CONTINUOUS DEPENDENCE IN FRONT PROPAGATION OF CONVECTIVE REACTION-DIFFUSION EQUATIONS

. Continuous dependence of the threshold wave speed and of the travelling wave proﬁles for reaction-diﬀusion-convection equations


1.
Introduction.We consider a scalar parabolic reaction-diffusion-convection equation u t + h(u)u x = d(u)u x x + f (u), with t ≥ 0, x ∈ R, and u(x, t) ∈ [0, 1] (1) where h ∈ C[0, 1] is an arbitrary nonlinear convective term, d ∈ C 1 [0, 1] stands for a diffusivity coefficient, i.e. d(u) > 0 for all u ∈ (0, 1), and f ∈ C[0, 1] is a Fisher-type reaction term, that is f (u) > 0 for every u ∈ (0, 1) and f (0) = f (1) = 0. ( This equation maintains a constant interest in mathematical literature since it is a model for the investigation of several problems in population dynamics, chemical processes, epidemiology, cancer growth, nerve pulses and ecology (see, e.g., [8] and [16]).In some processes, in addition to diffusion and reaction, motion is also due to convection forces.The monograph [16] contains several models concerning ecological control strategies, predator-prey pursuit and evasion, ion-exchange columns, chromatography etc.These models include, as a fundamental ingredient, a convective flux.Equation ( 1) is also used for the study of dispersion due to population pressure (see [17]) and for the study of chemotaxis behavior under some simplifying assumptions, and it appears when modeling the Gunn effect in semiconductors (see e.g., [4,9]).Special cases of (1) occur also in the investigation of the heat transfer with convective transport (see [5] and references therein).In all these areas, a particular relevance is held by the so called travelling wave solutions of equation (1).These solutions u satisfy u(x, t) = U (x − ct), for some sufficiently regular onevariable function U (the wave profile) and constant c ∈ R (the wave speed), and they connect the stationary states 0 and 1, that is satisfy the boundary conditions U (−∞) = 1, U (+∞) = 0. Put ξ = x − ct (the wave coordinate), U (ξ) is a solution of the following ordinary differential equation The case d(u) > 0 for u ∈ [0, 1] (non-degenerate case), has been investigated in [14].Under the following assumption it is proved that there exists a threshold value c * satisfying , such that (1) admits travelling wave solutions with speed c if and only if c ≥ c * .Moreover, any travelling wave solution is decreasing, hence 0 ≤ U (ξ) ≤ 1 for every ξ ∈ R, and for every admissible speed c the travelling wave solution is unique (up to a variable shift).The value c * is usually called minimal (or threshold) wave speed.
Travelling wave solutions play an important role in the investigation of (1).Indeed, it was proved (see [3] and [10]) that, for special cases of (1) and a wide class of initial conditions, any solution of (1) approaches the travelling wave solution having speed c * when t → ∞.
Nevertheless, the previous setting in which the travelling waves are actually defined and regular on the whole real line, is not satisfactory in various concrete situations.Indeed, for instance, when (1) models the spatial spreading of a population initially located in a bounded environment, since individuals diffuse with a finite speed, then equation (1) realistically should have the property of finite speed of propagation (see [7]), that is any solution satisfying a compactly supported initial condition, maintains a compact support in any time.This occurs if and only if the travelling wave solution having the threshold speed c * vanishes at a finite value of the wave variable (see [7] and [12]).As it is well known, when the diffusion coefficient is positive (as in the heat equation), in general the dynamics does not exhibit such a behavior, contrary to the degenerate parabolic equations, occurring when d(0) = 0. We also refer to [18] for some concrete models where the diffusion coefficient vanishes at both the equilibria 0 and 1 (doubly-degenerate case).
A prototype of equation (1) in the degenerate case is the porous media equation, with reaction and convection terms, where a, c, k > 0 and b ∈ R. The exact value of the threshold speed of its travelling wave solutions was recently obtained in [9].
In some models (see, e.g., [18]) the diffusion coefficient vanishes at both the equilibria 0 and 1 (doubly-degenerate case).In the degenerate case [doubly-degenerate case] (1) can support travelling wave solutions attaining the value 0 [both the values 0 and 1] at a finite value of the wave variable.
