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Invariance Principles for Hyperbolic Random Walk Systems
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
210, 360]374 Ž1997.
AY975411
Invariance Principles for Hyperbolic Random Walk Systems Thomas Hillen* Biomathematik, Un¨ ersity of Tubingen, Auf der Morgenstelle 10, ¨ D-72076, Tubingen, Germany ¨ Submitted by Howard A. Le¨ ine Received July 15, 1996
Reaction random walk systems are hyperbolic models for the description of spatial motion Žin one dimension. and reaction of particles. In contrast to reaction diffusion equations, particles have finite propagation speed. For parabolic systems invariance results and maximum principles are well known. A convex set is positively invariant if at each boundary point an outer normal is a left eigenvector of the diffusion matrix, and if the vector field defined by the pure reaction equation ‘‘points inward’’ at the boundary. Here we show a corresponding result for random walk systems. The model parameters are the particle speeds, the rates of change in direction, and the reaction vector field. A convex domain is invariant if at each boundary point an outer normal is a left eigenvector of the ‘‘speed matrix’’ and if a vector field given by the reaction equation combined with the turning rates points inward. Finally a positivity result is shown. Q 1997 Academic Press
1. INTRODUCTION Classical models to describe spatial motion and interaction of particles are reaction diffusion equations. Assume n types of particles denoted by y j , j s 1, . . . , n, move on the line and their interaction is given by a system of ordinary differential equations u ˙ j s f j Ž u1 , . . . , u n . ,
1 F j F n.
Ž 1.
* This work is supported by Deutsche Forschungsgemeinschaft, SFB 382.E-mail address: [email protected]. 360 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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Then the reaction diffusion equation is u t s Du x x q f Ž u . ,
Ž 2.
with u s Ž u1 , . . . , u n ., f s Ž f 1 , . . . , f n ., and a diagonal diffusion matrix D with positive entries d j , 1 F j F n. The subscripts t and x denote as usual the partial derivatives with respect to time and space. Many monographs cover qualitative properties of parabolic equations Žsee, e.g., Henry w14x, Rothe w25x, Smoller w27x, Britton w4x, and Ladyzhenskaja, Solonnikov, and ˇ Ural’ceva w21x.. Reaction diffusion equations suffer from the deficiencies of the linear heat equation, that is, infinitely fast propagation of particles. Many authors have discussed the underlying modeling problems. Einstein w7x criticized the fact that successive steps in Brownnian motion are uncorrelated and hence arbitrarily large speeds are possible. Taylor w28x, Furth ¨ w9x, Goldstein w10x, and Kac w20x introduced a model for correlated random walk in one dimension. Holmes w18x, Dunbar w5x, Hadeler w12x, and Hillen w16, 17x replaced Brownian motion in the reaction diffusion equation by correlated random walks and they defined hyperbolic reaction random walk systems. Holmes investigated traveling wave solutions of a reaction random walk equation corresponding to Fisher’s model of the advance of advantageous genes w8x. Hadeler w11, 13, 12x gave generalizations of this approach to birth]death processes and investigated motion in higher dimensions, i.e., the Cattaneo equations. Dunbar and Othmer w5, 6x combined correlated random walk with a branching process following the approach of McKean w22x in the parabolic case. In w17x a pattern formation of Turing type w29x was studied for reaction random walk systems. Qualitative analysis and global attractors of reaction random walk systems were investigated in w16x using a Lyapunov function. Here we prove an invariance result concerning reaction random walk systems and compare it to the results for parabolic equations. Roughly speaking we find a vector field Žnot f !. which is supposed to ‘‘point inward’’ at the boundary of the invariant region. A result of this form is well known for ordinary differential equations Žsee, e.g., Bony w3x.. Weinberger w31x and Amann w1x generalized it to parabolic systems. Moreover Weinberger w30, 24x considered an invariance principle for linear hyperbolic systems which says that in a characteristic cone, maxima are attained at the initial condition. His approach differs from the results presented here. In the next section we introduce reaction random walk systems and appropriate boundary conditions. Section 3 contains the known results on invariant sets for the parabolic equation Ž2. as far as they are needed here. In Section 4 the invariance principle for reaction random walk systems is
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proved and the condition for preserving positivity is deduced. In the closing Section 5 the results for hyperbolic and for parabolic systems are compared.
2. REACTION RANDOM WALK SYSTEMS Spatial motion in one dimension is modeled by a correlated random walk following Taylor, Goldstein, and Kac. Each particle density u j s y q uq and uy for right and left j q u j , 1 F j F n, is split into densities u j j moving particles, respectively. They move with constant speed g j and they change their direction with a constant rate m j . The correlated random walk equation for each particle type 1 F j F n is q y q uq jt q g j u j x s m j Ž u j y u j . y q y uy jt y g j u j x s m j Ž u j y u j . .
Let the reaction Žwithout motion. of the particles be given by Ž1. where f is at least C 1. First assume that the reaction is independent of the direction of the particles and that newborn particles choose either direction with the same probability. Then the reaction random walk system is, for 1 F j F n, 1 q y q uq jt q g j u j x s m j Ž u j y u j . q 2 f j Ž u1 , . . . , u n . 1 y q y uy jt y g j u j x s m j Ž u j y u j . q 2 f j Ž u1 , . . . , u n . .
Ž 3.
q. y y. Ž y Introduce vectors uq[ Ž uq and matrices 1 , . . . , u n , u [ u1 , . . . , u n G [ Žg j d i j . and M [ Ž m j d i j .. Then in vector notation the reaction random walk system Ž3. reads 1 q y q uq t q Gu x s M Ž u y u . q 2 f Ž u . 1 y q y uy t y Gu x s M Ž u y u . q 2 f Ž u . .
Ž 4.
If the reaction depends on the direction of the particles Ži.e., in birth]death processes, see w12x. a more general model is obtained q y q q y uq t q Gu x s M Ž u y u . q g Ž u , u . y q y q y uy t y Gu x s M Ž u y u . q h Ž u , u . ,
where g and h are C 1. In general g / h.
Ž 5.
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Written in terms of the vector of particle densities u s uqq uy and the vector of particle flows ¨ s uqy uy the system Ž4. is ut q G ¨ x s f Ž u. ¨ t q Gu x s y2 M¨ .
Ž 6.
System Ž6. is equivalent to Ž4.. With f s 0 system Ž6. is a Cattaneo system for one space dimension and the generalization to higher dimensions is obvious. The systems Ž4., Ž5., and Ž6. are of hyperbolic type Žin the sense of Friedrichs w19x.. Indeed each system can be written in the form yt s A y x q 0 . in the case of Ž4., Ž5., and A s Ž 0 yG . in the F Ž y ., with A s Ž yG 0 G yG 0 case of Ž6.. In some sense the reaction random walk equations are more general than reaction diffusion equations. Consider one particle type only Ž n s 1. and assume that solutions Ž u, ¨ . of Ž6. are smooth. Differentiate the first equation of Ž6. with respect to t and the second equation with respect to x. Eliminating ¨ and all its derivatives leads to a reaction telegraph equation Ž‘‘Krac’s trick’’.
ut t q Ž 2 m y f 9Ž u. . ut s g 2 u x x q 2 m f Ž u. .
Ž 7.
Divide Ž7. by 2 m and let the speed g and the turning rate m go to infinity in such a way that the quotient g 2rŽ2 m . ª d - `. As a formal limit the reaction diffusion equation u t s du x x q f Ž u. follows. Limits of this form are considered rigorously by Milani w23x with a singular perturbation approach. In the sequel we consider Ž5. on a compact interval w0, l x with three kinds of boundary conditions: Dirichlet. No particle can enter the domain from outside, i.e., for 1FjFn v
uq j Ž t , 0 . s 0,
uy j Ž t , l . s 0.
uq Ž t , 0 . s 0,
uy Ž t , l . s 0.
In vector notation
v
Ž 8.
Neumann. Particles are reflected at the boundary, i.e., uq Ž t , 0 . s uy Ž t , 0 . ,
uy Ž t , l . s uq Ž t , l . .
Ž 9.
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THOMAS HILLEN v
Periodic. uq Ž t , 0 . s uq Ž t , l . ,
uy Ž t , l . s uy Ž t , 0 . .
Ž 10 .
