A new estimate of the minimal wave speed for travelling fronts in reaction – diffusion – convection equations

In this paper we prove a new estimate of the threshold wave speed for travelling wavefronts of the reaction–diffusion–convection equations of the type vτ + h(v)vx = [D(v)vx]x + f (v) where h is a convective term, D is a positive (potentially degenerate) diffusive term and f stands for a monostable reaction term.


Introduction
Reaction-diffusion equations have been intensively investigated, since they model various biological and chemical phenomena.The simplest model in this context is the Fisher equation . This equations has been subsequently generalized involving a general reaction term f (v), vanishing at v = 0 and v = 1, a general diffusive term D(v), which can be positive in [0, 1] (non-degenerate case), or positive in (0, 1] with D(0) = 0 (degenerate case), and a term H(v) to include possible convective phenomena ∂v ∂τ A relevant class of solutions of equation (1.1) is that of travelling wave solutions (t.w.s.), that is solutions of the type v(τ, x) := u(x − cτ) for some one-variable function u and real constant c, the wave speed.
In the monostable case, that is when the reaction term f is positive in (0, 1), it is known (see [11]) that there exists a threshold wave speed c * such that equation (1.1) supports travelling wave solutions having speed c if and only if c ≥ c * .The importance of the value c * is due to the fact that in many cases the t.w.s.having speed c * is the limit profile, for large times, of the solutions of the equation (1.1) (see, e.g., [3,6,7,9]).
For the value c * it is known the following estimate (see [11]) where h is the derivative of the one-variable function H. Estimate (1.2) is valid provided that the product function D • f is differentiable at u = 0. Notice that, when the convection is constant and the product function D • f is concave, then the previous estimate reduces to an equality: In [1] the following upper estimate was achieved for the threshold value c * , valid just in the case of no convective effects which improves the previous upper estimate (1.2) in the particular case h ≡ 0. Such a result was achieved by means of a variational approach which seems to be not appropriate for equations involving convective effects.A different variational principle was proposed in [2], where, under the further assumption that h ∈ C 1 ([0, 1]), it was proved that which improves the upper estimate in (1.2), since the latter can be obtained by (1.4) taking The value of c * has been exactly determined for some special type of reaction-diffusionconvection equations.For instance, in the case when (see [15,16]).Other computations in presence of convective processes have been stated by Gibbs and Murray (see [14]) in the case D(u) = 1, f (u) = u(1 − u) and h(u) = ku, for which it was showed (see also [11]) that Moreover, in [5] it was considered the equation with α ≥ 0, β, γ ≥ 0 and k > 0 given constants, for which the authors proved that the value of c * is the following Finally, in the case when f (u) = u m (1 − u), D(u) = h(u) ≡ 0, the function c * = c * (m) has been studied in [4], by means of asymptotical expansions as m → +∞.However, the exact computation of c * , or the study of its properties related to some parameters of the equations, can be carried on just for special equations, for which the explicit solution is known.For general equations the value of c * has to be estimated.
The aim of this paper is to prove the following new upper estimate, valid also for equations involving convective effects, and we show it improves both (1.2) and (1.3).Moreover, we also show that when h is increasing, then the upper bound in (1.5) can be improved by the following We also present some extensions to other reaction-diffusion models, such as equations with bi-stable reaction terms (that is f is negative in (0, α) and positive in (α, 1)) or reactiondiffusion-aggregation equations, in which the diffusivity D can also assume negative values.For these type of equations, the estimate for the speed c * , obtained in the monostable reactiondiffusion case, plays a relevant role.
Assume that A function v(τ, x) is said to be a travelling wave solution (t.w.s.) of (1.1) if there exists a real constant c and an one-variable function u such that v(τ, x) = u(x − cτ) for every (τ, x) in the domain of existence of v.We are interested in t.w.s.connecting the two stationary states v = 0 and v = 1, that is satisfying u(−∞) = 1 and u(+∞) = 0.As it is immediate to verify, the profile u of a t.w.s. is a solution of the following second order boundary value problem Moreover, if a t.w.s. with speed c exists, then it is unique (up to shifts) and it is strictly decreasing whenever 0 < u < 1 (see [11]).
A travelling wave can be a solution in the classical sense (that is a C 1 -function such that the product D(u)u is C 1 as well, or in the weak sense (sharp solutions), that is solutions reaching one or both the equilibria at finite times and with slope possibly non zero.Sharp t.w.s.can appear when the diffusivity vanishes at u = 0 (degenerate case) and/or at u = 1 (doubly degenerate case).We refer to [10] for a discussion about the characterization of the appearing travelling fronts.
The study of the existence or non-existence of t.w.s. is carried on by investigating the solvability of the following associate singular boundary value problem where the dot stands for derivation with respect to the variable u.Indeed, since the profiles are strictly decreasing whenever 0 < u < 1, then equation in (2.2) can be handled as a typical autonomous equation, setting z(u) = D(u)u (t(u)), where t(u) is the inverse function of u(t).However, a careful analysis is needed regarding the behaviour at the equilibria, in order to obtain classical or sharp travelling fronts.The aim of this paper is to improve the estimate of the threshold speed c * , so from now on we investigate the solvability of the boundary value problem (2.3), referring to [10] for the classification of the related travelling fronts.A key result providing a sufficient condition for the solvability of (2.3) is the following.
So, in order to prove the solvability of (2.3), it suffices to show the existence of a negative upper-solution for the equation in (2.3), approaching the origin.

