Degree, instability and bifurcation of reaction–diffusion systems with obstacles near certain hyperbolas

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Abstract

For a reaction–diffusion system which is subject to Turing’s diffusion-driven instability and which is equipped with unilateral obstacles of various types, the nonexistence of bifurcation of stationary solutions near certain critical parameter values is proved. The result implies assertions about a related mapping degree which in turn implies for “small” obstacles the existence of a new branch of bifurcation points (spatial patterns) induced by the obstacle.

Introduction

Let ΩRN be a bounded domain with a Lipschitz boundary. Consider the reaction–diffusion system ut=d1Δu+g1(u,v)vt=d2Δv+g2(u,v)on  Ω with boundary conditions un=0on  ΩΓ1,unm1(u)on  Γ1,vn=0on  ΩΓ2,vnm2(v)on  Γ2 with measurable given sets Γ1,Γ2Ω.

We are interested in bifurcation of stationary solutions of (1.1), (1.2) near a constant equilibrium (u¯,v¯) which is linearly stable without diffusion terms (i.e. if d1=d2=0). We assume that (1.1), (1.2) already describe the problem after a substitution which shifts this equilibrium to 0, i.e. without loss of generality, we assume that (u¯,v¯)=(0,0), in particular g1(0,0)=g2(0,0)=0. Note that u and v then actually denote the difference to the equilibrium so that also negative values of u and v have a physical interpretation.

In case Γ1=Γ2=, Turing’s famous effect of “diffusion-driven instability”  [13] implies that for the above problem the equilibrium (0,0) becomes instable for certain values of diffusion speeds d1,d2>0. Moreover, if one considers d=(d1,d2)R+2 as a diffusion parameter, one typically has bifurcation of spatially nonconstant stationary solutions from the equilibrium when d crosses certain hyperbolas Cn (cf. Section  3 for details).

We are interested in the question of what changes in these observations when we impose on Γi obstacles described by the (in general multivalued) functions mi. More precisely, we will assume that mi=m or mi(u)=m(u) where m is (or behaves in a certain weak sense near 0 qualitatively similar to) the function m0(u){[0,)if  u=0,{0}if  u>0,if  u<0, see Fig. 1. For the particular case m2=m0, the boundary condition (1.2) becomes on Γ2 for v the well-known Signorini boundary condition v0,vn0,vnv=0 which is the “classical” way to describe an obstacle on Γ2: Physically, if u and v denote the difference of the concentration of two chemicals to equilibrium, this obstacle can mean that on Γ2 the concentration v of the second chemical cannot go under the equilibrium concentration, because there is some unilateral membrane or other source which produces the corresponding chemical as soon as this would be too low. We call this an obstacle “of inequality type”.

Actually, this model is physically too ideal, because the fact that m0(v) can be unbounded and empty means in the above interpretation that the source produces if necessary an unlimited amount of the chemicals and thus completely avoids the possibility that the concentration v goes under the equilibrium concentration. Therefore, a more realistic assumption would be that the source produces “much” for v0 but cannot guarantee necessarily in all cases that v0. This would mean that m(v), limv0infm(v)v=,infm(0)0andlimv0+sup±m(v)=0 see Fig. 2. We call this an obstacle “of infinite derivative type”.

Finally, one can of course also assume that the obstacle is even rather weak so that instead of the first limit of (1.3) one has only lim supv0infm(v)v<0, see Fig. 3, which we call an obstacle “of finite derivative type”.

For a slightly modified problem–assuming a Dirichlet condition for u and v on some part ΓD of Ω–and for obstacles of inequality or infinite derivative type, it has been proved in various forms in e.g. [2], [3], [16], [17], [18] that there is a bifurcation of stationary solutions of (1.1), (1.2) for values d=(d1,d2)R+2 for which the problem without the obstacle is linearly stable: Hence, the obstacle can break the stability and lead to a new type of stationary patterns.

