Minimal Travelling Wave Speed and Explicit Solutions in Monostable Reaction-Diffusion Equations

We investigate the connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a scalar monostable reaction-diffusion equation.


Introduction
In this short paper we investigate the somewhat puzzling connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a monostable reaction-diffusion equation. More precisely, it often happens that the explicitly computable travelling wave solution is the solution with minimal speed. Moreover, for parameter-dependent problems with a parameter-dependent family of explicit solutions, it is common for there to be in fact a switching between the minimal speed being given by the explicit solution for some parameters, while for others it is given by the so-called linear speed, defined as the minimal value for which the problem linearised about the unstable steady state has a suitable eigenvalue. For a particular set of equations, of a type often encountered in applications, we formulate sufficient conditions for each of these phenomena to occur.
The plan of the paper is as follows. In this section, we introduce scalar monostable reactiondiffusion equations, define what we mean by a minimal speed, and discuss the linear (pulled) and the non-linear (pushed) regimes.
In section 2, we define the set of exactly solvable equations and prove a result connecting the minimal wave speed and the speed of the explicit travelling wave solution.
Finally, in section 3 we consider conditions for the exchange of minimality between the linear minimal speed and the speed of the explicit travelling wave solution.
Throughout, in all our proofs we only use two tools: the variational principle due to Hadeler and Rothe [3] and the integrability characterisations of the minimal speed proved by Lucia, Muratov and Novaga in [5].
We consider reaction-diffusion equations of the form where β ∈ R is a parameter, and f is a monostable nonlinearity, i.e., In the travelling wave frame z = x − ct, c ≥ 0, setting U (z) = u(x, t), and denoting derivatives with respect to z by primes, (1) becomes We seek monotone fronts connecting 1 and 0, i.e., solutions U (z) of (2) such that lim z→−∞ U (z) = 1 and lim z→∞ U (z) = 0.
Linearisation around the rest point with U = 0 shows that there cannot be any monotone fronts connecting 1 and 0 for c < c l := 2 f ′ (0). Phase plane analysis shows that there exists c min ≥ c l such that there exists a monotone front for all c ≥ c min ≥ c l . Determining c min is often of interest in applications, see e.g. [1] for a discussion.
Definition 1 If c min = c l , we say that we are in the case of linear selection mechanism ("pulled case") and if c min > c l , of nonlinear selection mechanism ("pushed case").
The basis of almost all analysis of monotone fronts in the scalar monostable case (2) is the following construction: As U (z) is a monotone solution, its derivative is a well-defined function of U . Set F (U ) := −U ′ . Note that F (U ) is non-negative. Also, F (0) = F (1) = 0. Now, On the other hand, by the chain rule, Hence the problem of solving U ′′ + cU ′ + f (U ) = 0 with the conditions that lim z→−∞ U (z) = 1 and lim z→∞ U (z) = 0 is equivalent to solving Using this construction, we have the Hadeler-Rothe variational principle [3]: where

Exact solvability
We are interested in the situation when (2) has a solution U (z) that can be determined by quadratures. A sufficient condition is: The travelling wave equation of (3) is solvable by quadratures if f can be written in the form Proof. In this case a solution of (3) is from which U can be computed by quadratures.
In fact, we can compute the speed without solving for the front profile: We will describe as the solvable case the situation in which the nonlinearity f (u, β) satisfies the conditions of Lemma 2. In the solvable case, we have that Note that the fact that A(β) > B(β) follows from the conditions of Lemma 2.
Of course, by the definition of minimal speed, we always have that

