Degree, instability and bifurcation of reaction–diffusion systems with obstacles near certain hyperbolas
Introduction
Let be a bounded domain with a Lipschitz boundary. Consider the reaction–diffusion system with boundary conditions with measurable given sets .
We are interested in bifurcation of stationary solutions of (1.1), (1.2) near a constant equilibrium which is linearly stable without diffusion terms (i.e. if ). 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 , in particular . Note that and then actually denote the difference to the equilibrium so that also negative values of and have a physical interpretation.
In case , Turing’s famous effect of “diffusion-driven instability” [13] implies that for the above problem the equilibrium becomes instable for certain values of diffusion speeds . Moreover, if one considers as a diffusion parameter, one typically has bifurcation of spatially nonconstant stationary solutions from the equilibrium when crosses certain hyperbolas (cf. Section 3 for details).
We are interested in the question of what changes in these observations when we impose on obstacles described by the (in general multivalued) functions . More precisely, we will assume that or where is (or behaves in a certain weak sense near 0 qualitatively similar to) the function see Fig. 1. For the particular case , the boundary condition (1.2) becomes on for the well-known Signorini boundary condition which is the “classical” way to describe an obstacle on : Physically, if and denote the difference of the concentration of two chemicals to equilibrium, this obstacle can mean that on the concentration 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 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 goes under the equilibrium concentration. Therefore, a more realistic assumption would be that the source produces “much” for but cannot guarantee necessarily in all cases that . This would mean that , 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 see Fig. 3, which we call an obstacle “of finite derivative type”.
For a slightly modified problem–assuming a Dirichlet condition for and on some part of –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 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 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 , 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 is differentiable at with Jacobian we can rewrite the problem of finding stationary solutions of (1.1) (i.e. ) in the form with and . We assume that and thus are continuous and have subcritical growth near , i.e. with some constants where we assume in case that . In case , we need no hypothesis about , and in case , 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 be a real Hilbert space, and a parameter which will be fixed once and for all. We assume throughout: In particular, 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 ()–(), we note already that () is essentially a certain sign condition about the map close to . Unfortunately, this means that if the map satisfies () then nonlinear perturbations of will typically fail to satisfy (), 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 from Section 2 typically do not
Verification of the Hypotheses
In this section, we discuss how to verify the hypotheses ()–() of the previous section for given maps , , and . Hypotheses () and () 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 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 is such that the corresponding local degree 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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