The main tools for the mentioned bifurcation results were to prove that the mapping degree of an associated map is zero close to certain hyperbolas Cn mentioned above. The main result of the current paper is to prove that an analogous degree result (which is actually a non-bifurcation result) holds even without assuming any artificial Dirichlet conditions. Moreover, we show the result also for obstacles of finite derivative type.

Note that omitting the Dirichlet conditions is not a trivial step, in general. For instance, in case Γ1=, it was observed in  [4], [8] that for obstacles of inequality type one has a rather different bifurcation behavior than with a Dirichlet condition (and by  [22] this result carries over for obstacles of infinite derivative type).

To our knowledge, obstacles of finite derivative type have not yet been treated at all in connection with reaction–diffusion systems. For this reason, we also give a (rather simple) example showing how our main result (which is actually a result about the nonexistence of bifurcation) can be used to prove the existence of bifurcation for obstacles of finite derivative type.

We point out that, although a major feature of our result is the avoidance of any Dirichlet conditions, our main result covers also the case with additional Dirichlet conditions and thus contains the earlier mentioned degree results of e.g.  [2], [3] as a special case. In fact, our result applies in a much more general Hilbert space framework which we introduce in the next section.

Section snippets

Translation into a Hilbert space framework

Assuming that (g1,g2) is differentiable at (0,0) with Jacobian (bij)ij we can rewrite the problem of finding stationary solutions of (1.1) (i.e. ut=vt=0) in the form d1Δu=b11u+b12v+f1(u,v)d2Δv=b21u+b22v+f2(u,v)on  Ω, with fi(0,0)=0 and Dfi(0,0)=0. We assume that gi and thus fi are continuous and have subcritical growth near , i.e. |fi(u,v)|a(1+|u|+|v|)p with some constants a,p(0,) where we assume in case N>3 that p<NN2. In case N=2, we need no hypothesis about p, and in case N=1, we do

The main abstract result

For the rest of the paper, we consider an even more general abstract setting than in Section  2. However, we always have these and similar situations in mind.

Let H0 be a real Hilbert space, and η0 a parameter which will be fixed once and for all. We assume throughout: A0:H0H0  is compact linear symmetric, and every eigenvalues  λ  of  A0  satisfies  λ0  and  ηλ1. In particular, A0 is nonnegative definite. For reasons which will be clear later on (e.g. in the proof of Lemma 3.3), we write

A differentiable version of the main abstract result

Although we did not yet discuss how to verify (H1)–(H3), we note already that (H3) is essentially a certain sign condition about the map M0 close to (d0,0). Unfortunately, this means that if the map M0 satisfies (H3) then nonlinear perturbations of M0 will typically fail to satisfy (H3), even if the perturbation terms are small of arbitrarily large order. This is not acceptable for the applications to obstacle problem from the introduction, because the maps Fi from Section  2 typically do not

Verification of the Hypotheses

In this section, we discuss how to verify the hypotheses (H1)–(H5) of the previous section for given maps β, M0, and M. Hypotheses (H1) and (H4) will be verified in three important particular cases in Section  6, but we will also show in this section that these hypotheses will then actually hold for certain (not necessarily small) perturbations of the maps. This will be crucial for the calculation of the degree later on.

All arguments in this section center around the fact that we will assume

Degrees in three special cases

In this section we show how to verify the hypotheses of Theorem 4.1 in some important cases and use this to prove that some related mapping degrees are zero.

Applications

The knowledge about the degree leads to various type of results. For one, using  [6] (see also  [12]), we obtain for the case that the nonlinearity F is single-valued and satisfies some Lipschitz-type conditions for obstacles of variational inequality type that the corresponding time-dependent problem (1.1), (1.2) fails to be asymptotically stable if d=(d1,d2) is such that the corresponding local degree deg(id,A(d)+F(d,),PK,0) is not one: Recall that Theorem 6.2 states that this degree is zero

Acknowledgments

The authors want to thank the referees for their very careful reading and corrections of inaccuracies of the first version. The second author acknowledges financial support by RVO:67985840.

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