In w15x it is proved that the reaction random walk system Ž5. with each of the above boundary conditions defines a Žlocal. C0-semigroup on y. Ž L2 Žw0, l x.. 2 n, i.e., for each initial data Ž uq Ž 1, 2 Žw0, l x.. 2 n which 0 , u0 g W satisfies the prescribed boundary condition there exists a local ŽT F `. solution
Ž uq, uy . g C ž w 0, T . , Ž W 1, 2 Ž w 0, l x . .
2n
/ l C ž w0, T . , Ž L Ž w0, l x . . / . 1
2
2n
To consider invariant sets we assume that solutions of the boundary value problems Ž5., Ž8., and Ž5., Ž9. and Ž5., Ž10. are continuously differentiable y. with respect to t and x. Indeed, this holds if the initial data Ž uq 0 , u0 g 1 2 n Ž C Žw0, l x.. satisfy some compatibility conditions at the boundary w15x v
yŽ . Ž . Dirichlet. Dx uq 0 0 s 0 and Dx u 0 l s 0.
v
yŽ . qŽ . yŽ . Ž . Neumann. Dx uq 0 0 s yDx u 0 0 and Dx u 0 l s yDx u 0 l .
v
qŽ . yŽ . yŽ . Ž . Periodic. Dx uq 0 0 s Dx u 0 l and Dx u 0 0 s Dx u 0 l .
3. INVARIANT SETS REVISITED Let X be a Banach space of R k Ž k g N. valued functions on w0, l x and consider a dynamical system
˙y s Q Ž y .
Ž 11 .
on X . In the parabolic case Ž2., X s Ž L2 Žw0, l x.. n, in the hyperbolic case Ž5., X s Ž L2 Žw0, l x.. 2 n. DEFINITION. A closed set S ; R k is positively in¨ ariant with respect to Ž11. if for all initial values y 0 g X with y 0 Ž x . g S for all x g w0, l x the solution y Ž t . satisfies y Ž t .Ž x . g S for all x g w0, l x as long as the solution exists Ž t - T .. DEFINITION. Assume S ; R k is convex. An outer normal n g R k at z 0 g S satisfies
n ? z 0 s max n ? z, z g S 4 ,
Ž 12 .
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where the dot denotes the inner product in R k . This definition of an outer normal is used by Amann w1x. In general the outer normal is not unique. Condition Ž12. is equivalent to max n ? Ž z y z 0 . , z g S 4 s 0. The invariance principle for reaction random walk systems, stated in the next section, uses an appropriate vector field F which is supposed to point inward at the region, i.e., n Ž z . ? F Ž z . F 0 for all z g S. A condition of this type is used to prove invariance results for ordinary differential equations. A result of Bony w3x, for example, is not restricted to convex regions and the ‘‘inward’’ condition is needed only at boundary points which satisfy an outer sphere condition. Weinberger w31x and Amann w1x gave a similar result for convex sets for parabolic systems Žsee also Schaefer w26x.. Restricted to the reaction diffusion equation Ž2. their result can be stated as follows: Consider Ž2. on w0, l x with classical homogeneous Dirichlet, Neumann, or periodic boundary conditions, i.e., Dirichlet Neumann periodic
u Ž t , 0 . s 0,
u Ž t , l . s 0,
Ž 13 .
u x Ž t , 0 . s 0,
u x Ž t , l . s 0,
Ž 14 .
u x Ž t , 0. s u x Ž t , l . .
Ž 15 .
u Ž t , 0. s u Ž t , l . ,
Assume ŽP1. Let S ; R n be closed, convex Žwith 0 g S in the case of Dirichlet conditions. and suppose that z g S has an outer normal which is a left eigenvector of the diffusion matrix D. ŽP2. For all z g S and for each outer normal n Ž z . let
n Ž z . ? f Ž z . F 0.
Ž 16 .
THEOREM 1 ŽAmann.. Assume ŽP1. and ŽP2. then S is positi¨ ely in¨ ariant for Ž2. with each of Dirichlet Ž13., Neumann Ž14., or periodic Ž15. boundary conditions. To prove this theorem Amann writes the convex set S as S s x g R n : f Ž x . F 0, ;f g F 4 , where F is a family of linear affine functions. The sets f Ž x . s 04 for f g F are supporting hyperplanes of S. Using ŽP1. he proves for each of these functions, f g F a parabolic differential inequality. With assumption ŽP2. he can apply the parabolic maximum principle w24x to show that
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for each solution starting in S the inequality f Ž uŽ t, x .. F 0 holds for all 0 F t - T. Hence S is invariant. To find invariant sets for a given vector field f is a difficult task. One first looks for rectangles which satisfy ŽP1. and ŽP2. Žsee, e.g., Amann w2x.. An example is given at the end of the discussion in Section 5.
4. INVARIANCE PRINCIPLES FOR REACTION RANDOM WALK SYSTEMS Here we consider Ž5. with each of the prescribed boundary conditions Ž8., Ž9., or Ž10.. To rewrite Ž5. as a differential equation in X s Ž L2 Žw0, l x.. 2 n we define y [ Ž uq, uy .T , F Ž y . [ Ž g Ž y ., hŽ y ..T , and Ž2 n = 2 n.-matrix operators G[
ž
yGDx 0
0 GDx
/
and
B[
ž
yM M
M . yM
/
Then Ž5. is yt s Gy q By q F Ž y . . q
Ž 17 . y.
q
y ..
In the special case of Ž4., F Ž y . s Ž1r2.Ž f Ž u q u , f Ž u q u . The domain of the unbounded operator G depends on the boundary condition D Ž G . [ y g Ž C 1 Ž w 0, l x . . : y s Ž uq, uy . , uq and uy satisfy the given boundary and compatibility condition 4 . 2n
½
Assume ŽH1. Let L ; R n be closed, convex Žwith 0 g L in the case of Dirichlet conditions. and suppose that each z g L has an outer normal n which is a left eigenvector of G. ŽH2. Define S [ L = L. Assume for each y g S and for each outer normal h Ž y . at S that
h Ž y . ? Ž By q F Ž y . . F 0.
Ž 18 .
Remarks. Ž1. Condition Ž18. ensures that the set S is invariant with respect to the ordinary differential equation z t s Bz q F Ž z . ,
z g R2n.
Ž 19 .
If S is an invariant rectangle for Ž19. which can be written as a product of rectangles L, S s L = L then ŽP1. is automatically fulfilled. Indeed outer
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normals at L are the canonical vectors e j , 1 F j F n, which are left eigenvectors of the Ždiagonal. matrix G. Ž2. Since S s L = L, we have for y s Ž yq, yy . g S that yqg L or yyg L or both. Hence y has an outer normal h of the form h s Ž nq, 0. or h s Ž0, ny. , where nq and ny are outer normals at L. Hence condition Ž18. is equivalent with the following two conditions: For all yqg L, for all yyg L, and for each outer normal nq in yq let
nq? M Ž yyy yq . q nq? g Ž yq, yy . F 0 and for all yyg L, for all yqg L, and for each outer normal ny in yy let
ny? M Ž yqy yy . q ny? h Ž yq, yy . F 0. In the case of Ž4., where g Ž uq, uy . s hŽ uq, uy . s Ž1r2. f Ž uqq uy ., these two conditions are equivalent to the condition that for all w 0 g L and for each outer normal n Ž w 0 . sup n ? M Ž w y w 0 . q 12 n ? f Ž w q w 0 . : w g L 4 F 0.
Ž 20 .
Condition Ž20. has the following physical interpretation. Consider a particle in w g L with < w y w 0 < - « such that
n ? f Ž w q w 0 . ) 0.
Ž 21 .
Since w g L we have n ? Ž w y w 0 . - 0. If we assume that the turning rates m j are all of the same magnitude we expect that also n ? M Ž w y w 0 . - 0. Hence the turning rates must be large enough to compensate the drift of f as given by Ž21.. In other words, if the particle does not reverse its direction immediately it will be driven out of the domain. Roughly speaking condition Ž20. requires that f points inward strongly enough. n Ž3. As a special case assume that L s Ł is1 w a i , bi x is a product of nonempty bounded intervals w a i , bi x, 1 F i F n. Then outer normals to L are the canonical vectors "e i , 1 F i F n Žand in addition linear combinations at the vertices of L .. Then, for system Ž4., condition Ž20. is equivalent to the condition that for 1 F j F n. min m j Ž s y a j . q 12 f j Ž w 1 , . . . , s q a j , . . . , wn . : s g a j , bj , w k g w 2 a k , 2 bk x , k / j 4 G 0, max m j Ž s y bj . q 12 f j Ž w 1 , . . . , s q bj , . . . , wn . : s g a j , bj , w k g w 2 a k , 2 bk x , k / j 4 F 0.
