1 [, ] with t < t 1 and suppose that x 1, x 2 C([t, t 1 ), R 3 ) C 1 ((t, t 1 ), R 3 ). If x 1 is a subsolution of (4.3) with a 1, x 2 is a supesolution of (4.3), and x 1 (t ) x 2 (t ) (x 1 (t ) < x 2 (t )), then x 1 (t) x 2 (t) (x 1 (t) < x 2 (t)) fo all t [t, t 1 ). By a subsolution (supesolution) of (4.3) we mean, of couse, a function x = (a, b, c) satisfying the diffeential inequality obtained fom (4.3) by eplacing = with ( ). Also, given vectos x 1, x 2 R 3, we wite x 1 x 2 o x 2 x 1 (x 1 < x 2 o x 2 > x 1 ), if the espective inequality holds componentwise, and we call a vecto x R 3 nonnegative (positive) if x (x > ), whee := (,, ). Remak 4.6 Fo evey λ [, ], let λ denote the vecto (λ, λ, λ). Fo abitay µ R +, is a solution of (4.3), and 1 is a supesolution (a solution if µ = ). Moe geneally, fo evey λ [, 1], λ is a supesolution. Futhemoe, fo evey λ (1, ), thee exists a numbe τ [, ), depending only on λ and µ, such that ( λ+1 2, λ, λ) is a subsolution on the inteval (τ, ) (whee τ = if µ = ). 21

22 Poposition 4.7 Fo a given µ R +, let x be a nonnegative maximal fowad solution of (4.3). (a) If x is unbounded, then x blows up in finite time and appoaches. (b) If x is bounded, then x conveges to eithe o 1. Poof. Fix µ R + and let x = (a, b, c) be a nonnegative maximal fowad solution of (4.3). (a) Suppose that x = (a, b, c) is unbounded. Fist we will show that b is unbounded. By way of contadiction, suppose that b b fo some positive constant b. Then we have a a + µ e t 1 a = 3(b a) 3(b a), which implies that a is bounded. A simila agument then shows that c is bounded as well, and this contadicts the unboundedness of x = (a, b, c). Thus, b is unbounded. Now fix a numbe p > such that the solution (w, φ, ψ ) of the initial-value poblem (2.3) with p = p blows up at a point R < 1. The coesponding solution x = (a, b, c ) of the initial-value poblem (4.2) then blows up at ln(1 R ). Recall that the b-component of the tajectoy x = (a, b, c) is unbounded and choose a point τ in the inteval of existence of x such that b(τ) p /18; define x := x(τ + ). Then we have and x + µ e t 1 E( x) µ x + e τ+t E( x) = F ( x) 1 x() = x(τ) = ( a(τ), b(τ), c(τ) ) (, p /18, ) = x (). Thanks to the compaison pinciple in Remak 4.5, it follows that x x. In paticula, x blows up in finite time, and then, so does x. Reasoning as in the poof of Lemma 2.4(b), it is now easy to veify that all thee components of x appoach infinity. (b) Suppose that x is bounded. Fist, conside the autonomous case, µ =. The vecto field F is coopeative in the half-space a and, in paticula, in the nonnegative cone R 3 + ; moeove, div(f ) 12, and F has exactly two zeos, at and 1. Thus, F geneates a monotone, volume-contacting semiflow Φ in R 3 +, with exactly two equilibia, at and 1. The equilibium at is a stable node, the one at 1 is unstable, with a two-dimensional stable manifold (the eigenvalues ae 1 and 13/2 ± i 71/2). Moeove, the system can be embedded into a coopeative system in all of R 3 by eplacing the nonlinea tem a 2 in the thid component of the 22