Only recently a detailed discussion and a sharp classification of the qualitative properties of the solutions have been carried on for such a type of equations.In particular, it was shown that if d(0) = 0 (see [9] and [12]) and/or d(1) = 0 (see [12]), then the travelling wave having speed c * attains the equilibria 0 and/or 1 at finite values and the set J := {ξ ∈ R : 0 < U (ξ) < 1} is a halfline or a bounded interval.In this case, U, d(U )U ′ ∈ C 1 (J), U is a solution of ( 3) in the open interval J, and satisfies the boundary conditions lim together with the following ones: The previous conditions can be adopted as a unifying definition of travelling wave, both for the degenerate and the non-degenerate case, since ( 6) is trivially satisfied if J = R (see [12]).
For the special case d(u) = u k with k > 0, the relevant interpretation of c * as the asymptotic speed of propagation of any solution u(x, t) with a compactly supported initial condition was obtained in [10] for h ≡ 0, and in [15] for a wider family of models including convective terms.
Clearly, equation (5) depends on the constants a, b, k, and more in general in (1) the diffusivity, the convection and reaction terms, can be viewed as parameters.Hence, the interest about the dependence on the parameters arises very naturally and the aim of this paper is just to investigate the continuous dependence of the threshold speed c * and of the wave profiles U (ξ) on the nonlinear terms appearing in (1).
Recently, some researchers started this study.In [6] the continuous dependence and further regularity of the minimal speed c * was established in the particular nondegenerate case h(u) ≡ 0, d(u) ≡ 1 and f (u) = u m (1 − u), m ≥ 1. Subsequently, a general study of the continuous dependence of c * and of the corresponding profile U * was carried on in [1] recovering the degenerate equations, but again in absence of convection (h(u) ≡ 0).Such an investigation is based on a variational approach introduced in [2], where the following characterization of the minimal speed was obtained 1 where F (u) := u 0 f (s) ds and H 1 (e t ) := {u ∈ H 1 loc (R) : e t u(t) ∈ H 1 (R)}.By using this approach, in [1] the continuous dependence of the minimal speed with respect on the diffusion and reaction terms, and the continuous dependence of the corresponding profiles U * , have been proved.More in detail, given a sequence (d n ) n of positive diffusion terms, uniformly convergent to d 0 , and given a sequence (f n ) n of Fisher-type reaction terms, such that fn(u) u uniformly converges to f0(u) u in (0, 1], then c * (d n , f n ) → c * (d 0 , f 0 ).Moreover, as for the corresponding profiles U * n , in [1] it was showed that, when for every α > 0 if d 0 (0) = 0 but inf n≥0 ḋn (0) > 0. This approach also allowed to discuss the fastness of the rate of decay at 0 of the solutions u * , both in the case of constant diffusion, and in the case of non-constant diffusion (degenerate or non-degenerate).
Due to the presence of a non-constant term multiplying the first derivative u x , the variational technique used in [1] seems to be not appropriate for the study of the reaction-diffusion-convection equation (1).So, we introduce an alternative approach for this analysis based on differential inequalities applied to the following first order singular boundary value problem (7) to which the investigation can be reduced, due to the monotonicity of the wave profiles (see, e.g., [12]).In Section 2 we discuss the main properties of ( 7) and the most important comparison techniques used for its investigation.The study of the behavior of the minimal speeds and of the fronts in the case of monotone convergence of the nonlinear terms is treated in Section 3. The discussion about the convergence of c * n (h n , d n , f n ) to c * (h 0 , d 0 , f 0 ) in the general case is contained in Section 4. Firstly we prove that, under the sole conditions ensuring the existence of travelling wave solutions, the lower semi-continuity of c * is guaranteed (see Theorem 4.1 and Example 1).As showed in [1] for the case of no convective effects, further regularities have to be assumed to achieve the continuous dependence of c * on the nonlinear terms of the equation (see Theorem 4.2).The present investigation provides an extension to reaction-diffusion-convection equations of the study carried on in [1].However, Theorem 4.2 improves the analogous convergence result in [1], even in the particular case of a null convective effect (see Remark 1 and Example 2).The problem of the convergence of the wave profiles is studied in Section 5.The main result is the following, whose proof is presented at the end of the section.