Ž 22 .
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THEOREM 2. Assume ŽH1. and ŽH2.. Then S is positi¨ ely in¨ ariant for the reaction random walk system Ž5. with each of the Dirichlet Ž8., Neumann Ž9., and periodic Ž10. boundary conditions. Proof. The proof is divided into three steps. First we show invariance in the case where the initial value y 0 Ž x . g ˚ S for all x g w0, l x and the vector field B q F Ž?. points strictly inward Ži.e., -0 in Ž18... The case where F0 holds in Ž18. we trace back to the first case considering parallel sets. When initial values meet the boundary of S, then a continuity argument shows the invariance. Ž1. Let Ž uq Ž . y Ž .. ˚ w x 0 x , u 0 x g S for all x g 0, l . Assume that for all y g S and for each outer normal h in y
h ? Ž By q F Ž y . . - 0.
Ž 23 .
Assume that S is not positively invariant. Then we find Ž t*, x*. g Rq=w0, l x such that y* [ y Ž t*, x*. g S and y Ž t, x . g ˚ S for all 0 F t - t*, x g w0, l x. Let h * [ h Ž y*. be an outer normal at y*. Since S is convex the function w Ž t, x . [ h * ? y Ž t, x . satisfies at t s t* 0F
t
w Ž t*, x* . s h * ? yUt
Ž 24 .
Žsee Ž12... We consider two cases. Ža. x* g Ž0, l .. In x* the function w Ž t*, x . has a local maximum, i.e., U
uq x 0s w Ž t*, x* . s h * ? yU . ux x
ž /
Ž 25 .
As we pointed out before, each boundary point has an outer normal of the form h s Ž n , 0. or h s Ž0, n ., were n is an outer normal of L. As assumed in ŽH1., n can be chosen as a left eigenvector of G, hence from Ž25. it follows that U
h * ? Gy* s h * ?
ž
yGuq x U Guy x
/
s 0.
Multiply Eq. Ž17. by h * to obtain, with Ž24., a contradiction 0 F h * ? yUt s h * ? GyU q h * ? Ž By* q F Ž y* . . , since the first term is zero by Ž26. and the second is negative by Ž23..
Ž 26 .
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Žb. x* g 0, l 4 . We consider each boundary condition separately. ŽI. Dirichlet boundary condition. uqŽ t, 0. s 0, uyŽ t, l . s 0. If Ž . y Ž .. g ˚ x* s 0 then y* s Ž0, uy Ž t*, 0... The assumption Ž uq S 0 x , u0 x qŽ . yŽ . together with the boundary condition u 0 0 s 0, u 0 l s 0 implies that ˚ Hence an outer normal at S in y* is of the form h s Ž0, n ., 0 g L. where n is an outer normal to L and a left eigenvector of G. Since y Ž t*, x . g ˚ S for all 0 - x - l we obtain U
uq x 0G Ž h * ? y* . s h * ? yU . ux x
ž /
The outer normal n is a left eigenvector of G, hence U
yGuq x U h * ? Gy* s Ž 0, n . ? GUxy
ž
/
F 0.
Again multiply Ž17. with h * to obtain a contradiction 0 F h * ? yUt s h * ? Gy* q h * ? Ž By* q F Ž y* . . ,
Ž 27 .
since the first term is non-positive and the second is negative. With a similar argument x* s l is excluded. ŽII. Neumann boundary conditions. uqŽ t, 0. s uyŽ t, 0., uyŽ t, l . s qŽ u t, l .. Since S s L = L the point y* is a vertex of S. Hence y* has outer normals of the form h1U s Ž0, ny. and hU2 s Ž nq, 0.. If x* s 0 then we multiply Ž17. by h1U to obtain a contradiction as in ŽI.. If x* s l, multiplication of Ž17. by hU2 leads to a contradiction. ŽIII. Periodic boundary conditions. uqŽ t, 0. s uqŽ t, l ., uyŽ t, l . s yŽ u t, 0.. Then y* s y Ž t*, 0. s y Ž t*, l .. Since y Ž t*, x . g ˚ S for all 0 - x - l, it follows that
x
Ž h * ? y Ž t*, 0 . . F 0
and
x
Ž h * ? y Ž t*, l . . G 0.
Ž 28 .
If the outer normal h * has the form Ž0, n ., we multiply Ž17. with h * and evaluate at x s 0 to obtain a contradiction. If h * has the form Ž nq, 0., we again multiply Ž17. by h * and evaluate at x s l. With Ž28., a contradiction follows. Ž2. Consider Ž uq Ž . y Ž .. ˚ w x Ž . 0 x , u 0 x g S for all x g 0, l and assume 18 . 2n We endow R with the norm 5 z 5 ` s max Ž z1 , . . . , z n .
2,
Ž z nq1 , . . . , z 2 n . 2 4
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THOMAS HILLEN
and define for « ) 0 the parallel set S« [ z g R 2 n : dist`Ž z, S . F « 4 . ˜ = L, ˜ too. We show that for small « the set Then S« is of the form S« s L S« is positively invariant for a modified differential equation, which coincides with Ž17. on S. Consider z g S«rS. Since S is convex, there is a unique radius r Ž z . such that the 5 ? 5 2-ball Br Ž z . satisfies Br Ž z . l S s y4 . Then a surjective map y : Ž S«rS . ª S : z ¬ y Ž z . is defined. If w g S« then obviously 5 w y y Ž w .5 ` s « and r Ž w . G « . Observe that each outer normal h Ž w . of w at S« is an outer normal of Ž y w . at S and
hŽ w. ? Ž w y yŽ w. . G « .
Ž 29 .
Define a vector field on S« as HŽ z. [
½
Bz q F Ž z . , By Ž z . q F Ž y Ž z . . y z q y Ž z . ,
zgS z f S.
Then H is Lipschitz continuous with respect to z g S« , and for all w g S« it follows from Ž18. and Ž29. that
h Ž w . ? H Ž w . s h Ž w . ? Ž By Ž w . q F Ž y Ž w . . . q h Ž w . ? Ž yw q y Ž w . . F y« - 0. Hence the vector field H Ž z . satisfies a strong inward condition at the boundary and with the first part of this proof it follows that S« is invariant for the partial differential equation z t s Gz q H Ž z .. Then with « ª 0 it follows that the set S is invariant for Ž17.. Ž3. Assume Ž uq Ž . yŽ .. w x 0 x , u 0 x g S for all x g 0, l . As pointed out in Section 2 solutions of the boundary value problem Ž17. are given by C0-semigroups, hence they depend continuously on the initial data. Since S is invariant for data starting in the interior of S it has to be invariant for all initial data y 0 g S. An important property is preservation of positivity. Thus we consider the 2n n positive cone S s Rq . Then the boundary of S is S s D2js1 j S, where q y j S s uq j s 0, Ž u , u . g S 4 ,
1 F j F n,
q y j S s uy j s 0, Ž u , u . g S 4 ,
n q 1 F j F 2 n.
The outer normals are the negative coordinate vectors ye j , 1 F j F 2 n, n which are all left eigenvectors of the speed matrix G. The set L s Rq is unbounded but nevertheless we can use condition Ž22. with ‘‘inf’’ instead
HYPERBOLIC INVARIANCE PRINCIPLES
371
of ‘‘min’’ and ‘‘sup’’ instead of ‘‘max’’. Then Ž22. with a j s 0 and bj s ` for 1 F j F n reduces to n : f j Ž y . G y2 m j y j , ; j, 1 F j F n, ; y g Rq
Ž 30 .
which is ŽH2. in this case. y. COROLLARY 3 ŽPositivity.. Assume Ž uq is an initial condition 0 , u0 qŽ . yŽ . w with u 0 x G 0 and u 0 x G 0 for all x g 0, l x. If f satisfies Ž30. then uqŽ t, x . G 0 and uyŽ t, x . G 0 for all x g w0, l x as long as the solution exists.