23 vecto field F with a a ; the extended system still has negative divegence and hypebolic equilibia, at and ± 1. Mois Hisch poved (see [9], Theoem 1) that evey compact α- o ω -limit set of a coopeative o competitive system in R 3 is eithe a cycle o contains an equilibium. Anothe esult of Hisch (see [8], Theoem 7) guaantees that a coopeative system in R 3 with negative divegence cannot have any cycles. Moeove, if a limit set contains a hypebolic equilibium, then the limit set is a singleton. Combining these esults, we infe that in a coopeative system in R 3 with negative divegence and hypebolic equilibia, evey bounded (fowad o backwad) tajectoy conveges. Since by assumption, the tajectoy x is nonnegative, it follows that x conveges to eithe o 1. Now conside the nonautonomous case, µ >. As we obseved befoe, the system (4.3) is asymptotically autonomous. An old esult of Makus [13] implies that the ω -limit set K of the tajectoy x is a nonempty compact and connected subset of R 3 + ; moeove, dist(x(t), K) as t, and K is invaiant unde the semiflow Φ of the autonomous limit system, that is, (4.3) with µ =. A moe ecent esult by Mischaikow, Smith and Thieme (see [14], Theoem 1.8) implies that K is also chain-ecuent unde Φ. Roughly speaking, this means that the Φ-tajectoy stating at any given point z K will etun to the vicinity of z, time and again, as t. We claim that K {, 1}. By way of contadiction, suppose thee is a point z K \ {, 1}. In light of what we poved fo the autonomous case, since K is compact and Φ-invaiant, the tajectoy Φ(, z) must convege, both fowad and backwad in time. Backwad in time, Φ(, z) can only convege to 1 (since is stable). Thus, z belongs to the unstable manifold of 1, and it follows that, fowad in time, Φ(, z) can only convege to (see the emaks on the dynamics of Φ at the end of this section). Hence, K consists of the two equilibia, and 1, and a heteoclinic obit connecting the two; such a set is obviously not chain-ecuent. The contadiction poves that K {, 1}. In fact, since K is nonempty and connected, we have eithe K = { } o K = { 1}; that is, x conveges to eithe o 1. Coollay 4.8 Let µ R + and p R be such that the maximal solution (w, φ, ψ) of the Cauchy poblem (2.3) blows up at = 1. Then the maximal solution (a, b, c) of (4.2) conveges to 1. Poof. Unde the assumptions of the coollay, (w, φ, ψ) appoaches at = 1; thus, the coesponding solution (α, β, γ) of (4.1) is eventually positive and exists on [, 1) (see Remak 4.2). This means that x = (a, b, c) is eventually positive and exists on [, ). By Pat (a) of Poposition 4.7, 23

24 it follows that x is bounded, and then, Pat (b) of the same poposition implies that x conveges to eithe o 1. Now, suppose that x(t) as t, and choose t (, ) such that < x(t ) < 1. Since the solution of (4.2) depends continuously on p, we can find a value p, close to p, with p > p such that the coesponding maximal solution x of (4.2) exists at t = t and satisfies < x(t ) < 1. Since is a solution and 1 is a supesolution of (4.3), the compaison pinciple in Remak 4.5 implies that < x(t) < 1 fo all t t, as long as x exists. Fom Remak 4.3, howeve, we know that x goes to (in finite time). This contadiction poves that x does not convege to, and so, it must convege to 1. Poof of Poposition 4.1. Due to the scaling popety of the system (2.3) (see Remak 2.1), it suffices to pove the assetion of the poposition fo R = 1; in this case, it is an immediate consequence of Coollay 4.8 (see Remak 4.2). Poof of Theoem 1.5. Since evey lage adial solution (v, φ) of the system (1.3) on BR N () coesponds to a solution (w, φ, ψ) of (2.3) that blows up at R, the asymptotic behavio of φ is clea fom Poposition 4.1. Moeove, the asymptotic behavio of v, given by v() = w(s) ds fo < R, follows eadily fom that of w. In closing, we note that Hisch s esults on coopeative systems in R 3 (see [8, 9]) allow us to completely descibe the dynamics of the monotone, volume-contacting semiflow Φ in R 3 +, induced by the vecto field F. Fist, it can be easily veified that Φ is, in fact, stongly monotone (even though F is ieducible only in the open half-space a > ). As shown in the fist pat of the poof of Poposition 4.7(b), Hisch s esults imply that evey fowad tajectoy of Φ eithe conveges to (a stable node) o to 1 (a saddle point), o it appoaches, necessaily in finite time. Clealy, both and ae stable attactos. In fact, using the sub and supesolutions constucted in Remak 4.6, we see that the open ode intevals (, 1) and ( 1, ) ae positively invaiant and contained in the basins of attaction of and, espectively. The two basins of attaction ae sepaated by the (two-dimensional) stable manifold W s ( 1) of the saddle point. The (onedimensional) unstable manifold W u ( 1) has a positive tangent vecto at 1, which implies that W u ( 1)\{ 1} is contained in the union of the ode intevals (, 1) and ( 1, ). Thus, evey fowad tajectoy on W u ( 1) \ { 1} eithe conveges to o appoaches. It follows that W u ( 1) \ { 1} consists of two heteoclinic obits, connecting 1 to and, espectively. 24

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