Let U n (ξ) be the profile of the corresponding travelling wave solution with speed The choice of the sequential (discrete) point of view to study the continuous dependence on the parameters of the problem, is essentially due to a simpler notation requirement.Of course, all the results presented in this paper could be rewritten also in the setting of terms depending on a continuous parameter k ∈ R, assuming that h, d, f are continuous functions of the two variables (u, k) ∈ [0, 1] × R. In this framework, the continuous dependence of the minimal speed means the continuity of c * (k) as a function of the parameter k and the convergence of the profiles in C 1 loc (J) means that U (t; k) and U ′ (t; k) are continuous with respect to both the variables.This last statement is guaranteed by the uniform convergence on compact sets of R.
2. Notations and preliminary results.This section is devoted to the statement of preliminary results which will be used in the following.Most of these results are generalizations to the present case of analogous ones proved by the authors in [13] and [14], see also the discussions before the statements.
Till Section 4 we focus our attention on the following singular first order boundary value problem where c is a given constant, and h, g : [0, 1] → R are continuous functions.
Comparing problem ( 8) with (7), notice that the function g(u) replaces the product f (u)g(u).Since f and d are positive in (0, 1) and f vanishes at 0 and 1, throughout Sections 2-4 we always assume that The investigation of the solvability of problem ( 8) has been carried on in [14].In particular, the following Proposition is consequence of [14, Theorem 1.4] (for d(u) ≡ 1 and h replaced by −h), combined to [14, Lemma 2.2].
Proposition 1. Assume that Then, there exists a real value c * such that (8) is solvable if and only if c ≥ c * , and the solution is unique.Moreover, c * satisfies where D + g(0) := lim inf u→0 + g(u) u .In our study we will deal with equations having various nonlinear terms and speeds.So, from now on, we will use the notation z(u; c, h, g) to denote the (unique) solution of (8), in order to avoid misunderstandings.Similarly, the notation c * (h, g) will stand for the minimal admissible speed.
Our approach for handling problem ( 8) is based on differential inequalities and upper and lower solutions techniques.The following Lemma is a key result in this matter, and it will be also used for proving the main results in the subsequent sections.
Lemma 2.1.For a fixed constant c ∈ R, assume that there exists and ζ(0 Proof.The proof of this Lemma is based on a preliminary comparison result for strict inequalities, which was proved in [13] (see Lemma 8) in the special case h(u) ≡ 0. The same argument works also in the present general context.From [13, Lemma 8] it follows that, if (11) holds with strict inequality, then the statement of Lemma 2.1 holds with both the inequalities in ( 12) strict.Assume now that (11) holds.Then, for every ǫ > 0 we have for every u ∈ (0, 1).
Hence, the boundary value problem possesses a solution.If c * denotes the threshold value for problem (8), we have c * ≤ c + ǫ.By the arbitrariness of ǫ > 0, we get c * ≤ c, that is problem (8) has a solution z(u).Notice that if z(u) < ζ(u) for some u ∈ (0, 1) then , for every u ∈ (0, 1).
As an immediate consequence, the following monotonicity result holds.
Lemma 2.2.Let h 1 , h 2 , g 1 , g 2 be continuous functions, with g 1 , g 2 satisfying (9).Assume that h 1 (u) ≤ h 2 (u) and g 1 (u) ≤ g 2 (u) for every u ∈ [0, 1].Consider problem (8) for h = h i , and g = g i , i = 1, 2 and let c * i , i = 1, 2 be the threshold value of the wave speed.Then c * 1 ≤ c * 2 .Moreover, let c 1 ≥ c 2 ≥ c * 2 , and denote by z i the solution of problem (8) for h = h i , c = c i and g = g i , i = 1, 2. Then we have for every u ∈ (0, 1), by Lemma 2.1 we get that problem (8), for h = h 1 and g = g 1 , is solvable for c = c * 2 , and this implies c for every u ∈ (0, 1), implying that there exists a solution ζ of problem (8 . By the uniqueness of the solution, we conclude that ζ = z 1 .