5. DISCUSSION The hyperbolic model Ž4. is related to the parabolic equation by the scaling g j2rŽ2 m j . f d j for large values of g j and m j for 1 F j F n. But we cannot use Kac’s trick for the more general model Ž5.. Hence relations of Ž5. to parabolic equations are not obvious. To compare the results for hyperbolic and for parabolic equations we investigate Ž4. only. PROPOSITION 4. Assume S s L = L satisfies conditions Ž H 1. and Ž H 2., i.e., S is in¨ ariant for the hyperbolic system Ž4.. Moreo¨ er let each outer normal of L be a left eigen¨ ector of M. Then Q [ L q L is in¨ ariant for the parabolic problem Ž2. with D [ Ž2 M .y1 G 2 . Proof. Assume ŽH1. and ŽH2. with condition Ž20. instead of Ž18.. From Ž20. it follows that for w ª w 0
n ? f Ž 2 w 0 . F 0. Hence Q [ L q L satisfies condition ŽP2. Žnote that for each z g Q there is a y g L such that z s 2 y .. Since each z g L has an outer normal which is left eigenvector of G and of M this also holds for Q. Then each y g Q has an outer normal which is a left eigenvector of D s Ž2 M .y1 G 2 and condition ŽP1. holds. PROPOSITION 5. Assume the closed, con¨ ex set Q g R n satisfies ŽP1., ŽP2. with - instead of F . Then there is a hyperbolic model with Ž2 M .y1 G 2 s D such that L [ 12 Q satisfies ŽH1. and ŽH2.. Proof. Since n Ž z 0 . ? f Ž z 0 . - 0 for all z 0 g Q and since f is continuous there is a « ) 0 such that for all z g Q with 5 z y z 0 5 - « also n Ž z 0 . ? f Ž z . - 0 holds. Define a closed subset of L [ 12 Q by
˜ [ w g L : w y w 0 G « , ; w 0 g L 4 . L
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THOMAS HILLEN
˜ Let From the choice of « it is obvious that Ž20. holds for all w f L. ˜ m [ sup f Ž w q w 0 . : w 0 g L, w g L4 . Since f is continuous and L is compact it follows that m - `. Hence there are turning rates Ž m 1 , . . . , m n . such that for all w 0 g L, for each outer normal n Ž w 0 . and for each ˜ wgL n ? M Ž w y w 0 . - ym, ˜ Moreover we can choose M and G hence condition Ž20. holds with w g L. in such a way that G 2 s 2 DM and that each boundary point of L has an outer normal which is a left eigenvector of G. Remarks. Ž1. In Propositions 4 and 5 nothing is assumed on the convergence of the solutions of the hyperbolic to the solutions of the parabolic system. They just clarify how the model parameters of the hyperbolic and the parabolic models are related. Ž2. As pointed out in Remark 2 of Section 4 the vector field f is supposed to point inward strongly enough such that the region is invariant for the hyperbolic model. Hence we cannot expect that Proposition 5 holds in the case of F in ŽP2.. EXAMPLE. We consider one particle type only Ž n s 1.. In Fisher’s equation Ž2. with logistic growth f Ž u. s a uŽ1 y u., a ) 0 it is supposed that 0 F u F 1. Thus for the hyperbolic model Ž4. we require 0 F uq, uyF 1. If we consider the parabolic equation Ž2. on w0, l x with Neumann boundary conditions u x Ž t, 0. s u x Ž t, l . s 0 then the parabolic maximum principle shows that the set w0, 1x is positively invariant without any restriction on a . For the hyperbolic system we chose S s w0, 12 x = w0, 12 x. Then uqq uyg w0, 1x. The boundary of S and outer normals are
1 S s uqs 0, 0 F uyF
1 2
4, 2 S s uqs 21 , 0 F uyF 21 4 , 3 S s uys 0, 0 F uqF 12 4 , 4 S s uys 21 , 0 F uqF 21 4 ,
n 1 s Ž y1, 0 . n 2 s Ž 1, 0 . n 3 s Ž 0,y1 . n4 s Ž 0, 1 . .
Condition Ž20. becomes
1 S: 2 S:
m Ž uyy
1 2
3 S: 4 S:
m Ž uqy
1 2
m uyq 12 f Ž uy . G 0,
0 F uyF 12 ,
. q 12 f Ž uyq 12 . F 0,
0 F uyF 12 ,
m uqq 21 f Ž uq . G 0,
0 F uqF 21 ,
. q 12 f Ž uqq 12 . F 0,
0 F uqF 12 .
HYPERBOLIC INVARIANCE PRINCIPLES
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Hence we have to assume that f satisfies the condition f Ž z . G y2 m z,
0 F z F 12 ,
f Ž z. F 2mŽ1 y z. ,
1 2
F z F 1.
Ž 31 .
For f Ž z . s a z Ž1 y z . this condition reads
a F 2 m. COROLLARY 6. ¨ ariant for
For a - 2 m the set S s w0, 12 x = w0, 12 x is positi¨ ely in-
1 q y q q y q y uq t q g ux s mŽ u y u . q 2 a Ž u q u . Ž1 y u y u . , 1 y q y q y q y uy t y g ux s mŽ u y u . q 2 a Ž u q u . Ž1 y u y u . ,
with each of the Dirichlet Ž8., Neumann Ž9., and periodic Ž10. boundary conditions. Remark. In the parabolic case invariance principles can be used to show global existence Žsee, e.g., Amann w1x or Smoller w27x.. Since the solution semigroup of a parabolic system has the compactness property Ži.e., weak solutions become classical solutions instantaneously ., an L` a priori bound for the solutions is sufficient to show global existence. In the hyperbolic case considered here we have no compactness property w15x. To deduce a global existence result from the existence of an invariant region we need an a priori estimate of the form 5Ž uq, uy .5 C 1 F K 5Ž uq, uy .5 L` . In w15x regularity of solutions of Ž4. is considered in the linear case Ž f s 0.. Together with compatibility assumptions at the boundary x s 0 and x s 1 solutions with smooth initial conditions are smooth.
REFERENCES 1. H. Amann, Invariant sets and existence theorems for semilinear parabolic and elliptic systems, J. Math. Anal. Appl. 65 Ž1978., 432]467. 2. H. Amann, ‘‘Gewohnliche Differentialgleichungen,’’ de Gruyter, BerlinrNew York, 1983. ¨ 3. J.-M. Bony. Principe du maximum, inegalite de ´ ´ de Harnack et unicite´ du probleme ` Cauchy pour les operateurs elliptique degeneres, ´ ´ ´ ´ ´ Ann. Inst. Fourier Ž Grenoble. 19 Ž1969., 277]304. 4. N. F. Britton, ‘‘Reaction-Diffusion Equations and Their Applications to Biology,’’ Academic Press, London, 1986. 5. S. Dunbar, A branching random walk evolution and a nonlinear hyperbolic equation, SIAM J. Appl. Math. 48 Ž1988., 1510]1526. 6. S. Dunbar and H. Othmer, On a nonlinear hyperbolic equation describing transmission lines, cell movement, and branching random walks, in ‘‘Nonlinear Oscillations in Biology
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7. 8. 9. 10. 11. 12.
13. 14. 15.
16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.