Minimality Exchange
In this section, for a nonlinearity f (u, β) of solvable type, we investigate conditions under which there exists a value β * , such that for values β to one side of β * , c min (β) = c l (β), and for values of β to the other side of β * , c min (β) = c nl (β), so that at β * minimality is exchanged between c l (β) and c nl (β). This is what we call a minimality exchange. Examples, two of which we outline below, are discussed in [3,5] and the isotropic case of [1], which is also investigated in [2,7].
First note that for a minimality exchange, c l (β) and c nl (β) must clearly intersect. Therefore the equation must have a solution, which is equivalent to demanding the existence of β * such that A(β * ) = 2B(β * ).
Hence, for instance, in any equation (1) with solvable f (u, β) such that A(β) = 2B(β) + 1, there can never be a minimality exchange between the linear and the nonlinear speeds.
Before continuing with the analysis, we present two concrete examples of minimality exchange. In [3,Eq. (27)], Hadeler and Rothe consider the nonlinearity which can be put into the framework of Lemma 2 by setting The solution of A(β) = 2B(β) is therefore β * = 2, the nonlinear speed is and it is shown in [3] that a minimality exchange occurs at β = β * , with c min (β) = c l (β) for β < β * and c min (β) = c nl (β) for β > β * .
We now establish our general results, starting with a sufficient condition for nonlinear selection.
Proof. For any c > 0, denote by H 1 c (R) the completion of C ∞ 0 (R) with respect to the norm If U (z) is an explicit travelling front with −U ′ = F (U ) = γh(U ), by L'Hôpital's rule Hence for those values of the parameter β for which c nl (β) < 2γ, U ∈ H 1 c nl (β) (R) and hence for such β, by Corollary 2.7 of [5] (see also Proposition 2 of [1]), c(β) is the (nonlinear) minimal wave speed. The claim then follows by (6) and (7).
To formulate our next results, we set We adapt some arguments from [1].
there is a minimality exchange at β = β * .
Theorem 5 fully characterises minimality exchange when L = 1, that is, when h ′ (u) attains its supremum L at u = 0, which holds in particular when h is concave. If L > 1, however, the situation is less clear. Lemma 3 clearly still implies that c min (β) = c nl (β) > c l (β), so in particular nonlinear selection holds, if A(β) < 2B(β), and linear selection holds, with c min (β) = c nl (β) = c l (β) if A(β) = 2B(β), but whether it is possible to have again nonlinear selection for some β with A(β) > 2B(β), either with the minimal speed corresponding to the explicit solution or another value, is not obvious. The estimate (13) only applies when A(β) > 2LB(β), and even in that range, (13) is no longer sufficient to imply linear selection if L > 1.
In Theorem 8 below, we present a result complementary to Theorem 5 that makes no assumption on h beyond the hypotheses in Lemma 2, but instead imposes monotonicity conditions on the dependence of A and B on β. This yields a partial answer to what happens when L > 1 and A(β) > 2B(β). We begin with the following preliminary result, based on [5, Theorem 2.8], which forms the basis for the alternative sufficient condition for minimality exchange in Theorem 8.
The following is an immediate consequence of Lemma 6.
We can now prove our second set of sufficient conditions for minimality exchange.
Note that for the two concrete examples of minimality exchange discussed in Section 3, both Theorem 5 and Theorem 8 apply.

Conclusions
In this article we have focussed on a class of parameter-dependent monostable reaction-diffusion equations with explicit travelling-wave solutions and used this class to explore the phenomenon of minimality exchange, when the minimal wave speed switches from a linearly determined value to the speed of the explicitly determined front as a parameter changes. Two alternative sets of sufficient conditions for minimality exchange are proved, in Theorems 5 and 8. Why there should be such an exchange, not only from linear selection to nonlinear selection, but to nonlinear selection given by an explicit solution, is quite puzzling at first sight. Our framework here provides insight into why minimality exchange of this type occurs, and includes concrete examples from [1,2,3,5]. The proofs owe much to a variety of tools for determining whether there is linear or nonlinear selection -in particular, ideas developed previously in the special case of an isotropic liquid-crystal model [1], as well as general results from [3,5]. Some additional interesting material about minimal wave speeds is given in [2, Section 10.1.1], including Theorem 10.12, which provides sufficient criteria that can be used to identify cases when a given explicit solution has the minimal wave speed, and the examples that follow.