We will use also the following auxiliary result about the interval of existence of the solutions.Lemma 2.3.Each negative solution z(u) of the equation Proof.First observe that the solution z(u) of equation ( 14) can not blow up at a finite value.Moreover, if there exists a value u 0 ∈ (0, 1) such that z(u) → 0 as u → u + 0 , then since g(u 0 ) > 0, we deduce ż(u) → +∞ as u → u + 0 , a contradiction.Therefore, z(u) can be extended on (0, b]. z(ū) = ż(ū), a contradiction.Hence, ū = 0.The last result of this section concerns the regularity of the solution of (8) at 0. Corollary 1.Let ġ(0) exist (finite).Then there exists ż(0) and Moreover, if ġ(0) = 0 and z * denotes the solution corresponding to the threshold value c * , we have ż * (0) = h(0) − c * .

Monotone convergence.
We begin the study of the continuous dependence from the parameters for problem (8) in the case of monotone sequences.Theorem 3.1.Let (h n ) n≥1 and (g n ) n≥1 be increasing sequences of continuous functions defined in [0, 1], convergent to continuous functions h 0 and g 0 , respectively.Assume that condition (9) holds for g = g n , n ≥ 0. Then the sequence of the corresponding threshold values (c * n ) n≥1 given by Proposition 1 is increasing and converges to c * 0 .Moreover, let (c n ) n≥1 be a decreasing sequence of real numbers converging to a value c 0 ≥ c * 0 .Then, denoted by z n (u) the solution z(u; c n , h n , g n ), the sequence (z n ) n≥1 is decreasing and uniformly convergent to z 0 .

Proof. By virtue of Lemma 2.2 we get c
Let us consider a decreasing sequence (c n ) n converging to a value c 0 ≥ c * 0 ≥ ĉ.By Lemma 2. (z 1 (u)) .
Moreover, being h 1 (u) ≤ h n (u) ≤ h 0 (u) for every n ≥ 1 and u ∈ [a, b], we can apply the dominated convergence theorem obtaining that is z is a solution in [a, b] of the differential equation in (8), for h = h 0 , c = c 0 , g = g 0 .By the arbitrariness of the interval [a, b], we get that z solves the differential equation on the whole interval (0, 1).Finally, since z 0 (u) ≤ z(u) ≤ z 1 (u) for every u ∈ (0, 1), we obtain that z(0 + ) = z(1 − ) = 0, and then z is a solution of problem ( 8) for h = h 0 , c = c 0 , g = g 0 .By the uniqueness of the solution, we conclude that z = z 0 , i.e. (z n ) n≥1 is a decreasing sequence convergent to z 0 .The uniformity follows from the Dini's theorem.Now consider the decreasing sequence (ĉ+ 1 n ) n , where ĉ is defined in (16).Observe that The function z(u) = inf n z n (u) is therefore well defined and reasoning as before we get that it is a solution of the equation ż(u) = h 0 (u) − ĉ − g0(u) z(u) in (0, 1).Since z(u) ≥ u 0 h 1 (s) ds−c 1 u for every u ∈ (0, 1), we get z(0 + ) = 0. Therefore, according to Lemma 2.1, the boundary value problem is solvable.This implies that ĉ ≥ c * 0 .Taking ( 16) into account, we conclude that ĉ = c * 0 , i.e. c * n → c * 0 .An analogous result for the reversed monotonicity does not hold, as the following example shows.
Since ġn (0) = 2 for n ≥ 1, while ġ0 (0) = 1, from the convergence of (h n ) n to h 0 we conclude that Nevertheless, a partial continuous dependence result for the reversed type of monotonicity holds.Theorem 3.2.Let (h n ) n≥1 and (g n ) n≥1 be decreasing sequences of continuous functions defined in [0, 1], convergent to continuous functions h 0 and g 0 , respectively.Assume that condition (9) holds for g = g n , n ≥ 0. Then the sequence of the corresponding threshold values (c * n ) n≥1 given by Proposition 1 is decreasing and inf c * n ≥ c * 0 .Moreover, let (c n ) n≥1 be an increasing sequence of real numbers, converging to c 0 , and such that c 1 ≥ c * 1 .Then, denoted by z n (u) the solution z(u; c n , h n , g n ), n ≥ 0, the sequence (z n ) n≥1 is increasing and uniformly convergent to z 0 .
and vanishes at u = 0 and u = 1.Moreover, for a fixed closed interval [a, b] ⊂ (0, 1) we have Hence, being h 0 (u) ≤ h n (u) ≤ h 1 (u) for every n ≥ 1 and u ∈ [a, b], we can apply the dominated convergence theorem obtaining, as in the proof of Theorem 3.1, that z is a solution of problem ( 8) for h = h 0 , c = c 0 , g = g 0 .By the uniqueness of the solution, we conclude that z = z 0 .