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and Chemistry’’ ŽH. Othmer, Ed.., Lecture Notes in Biomath., Vol. 66, Springer-Verlag, New YorkrBerlin, 1986. A. Einstein, Zur Theorie der Brownschen Bewegung, Ann. Physik 19 Ž1906., 371]381. R. A. Fisher, The advance of advantageous genes, Ann. Eugenics 7 Ž1937., 355]369. R. Furth, Die Brownsche Bewegung bei Berucksichtigung einer Persistenz der ¨ ¨ Bewegungsrichtung, Z. Phys. 2 Ž1920., 244]256. S. Goldstein, On diffusion by discontinuous movements and the telegraph equation, Quart. J. Mech. Appl. Math. 4 129]156. K. P. Hadeler, Traveling fronts for correlated random walks, Canad. Appl. Math. Quart. 2 Ž1994.. K. P. Hadeler, Random walk systems and reaction telegraph equations, in ‘‘Dynamical Systems and Their Applications in Science’’ ŽS. V. Strien and S. V. Lunel, Eds.., Royal Academie of the Netherlands, 1995. K. P. Hadeler, Travelling fronts in random walk systems, FORMA Soc. Sci. Form 10 Ž1995., 223]233. D. Henry, Geometric theory of semilinear parabolic equations, in ‘‘Lecture Notes in Math.,’’ Vol. 840, Springer-Verlag Berlin, 1981. T. Hillen, ‘‘Nichtlineare hyperbolische Systeme zur Modellierung von Ausbreitungsvorgangen und Anwendung auf das Turing Modell,’’ Dissertation, University of Tubingen, ¨ ¨ 1995. T. Hillen, ‘‘Qualitative Analysis of Hyperbolic Random Walk Systems,’’ SFB 382, Report No. 43, 1996. T. Hillen, A Turing model with correlated random walk, J. Math. Biol. 35 Ž1996., 49]72. E. E. Holmes, Are diffusion models too simple? A comparison with telegraph models of invasion, Amer. Natur. 142 Ž1993., 779]795. K. Ito, ‘‘Encyclopedic Dictionary of Mathematics,’’ 3rd ed., Math. Soc. of Japan, 1954. M. Kac, A stochastic model related to the telegrapher’s equation, Rocky Mountain J. Math. 4 Ž1956., 497]509. O. A. Ladyzhenskaja, V. A. Solonnikov, and N. N. Ural’ceva, ‘‘Linear and Quasilinear ˇ Equations of Parabolic Type,’’ Amer. Math. Soc., Providence, 1968. H. P. McKean, Applications of Brownian motion to the equation of Kolmogorov] Petrovskij]Piskunov, Comm. Pure Appl. Math. 28 Ž1975., 323]331. A. Milani, Global existence via singular perturbations for quasiliner evolution equations, Ad¨ . Math. Sci. Appl. 6 Ž1996., 419]444. M. H. Protter and H. F. Weinberger, ‘‘Maximum Principles in Differential Equations,’’ Springer-Verlag, New York, 1984. F. Rothe, Global solutions of reaction diffusion systems, in ‘‘Lecture Notes in Math,’’ Vol. 1072, Springer-Verlag, New YorkrBerlin, 1979. C. Schaefer, Invariant sets and contractions for weakly coupled systems of parabolic differential equations, Rend. Mar. Roma 13 Ž1980., 337]357. J. Smoller, ‘‘Shock Waves and Reaction-Diffusion Equations,’’ Springer-Verlag, New YorkrBerlin, 1983. G. I. Taylor, Diffusion by discontinuous movements, Proc. London Math. Soc. 20 Ž1920., 196]212. A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London. Ser. B 237 Ž1952., 5]72. H. F. Weinberger, A maximum property of Cauchy’s problem, Ann. of Math. 64 Ž1955., 505]513. H. F. Weinberger, Invariant sets for weakly coupled parabolic and elliptic systems, Rend. Mat. Roma 8 Ž1975., 295]310.
210, 360]374 Ž1997.
AY975411
Invariance Principles for Hyperbolic Random Walk Systems Thomas Hillen* Biomathematik, Un¨ ersity of Tubingen, Auf der Morgenstelle 10, ¨ D-72076, Tubingen, Germany ¨ Submitted by Howard A. Le¨ ine Received July 15, 1996
Reaction random walk systems are hyperbolic models for the description of spatial motion Žin one dimension. and reaction of particles. In contrast to reaction diffusion equations, particles have finite propagation speed. For parabolic systems invariance results and maximum principles are well known. A convex set is positively invariant if at each boundary point an outer normal is a left eigenvector of the diffusion matrix, and if the vector field defined by the pure reaction equation ‘‘points inward’’ at the boundary. Here we show a corresponding result for random walk systems. The model parameters are the particle speeds, the rates of change in direction, and the reaction vector field. A convex domain is invariant if at each boundary point an outer normal is a left eigenvector of the ‘‘speed matrix’’ and if a vector field given by the reaction equation combined with the turning rates points inward. Finally a positivity result is shown. Q 1997 Academic Press
1. INTRODUCTION Classical models to describe spatial motion and interaction of particles are reaction diffusion equations. Assume n types of particles denoted by y j , j s 1, . . . , n, move on the line and their interaction is given by a system of ordinary differential equations u ˙ j s f j Ž u1 , . . . , u n . ,
1 F j F n.
Ž 1.
* This work is supported by Deutsche Forschungsgemeinschaft, SFB 382.E-mail address: [email protected]. 360 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
HYPERBOLIC INVARIANCE PRINCIPLES
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Then the reaction diffusion equation is u t s Du x x q f Ž u . ,
Ž 2.
with u s Ž u1 , . . . , u n ., f s Ž f 1 , . . . , f n ., and a diagonal diffusion matrix D with positive entries d j , 1 F j F n. The subscripts t and x denote as usual the partial derivatives with respect to time and space. Many monographs cover qualitative properties of parabolic equations Žsee, e.g., Henry w14x, Rothe w25x, Smoller w27x, Britton w4x, and Ladyzhenskaja, Solonnikov, and ˇ Ural’ceva w21x.. Reaction diffusion equations suffer from the deficiencies of the linear heat equation, that is, infinitely fast propagation of particles. Many authors have discussed the underlying modeling problems. Einstein w7x criticized the fact that successive steps in Brownnian motion are uncorrelated and hence arbitrarily large speeds are possible. Taylor w28x, Furth ¨ w9x, Goldstein w10x, and Kac w20x introduced a model for correlated random walk in one dimension. Holmes w18x, Dunbar w5x, Hadeler w12x, and Hillen w16, 17x replaced Brownian motion in the reaction diffusion equation by correlated random walks and they defined hyperbolic reaction random walk systems. Holmes investigated traveling wave solutions of a reaction random walk equation corresponding to Fisher’s model of the advance of advantageous genes w8x. Hadeler w11, 13, 12x gave generalizations of this approach to birth]death processes and investigated motion in higher dimensions, i.e., the Cattaneo equations. Dunbar and Othmer w5, 6x combined correlated random walk with a branching process following the approach of McKean w22x in the parabolic case. In w17x a pattern formation of Turing type w29x was studied for reaction random walk systems. Qualitative analysis and global attractors of reaction random walk systems were investigated in w16x using a Lyapunov function. Here we prove an invariance result concerning reaction random walk systems and compare it to the results for parabolic equations. Roughly speaking we find a vector field Žnot f !. which is supposed to ‘‘point inward’’ at the boundary of the invariant region. A result of this form is well known for ordinary differential equations Žsee, e.g., Bony w3x.. Weinberger w31x and Amann w1x generalized it to parabolic systems. Moreover Weinberger w30, 24x considered an invariance principle for linear hyperbolic systems which says that in a characteristic cone, maxima are attained at the initial condition. His approach differs from the results presented here. In the next section we introduce reaction random walk systems and appropriate boundary conditions. Section 3 contains the known results on invariant sets for the parabolic equation Ž2. as far as they are needed here. In Section 4 the invariance principle for reaction random walk systems is
362
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proved and the condition for preserving positivity is deduced. In the closing Section 5 the results for hyperbolic and for parabolic systems are compared.
2. REACTION RANDOM WALK SYSTEMS Spatial motion in one dimension is modeled by a correlated random walk following Taylor, Goldstein, and Kac. Each particle density u j s y q uq and uy for right and left j q u j , 1 F j F n, is split into densities u j j moving particles, respectively. They move with constant speed g j and they change their direction with a constant rate m j . The correlated random walk equation for each particle type 1 F j F n is q y q uq jt q g j u j x s m j Ž u j y u j . y q y uy jt y g j u j x s m j Ž u j y u j . .
Let the reaction Žwithout motion. of the particles be given by Ž1. where f is at least C 1. First assume that the reaction is independent of the direction of the particles and that newborn particles choose either direction with the same probability. Then the reaction random walk system is, for 1 F j F n, 1 q y q uq jt q g j u j x s m j Ž u j y u j . q 2 f j Ž u1 , . . . , u n . 1 y q y uy jt y g j u j x s m j Ž u j y u j . q 2 f j Ž u1 , . . . , u n . .
Ž 3.
q. y y. Ž y Introduce vectors uq[ Ž uq and matrices 1 , . . . , u n , u [ u1 , . . . , u n G [ Žg j d i j . and M [ Ž m j d i j .. Then in vector notation the reaction random walk system Ž3. reads 1 q y q uq t q Gu x s M Ž u y u . q 2 f Ž u . 1 y q y uy t y Gu x s M Ž u y u . q 2 f Ž u . .
Ž 4.
If the reaction depends on the direction of the particles Ži.e., in birth]death processes, see w12x. a more general model is obtained q y q q y uq t q Gu x s M Ž u y u . q g Ž u , u . y q y q y uy t y Gu x s M Ž u y u . q h Ž u , u . ,
where g and h are C 1. In general g / h.