The new estimates
We now state our main results.
Then problem (2.3) admits a solution. Proof.Put The function G is continuous in the compact triangle T. Indeed, the continuity is obvious at every point (x 0 , u 0 ) ∈ T with x 0 < u 0 , whereas as for the continuity at the points of bisector u = x observe that for every pair (x, u) ∈ T there exists a value Let H : T → R be the function defined by Similarly to what we have done above, it is easy to show that H is continuous on the triangle T.
Put M := max u∈[0,1] G(0, u) and N := max u∈[0,1] H(0, u).We have By (3.1) we have 1 4 (c − N) 2 > M and c > N, so for some ε > 0 we have 1 4 (c − (N + ε)) 2 > M + ε and c > N + ε.By the uniform continuity of the functions G and H in the triangle T, there exists a real δ = δ > 0 such that Let us choose a costant L such that M + ε < L < 1 4 (c − (N + ε)) 2 and put Therefore, for every n > 1 δ and every u ∈ 1 n , 1 we have Moreover, by (3.5) we get −Ks ds that is, put φ(u) := −Ku, for every n > 1 δ and every u ∈ 1 n , 1 , we have implying that ζ n (u) > z n (u) for every u ∈ 1 n , 1 , where z n is the (unique) solution of problem (3.8), in the interval 1  n , 1 .So, we conclude that Let us now continue each function z n in the whole interval [0, 1] by setting z n (u) := φ(u) = −Ku for every u ∈ [0, 1 n ), denoting them again z n .Let us now prove that z n+1 (u) ≤ z n (u) for every u ∈ [0, 1].By the construction of z n , this is obvious for 0 ≤ u ≤ 1 n , while, for u > 1 n , since z n+1 n , 1 , this contradicts the uniqueness of the solution of the Cauchy problem Let Z(u) := lim n→+∞ z n (u).Notice that the sequence (z n ) n is equibounded and equicontinuous in each compact subinterval [ ũ, 1], with ũ > 0. Indeed, for n sufficiently large and every u ∈ [ ũ, 1] we have hence the convergence of the sequence (z n ) n towards the function Z is uniform in every compact subinterval [ ũ, 1], implying that for every u ∈ (0, 1).
When h is increasing, then the upper bound given by (1.5) can be improved by (1.6), as stated in the following result.

Theorem 3.2. Assume (2.1). If h is increasing and
then problem (2.3) admits a solution.
Remark 3.3.Notice that estimates (1.5) and (1.6) are not comparable, since when h is increasing the upper bound given by (1.6) can be strictly less than the one given by (1.5), as Example 4.4 in the next section shows.Instead, the validity of (1.6) for a generic convective term h, not necessarily increasing, remains an open problem.

Comparisons with the previous known estimates
In order to compare the various available estimates for c * , it is convenient to discuss separately the case of equations without convection and those with convection.

Comparison for reaction-diffusion equation (no convective effects)
Observe that in this case (1.5) and (1.6) provide the same upper bound: which obviously improves the one given by (1.2), by the mean value theorem.Moreover, in this case the upper bound in (1.4) becomes that is (1.4) furnishes the same upper bound as (1.2).Finally, the next result states that (4.1) improves also the upper bound given by (1.3) (which has not an analogous version for reaction-diffusion-convection equations).

Comparison for reaction-diffusion-convection equations
Obviously, estimate (1.5) improves (1.2), by the mean value theorem, and (1.6) improves (1.5) in the case when h is increasing.In this case, (1.6) provides a better upper bound than (1.5), as the following example shows.Moreover, the relation between (1.6) and (1.4) is unclear, but in the following example, (1.6) provides a better upper bound also with respect to (1.4).

Extension to other reaction-diffusion models
The estimate for the threshold wave c * for the reaction-diffusion-convection equations plays a relevant role also for other type of models.So, we now present the natural extension of known results which can be achieved taking estimate (1.5) into account.Of course, analogous results can be derived using (1.6) when h is increasing.

Reaction-diffusion-convection with bi-stable reaction terms
Another well-known model for reaction-diffusion equations concerns the case of bi-stable reaction terms, that is when the continuous function f is assumed to be negative in (0, α) and positive in (α, 1) for some α ∈ (0, 1).In this case, it was shown that there exists a unique value c such that equation (1.1) supports t.w.s.having speed c = c.As for the estimate of c, in [12] it was proved the following estimate h(u).
Such a relation has been obtained starting from estimate (1.2) for the mono-stable case.Using the same proof as in [12], taking the present estimate (1.5) into account, the estimate of c can be improved by the following

Extension to diffusion-aggregation models
In [8] it was proposed a model of diffusion-aggregation-reaction process (without convective effects), in which the diffusive term can assume negative values too, in order to describe the behaviour of a population tending to cluster into groups for some value of the density u.Assuming the existence of a value β ∈ (0, 1) such that D(u)(β − u) > 0 for every u ∈ (0, 1), u = β, it was proved that there exists a threshold value c * such that equation