Ž 5.
HYPERBOLIC INVARIANCE PRINCIPLES
363
Written in terms of the vector of particle densities u s uqq uy and the vector of particle flows ¨ s uqy uy the system Ž4. is ut q G ¨ x s f Ž u. ¨ t q Gu x s y2 M¨ .
Ž 6.
System Ž6. is equivalent to Ž4.. With f s 0 system Ž6. is a Cattaneo system for one space dimension and the generalization to higher dimensions is obvious. The systems Ž4., Ž5., and Ž6. are of hyperbolic type Žin the sense of Friedrichs w19x.. Indeed each system can be written in the form yt s A y x q 0 . in the case of Ž4., Ž5., and A s Ž 0 yG . in the F Ž y ., with A s Ž yG 0 G yG 0 case of Ž6.. In some sense the reaction random walk equations are more general than reaction diffusion equations. Consider one particle type only Ž n s 1. and assume that solutions Ž u, ¨ . of Ž6. are smooth. Differentiate the first equation of Ž6. with respect to t and the second equation with respect to x. Eliminating ¨ and all its derivatives leads to a reaction telegraph equation Ž‘‘Krac’s trick’’.
ut t q Ž 2 m y f 9Ž u. . ut s g 2 u x x q 2 m f Ž u. .
Ž 7.
Divide Ž7. by 2 m and let the speed g and the turning rate m go to infinity in such a way that the quotient g 2rŽ2 m . ª d - `. As a formal limit the reaction diffusion equation u t s du x x q f Ž u. follows. Limits of this form are considered rigorously by Milani w23x with a singular perturbation approach. In the sequel we consider Ž5. on a compact interval w0, l x with three kinds of boundary conditions: Dirichlet. No particle can enter the domain from outside, i.e., for 1FjFn v
uq j Ž t , 0 . s 0,
uy j Ž t , l . s 0.
uq Ž t , 0 . s 0,
uy Ž t , l . s 0.
In vector notation
v
Ž 8.
Neumann. Particles are reflected at the boundary, i.e., uq Ž t , 0 . s uy Ž t , 0 . ,
uy Ž t , l . s uq Ž t , l . .
Ž 9.
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THOMAS HILLEN v
Periodic. uq Ž t , 0 . s uq Ž t , l . ,
uy Ž t , l . s uy Ž t , 0 . .
Ž 10 .
In w15x it is proved that the reaction random walk system Ž5. with each of the above boundary conditions defines a Žlocal. C0-semigroup on y. Ž L2 Žw0, l x.. 2 n, i.e., for each initial data Ž uq Ž 1, 2 Žw0, l x.. 2 n which 0 , u0 g W satisfies the prescribed boundary condition there exists a local ŽT F `. solution
Ž uq, uy . g C ž w 0, T . , Ž W 1, 2 Ž w 0, l x . .
2n
/ l C ž w0, T . , Ž L Ž w0, l x . . / . 1
2
2n
To consider invariant sets we assume that solutions of the boundary value problems Ž5., Ž8., and Ž5., Ž9. and Ž5., Ž10. are continuously differentiable y. with respect to t and x. Indeed, this holds if the initial data Ž uq 0 , u0 g 1 2 n Ž C Žw0, l x.. satisfy some compatibility conditions at the boundary w15x v
yŽ . Ž . Dirichlet. Dx uq 0 0 s 0 and Dx u 0 l s 0.
v
yŽ . qŽ . yŽ . Ž . Neumann. Dx uq 0 0 s yDx u 0 0 and Dx u 0 l s yDx u 0 l .
v
qŽ . yŽ . yŽ . Ž . Periodic. Dx uq 0 0 s Dx u 0 l and Dx u 0 0 s Dx u 0 l .
3. INVARIANT SETS REVISITED Let X be a Banach space of R k Ž k g N. valued functions on w0, l x and consider a dynamical system
˙y s Q Ž y .
Ž 11 .
on X . In the parabolic case Ž2., X s Ž L2 Žw0, l x.. n, in the hyperbolic case Ž5., X s Ž L2 Žw0, l x.. 2 n. DEFINITION. A closed set S ; R k is positively in¨ ariant with respect to Ž11. if for all initial values y 0 g X with y 0 Ž x . g S for all x g w0, l x the solution y Ž t . satisfies y Ž t .Ž x . g S for all x g w0, l x as long as the solution exists Ž t - T .. DEFINITION. Assume S ; R k is convex. An outer normal n g R k at z 0 g S satisfies
n ? z 0 s max n ? z, z g S 4 ,
Ž 12 .
HYPERBOLIC INVARIANCE PRINCIPLES
365
where the dot denotes the inner product in R k . This definition of an outer normal is used by Amann w1x. In general the outer normal is not unique. Condition Ž12. is equivalent to max n ? Ž z y z 0 . , z g S 4 s 0. The invariance principle for reaction random walk systems, stated in the next section, uses an appropriate vector field F which is supposed to point inward at the region, i.e., n Ž z . ? F Ž z . F 0 for all z g S. A condition of this type is used to prove invariance results for ordinary differential equations. A result of Bony w3x, for example, is not restricted to convex regions and the ‘‘inward’’ condition is needed only at boundary points which satisfy an outer sphere condition. Weinberger w31x and Amann w1x gave a similar result for convex sets for parabolic systems Žsee also Schaefer w26x.. Restricted to the reaction diffusion equation Ž2. their result can be stated as follows: Consider Ž2. on w0, l x with classical homogeneous Dirichlet, Neumann, or periodic boundary conditions, i.e., Dirichlet Neumann periodic
u Ž t , 0 . s 0,
u Ž t , l . s 0,
Ž 13 .
u x Ž t , 0 . s 0,
u x Ž t , l . s 0,
Ž 14 .
u x Ž t , 0. s u x Ž t , l . .
Ž 15 .
u Ž t , 0. s u Ž t , l . ,
Assume ŽP1. Let S ; R n be closed, convex Žwith 0 g S in the case of Dirichlet conditions. and suppose that z g S has an outer normal which is a left eigenvector of the diffusion matrix D. ŽP2. For all z g S and for each outer normal n Ž z . let
n Ž z . ? f Ž z . F 0.
Ž 16 .
THEOREM 1 ŽAmann.. Assume ŽP1. and ŽP2. then S is positi¨ ely in¨ ariant for Ž2. with each of Dirichlet Ž13., Neumann Ž14., or periodic Ž15. boundary conditions. To prove this theorem Amann writes the convex set S as S s x g R n : f Ž x . F 0, ;f g F 4 , where F is a family of linear affine functions. The sets f Ž x . s 04 for f g F are supporting hyperplanes of S. Using ŽP1. he proves for each of these functions, f g F a parabolic differential inequality. With assumption ŽP2. he can apply the parabolic maximum principle w24x to show that
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THOMAS HILLEN
for each solution starting in S the inequality f Ž uŽ t, x .. F 0 holds for all 0 F t - T. Hence S is invariant. To find invariant sets for a given vector field f is a difficult task. One first looks for rectangles which satisfy ŽP1. and ŽP2. Žsee, e.g., Amann w2x.. An example is given at the end of the discussion in Section 5.
4. INVARIANCE PRINCIPLES FOR REACTION RANDOM WALK SYSTEMS Here we consider Ž5. with each of the prescribed boundary conditions Ž8., Ž9., or Ž10.. To rewrite Ž5. as a differential equation in X s Ž L2 Žw0, l x.. 2 n we define y [ Ž uq, uy .T , F Ž y . [ Ž g Ž y ., hŽ y ..T , and Ž2 n = 2 n.-matrix operators G[
ž
yGDx 0
0 GDx
/
and
B[
ž
yM M
M . yM
/
Then Ž5. is yt s Gy q By q F Ž y . . q
Ž 17 . y.
q
y ..
In the special case of Ž4., F Ž y . s Ž1r2.Ž f Ž u q u , f Ž u q u . The domain of the unbounded operator G depends on the boundary condition D Ž G . [ y g Ž C 1 Ž w 0, l x . . : y s Ž uq, uy . , uq and uy satisfy the given boundary and compatibility condition 4 . 2n
½
Assume ŽH1. Let L ; R n be closed, convex Žwith 0 g L in the case of Dirichlet conditions. and suppose that each z g L has an outer normal n which is a left eigenvector of G. ŽH2. Define S [ L = L. Assume for each y g S and for each outer normal h Ž y . at S that
h Ž y . ? Ž By q F Ž y . . F 0.
Ž 18 .
Remarks. Ž1. Condition Ž18. ensures that the set S is invariant with respect to the ordinary differential equation z t s Bz q F Ž z . ,
z g R2n.
Ž 19 .
If S is an invariant rectangle for Ž19. which can be written as a product of rectangles L, S s L = L then ŽP1. is automatically fulfilled. Indeed outer
HYPERBOLIC INVARIANCE PRINCIPLES
367
normals at L are the canonical vectors e j , 1 F j F n, which are left eigenvectors of the Ždiagonal. matrix G. Ž2. Since S s L = L, we have for y s Ž yq, yy . g S that yqg L or yyg L or both. Hence y has an outer normal h of the form h s Ž nq, 0. or h s Ž0, ny. , where nq and ny are outer normals at L. Hence condition Ž18. is equivalent with the following two conditions: For all yqg L, for all yyg L, and for each outer normal nq in yq let
nq? M Ž yyy yq . q nq? g Ž yq, yy . F 0 and for all yyg L, for all yqg L, and for each outer normal ny in yy let
ny? M Ž yqy yy . q ny? h Ž yq, yy . F 0. In the case of Ž4., where g Ž uq, uy . s hŽ uq, uy . s Ž1r2. f Ž uqq uy ., these two conditions are equivalent to the condition that for all w 0 g L and for each outer normal n Ž w 0 . sup n ? M Ž w y w 0 . q 12 n ? f Ž w q w 0 . : w g L 4 F 0.
Ž 20 .
Condition Ž20. has the following physical interpretation. Consider a particle in w g L with < w y w 0 < - « such that
n ? f Ž w q w 0 . ) 0.
Ž 21 .
Since w g L we have n ? Ž w y w 0 . - 0. If we assume that the turning rates m j are all of the same magnitude we expect that also n ? M Ž w y w 0 . - 0. Hence the turning rates must be large enough to compensate the drift of f as given by Ž21.. In other words, if the particle does not reverse its direction immediately it will be driven out of the domain. Roughly speaking condition Ž20. requires that f points inward strongly enough. n Ž3. As a special case assume that L s Ł is1 w a i , bi x is a product of nonempty bounded intervals w a i , bi x, 1 F i F n. Then outer normals to L are the canonical vectors "e i , 1 F i F n Žand in addition linear combinations at the vertices of L .. Then, for system Ž4., condition Ž20. is equivalent to the condition that for 1 F j F n. min m j Ž s y a j . q 12 f j Ž w 1 , . . . , s q a j , . . . , wn . : s g a j , bj , w k g w 2 a k , 2 bk x , k / j 4 G 0, max m j Ž s y bj . q 12 f j Ž w 1 , . . . , s q bj , . . . , wn . : s g a j , bj , w k g w 2 a k , 2 bk x , k / j 4 F 0.
Ž 22 .
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THOMAS HILLEN
THEOREM 2. Assume ŽH1. and ŽH2.. Then S is positi¨ ely in¨ ariant for the reaction random walk system Ž5. with each of the Dirichlet Ž8., Neumann Ž9., and periodic Ž10. boundary conditions. Proof. The proof is divided into three steps. First we show invariance in the case where the initial value y 0 Ž x . g ˚ S for all x g w0, l x and the vector field B q F Ž?. points strictly inward Ži.e., -0 in Ž18... The case where F0 holds in Ž18. we trace back to the first case considering parallel sets. When initial values meet the boundary of S, then a continuity argument shows the invariance. Ž1. Let Ž uq Ž . y Ž .. ˚ w x 0 x , u 0 x g S for all x g 0, l . Assume that for all y g S and for each outer normal h in y
h ? Ž By q F Ž y . . - 0.
Ž 23 .
Assume that S is not positively invariant. Then we find Ž t*, x*. g Rq=w0, l x such that y* [ y Ž t*, x*. g S and y Ž t, x . g ˚ S for all 0 F t - t*, x g w0, l x. Let h * [ h Ž y*. be an outer normal at y*. Since S is convex the function w Ž t, x . [ h * ? y Ž t, x . satisfies at t s t* 0F
t
w Ž t*, x* . s h * ? yUt
Ž 24 .
Žsee Ž12... We consider two cases. Ža. x* g Ž0, l .. In x* the function w Ž t*, x . has a local maximum, i.e., U
uq x 0s w Ž t*, x* . s h * ? yU . ux x
ž /
Ž 25 .
As we pointed out before, each boundary point has an outer normal of the form h s Ž n , 0. or h s Ž0, n ., were n is an outer normal of L. As assumed in ŽH1., n can be chosen as a left eigenvector of G, hence from Ž25. it follows that U
h * ? Gy* s h * ?
ž
yGuq x U Guy x
/
s 0.
Multiply Eq. Ž17. by h * to obtain, with Ž24., a contradiction 0 F h * ? yUt s h * ? GyU q h * ? Ž By* q F Ž y* . . , since the first term is zero by Ž26. and the second is negative by Ž23..
Ž 26 .
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Žb. x* g 0, l 4 . We consider each boundary condition separately. ŽI. Dirichlet boundary condition. uqŽ t, 0. s 0, uyŽ t, l . s 0. If Ž . y Ž .. g ˚ x* s 0 then y* s Ž0, uy Ž t*, 0... The assumption Ž uq S 0 x , u0 x qŽ . yŽ . together with the boundary condition u 0 0 s 0, u 0 l s 0 implies that ˚ Hence an outer normal at S in y* is of the form h s Ž0, n ., 0 g L. where n is an outer normal to L and a left eigenvector of G. Since y Ž t*, x . g ˚ S for all 0 - x - l we obtain U
uq x 0G Ž h * ? y* . s h * ? yU . ux x
ž /
The outer normal n is a left eigenvector of G, hence U
yGuq x U h * ? Gy* s Ž 0, n . ? GUxy
ž
/
F 0.
Again multiply Ž17. with h * to obtain a contradiction 0 F h * ? yUt s h * ? Gy* q h * ? Ž By* q F Ž y* . . ,
Ž 27 .
since the first term is non-positive and the second is negative. With a similar argument x* s l is excluded. ŽII. Neumann boundary conditions. uqŽ t, 0. s uyŽ t, 0., uyŽ t, l . s qŽ u t, l .. Since S s L = L the point y* is a vertex of S. Hence y* has outer normals of the form h1U s Ž0, ny. and hU2 s Ž nq, 0.. If x* s 0 then we multiply Ž17. by h1U to obtain a contradiction as in ŽI.. If x* s l, multiplication of Ž17. by hU2 leads to a contradiction. ŽIII. Periodic boundary conditions. uqŽ t, 0. s uqŽ t, l ., uyŽ t, l . s yŽ u t, 0.. Then y* s y Ž t*, 0. s y Ž t*, l .. Since y Ž t*, x . g ˚ S for all 0 - x - l, it follows that
x
Ž h * ? y Ž t*, 0 . . F 0
and
x
Ž h * ? y Ž t*, l . . G 0.
Ž 28 .
If the outer normal h * has the form Ž0, n ., we multiply Ž17. with h * and evaluate at x s 0 to obtain a contradiction. If h * has the form Ž nq, 0., we again multiply Ž17. by h * and evaluate at x s l. With Ž28., a contradiction follows. Ž2. Consider Ž uq Ž . y Ž .. ˚ w x Ž . 0 x , u 0 x g S for all x g 0, l and assume 18 . 2n We endow R with the norm 5 z 5 ` s max Ž z1 , . . . , z n .
2,
Ž z nq1 , . . . , z 2 n . 2 4
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and define for « ) 0 the parallel set S« [ z g R 2 n : dist`Ž z, S . F « 4 . ˜ = L, ˜ too. We show that for small « the set Then S« is of the form S« s L S« is positively invariant for a modified differential equation, which coincides with Ž17. on S. Consider z g S«rS. Since S is convex, there is a unique radius r Ž z . such that the 5 ? 5 2-ball Br Ž z . satisfies Br Ž z . l S s y4 . Then a surjective map y : Ž S«rS . ª S : z ¬ y Ž z . is defined. If w g S« then obviously 5 w y y Ž w .5 ` s « and r Ž w . G « . Observe that each outer normal h Ž w . of w at S« is an outer normal of Ž y w . at S and
hŽ w. ? Ž w y yŽ w. . G « .
Ž 29 .
Define a vector field on S« as HŽ z. [
½
Bz q F Ž z . , By Ž z . q F Ž y Ž z . . y z q y Ž z . ,
zgS z f S.
Then H is Lipschitz continuous with respect to z g S« , and for all w g S« it follows from Ž18. and Ž29. that
h Ž w . ? H Ž w . s h Ž w . ? Ž By Ž w . q F Ž y Ž w . . . q h Ž w . ? Ž yw q y Ž w . . F y« - 0. Hence the vector field H Ž z . satisfies a strong inward condition at the boundary and with the first part of this proof it follows that S« is invariant for the partial differential equation z t s Gz q H Ž z .. Then with « ª 0 it follows that the set S is invariant for Ž17.. Ž3. Assume Ž uq Ž . yŽ .. w x 0 x , u 0 x g S for all x g 0, l . As pointed out in Section 2 solutions of the boundary value problem Ž17. are given by C0-semigroups, hence they depend continuously on the initial data. Since S is invariant for data starting in the interior of S it has to be invariant for all initial data y 0 g S. An important property is preservation of positivity. Thus we consider the 2n n positive cone S s Rq . Then the boundary of S is S s D2js1 j S, where q y j S s uq j s 0, Ž u , u . g S 4 ,
1 F j F n,
q y j S s uy j s 0, Ž u , u . g S 4 ,
n q 1 F j F 2 n.
The outer normals are the negative coordinate vectors ye j , 1 F j F 2 n, n which are all left eigenvectors of the speed matrix G. The set L s Rq is unbounded but nevertheless we can use condition Ž22. with ‘‘inf’’ instead
HYPERBOLIC INVARIANCE PRINCIPLES
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of ‘‘min’’ and ‘‘sup’’ instead of ‘‘max’’. Then Ž22. with a j s 0 and bj s ` for 1 F j F n reduces to n : f j Ž y . G y2 m j y j , ; j, 1 F j F n, ; y g Rq
Ž 30 .
which is ŽH2. in this case. y. COROLLARY 3 ŽPositivity.. Assume Ž uq is an initial condition 0 , u0 qŽ . yŽ . w with u 0 x G 0 and u 0 x G 0 for all x g 0, l x. If f satisfies Ž30. then uqŽ t, x . G 0 and uyŽ t, x . G 0 for all x g w0, l x as long as the solution exists.
5. DISCUSSION The hyperbolic model Ž4. is related to the parabolic equation by the scaling g j2rŽ2 m j . f d j for large values of g j and m j for 1 F j F n. But we cannot use Kac’s trick for the more general model Ž5.. Hence relations of Ž5. to parabolic equations are not obvious. To compare the results for hyperbolic and for parabolic equations we investigate Ž4. only. PROPOSITION 4. Assume S s L = L satisfies conditions Ž H 1. and Ž H 2., i.e., S is in¨ ariant for the hyperbolic system Ž4.. Moreo¨ er let each outer normal of L be a left eigen¨ ector of M. Then Q [ L q L is in¨ ariant for the parabolic problem Ž2. with D [ Ž2 M .y1 G 2 . Proof. Assume ŽH1. and ŽH2. with condition Ž20. instead of Ž18.. From Ž20. it follows that for w ª w 0
n ? f Ž 2 w 0 . F 0. Hence Q [ L q L satisfies condition ŽP2. Žnote that for each z g Q there is a y g L such that z s 2 y .. Since each z g L has an outer normal which is left eigenvector of G and of M this also holds for Q. Then each y g Q has an outer normal which is a left eigenvector of D s Ž2 M .y1 G 2 and condition ŽP1. holds. PROPOSITION 5. Assume the closed, con¨ ex set Q g R n satisfies ŽP1., ŽP2. with - instead of F . Then there is a hyperbolic model with Ž2 M .y1 G 2 s D such that L [ 12 Q satisfies ŽH1. and ŽH2.. Proof. Since n Ž z 0 . ? f Ž z 0 . - 0 for all z 0 g Q and since f is continuous there is a « ) 0 such that for all z g Q with 5 z y z 0 5 - « also n Ž z 0 . ? f Ž z . - 0 holds. Define a closed subset of L [ 12 Q by
˜ [ w g L : w y w 0 G « , ; w 0 g L 4 . L
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˜ Let From the choice of « it is obvious that Ž20. holds for all w f L. ˜ m [ sup f Ž w q w 0 . : w 0 g L, w g L4 . Since f is continuous and L is compact it follows that m - `. Hence there are turning rates Ž m 1 , . . . , m n . such that for all w 0 g L, for each outer normal n Ž w 0 . and for each ˜ wgL n ? M Ž w y w 0 . - ym, ˜ Moreover we can choose M and G hence condition Ž20. holds with w g L. in such a way that G 2 s 2 DM and that each boundary point of L has an outer normal which is a left eigenvector of G. Remarks. Ž1. In Propositions 4 and 5 nothing is assumed on the convergence of the solutions of the hyperbolic to the solutions of the parabolic system. They just clarify how the model parameters of the hyperbolic and the parabolic models are related. Ž2. As pointed out in Remark 2 of Section 4 the vector field f is supposed to point inward strongly enough such that the region is invariant for the hyperbolic model. Hence we cannot expect that Proposition 5 holds in the case of F in ŽP2.. EXAMPLE. We consider one particle type only Ž n s 1.. In Fisher’s equation Ž2. with logistic growth f Ž u. s a uŽ1 y u., a ) 0 it is supposed that 0 F u F 1. Thus for the hyperbolic model Ž4. we require 0 F uq, uyF 1. If we consider the parabolic equation Ž2. on w0, l x with Neumann boundary conditions u x Ž t, 0. s u x Ž t, l . s 0 then the parabolic maximum principle shows that the set w0, 1x is positively invariant without any restriction on a . For the hyperbolic system we chose S s w0, 12 x = w0, 12 x. Then uqq uyg w0, 1x. The boundary of S and outer normals are
1 S s uqs 0, 0 F uyF
1 2
4, 2 S s uqs 21 , 0 F uyF 21 4 , 3 S s uys 0, 0 F uqF 12 4 , 4 S s uys 21 , 0 F uqF 21 4 ,
n 1 s Ž y1, 0 . n 2 s Ž 1, 0 . n 3 s Ž 0,y1 . n4 s Ž 0, 1 . .
Condition Ž20. becomes
1 S: 2 S:
m Ž uyy
1 2
3 S: 4 S:
m Ž uqy
1 2
m uyq 12 f Ž uy . G 0,
0 F uyF 12 ,
. q 12 f Ž uyq 12 . F 0,
0 F uyF 12 ,
m uqq 21 f Ž uq . G 0,
0 F uqF 21 ,
. q 12 f Ž uqq 12 . F 0,
0 F uqF 12 .
HYPERBOLIC INVARIANCE PRINCIPLES
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Hence we have to assume that f satisfies the condition f Ž z . G y2 m z,
0 F z F 12 ,
f Ž z. F 2mŽ1 y z. ,
1 2
F z F 1.
Ž 31 .
For f Ž z . s a z Ž1 y z . this condition reads
a F 2 m. COROLLARY 6. ¨ ariant for
For a - 2 m the set S s w0, 12 x = w0, 12 x is positi¨ ely in-
1 q y q q y q y uq t q g ux s mŽ u y u . q 2 a Ž u q u . Ž1 y u y u . , 1 y q y q y q y uy t y g ux s mŽ u y u . q 2 a Ž u q u . Ž1 y u y u . ,
with each of the Dirichlet Ž8., Neumann Ž9., and periodic Ž10. boundary conditions. Remark. In the parabolic case invariance principles can be used to show global existence Žsee, e.g., Amann w1x or Smoller w27x.. Since the solution semigroup of a parabolic system has the compactness property Ži.e., weak solutions become classical solutions instantaneously ., an L` a priori bound for the solutions is sufficient to show global existence. In the hyperbolic case considered here we have no compactness property w15x. To deduce a global existence result from the existence of an invariant region we need an a priori estimate of the form 5Ž uq, uy .5 C 1 F K 5Ž uq, uy .5 L` . In w15x regularity of solutions of Ž4. is considered in the linear case Ž f s 0.. Together with compatibility assumptions at the boundary x s 0 and x s 1 solutions with smooth initial conditions are smooth.
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