4. Continuity of the threshold values c * .As we showed in Example 1, when dealing with a decreasing sequence of reaction terms (g n ) n , one can not expect the convergence of the threshold values c * n , but at most a semicontinuity property: inf c * n ≥ c * 0 .The lower semicontinuity of c * n is a general property, as the following result shows.Theorem 4.1.Let (h n ) n and (g n ) n be sequences of continuous functions uniformly convergent to functions h 0 and g 0 respectively.Assume that condition (9) holds for every n ≥ 0. Then lim inf Proof.For every u ∈ [0, 1] and n ≥ 1, let ĥn (u) := min{h 0 (u), inf Due to the uniform convergence of the sequence (h n ) n to h 0 , it is easy to check that the functions ĥn are well defined and the sequence ( ĥn ) n is increasing and uniformly convergent to h 0 in [0, 1].
Let us now show that each function ĥn is continuous.To this aim, let us fix n ∈ N. First of all, notice that ĥn is upper semicontinuous, since it is the infimum of a family of continuous functions.Let us assume, by contradiction, that ĥn (u 0 ) > lim s→+∞ ĥn (u s ) =: L for some u 0 ∈ [0, 1] and some sequence (u s ) s converging to u 0 in [0, 1].Let ǫ > 0 be such that ĥn (u 0 ) − 3ǫ > L.Then, by virtue of the definition of L, the continuity of h 0 and the uniform convergence of (h n ) n towards h 0 , an integer n * ≥ n exists such that for every n, s ≥ n * .Hence, we deduce that ĥn (u s ) = min{h 0 (u s ), min n≤k≤n * h k (u s )} for every s ≥ n * .Since the function u → min{h 0 (u), min n≤k≤n * h k (u)} is continuous on all [0, 1], we get ĥn (u s ) → min{h 0 (u 0 ), min n≤k≤n * h k (u 0 )} ≥ ĥn (u 0 ), in contradiction with (19).
Similarly one can show that each function ĝn is well defined and continuous, and that the sequence (ĝ n ) n is increasing and uniformly convergent to g 0 .Moreover, being g n (0) = g n (1) = 0 for every n ≥ 0, we get ĝn (0) = ĝn (1) = 0 for every n ≥ 1. Similarly we get g n (u) > 0 in (0, 1) for every n ≥ 1.Finally, since ĝn (u) ≤ g n (u) in [0, 1], we get D + ĝn (0) < +∞ for every n ≥ 1.Therefore, according to Proposition 1, for every n ≥ 1 there exists a real value σ * n such that the boundary value problem is solvable if and only if c ≥ σ * n .Moreover, we can apply Theorem 3.1 to deduce that the sequence (σ * n ) n≥1 is increasing and convergent to c * 0 .Since ĥn (u) ≤ h n (u) for every n ≥ 1 and every u ∈ [0, 1], by virtue of Lemma 2.2, we get σ * n ≤ c * n , which implies lim infc * n ≥ c * 0 .
In view of Example 1, in order to obtain the continuity of the threshold value c * , we need to add some further requirements, concerning the infinitesimal asymptotic of g n (u) − g 0 (u) as u → 0. Theorem 4.2.Let (h n ) n and (g n ) n be sequences of continuous functions uniformly convergent to functions h 0 and g 0 respectively.Assume that condition (9) holds for every n ≥ 0 and let ġ0 (0) exists finite.
Let us now consider the initial value problem and let ψ n denote its unique solution.By Lemma 2.3 we get that ψ n (u) is defined on (0, δ], and taking (24) into account, z 0 (u) < ψ n (u) for every n ≥ n and u ∈ (0, δ].Thus, ψ n (0 + ) = 0 for every n ≥ n.From the continuous dependence on the data for problem (26), we get the existence of an integer ñ ≥ n such that ψ n (1) < 0 for every n ≥ ñ.Then, by applying Lemma 2.1, we deduce that problem ( 8) is solvable for c = c * 0 + ǫ, h = h n and g = g n .This yields c * n ≤ c * 0 + ǫ for every n ≥ ñ and the assertion follows from Theorem 4.1.
Assume now ż0 (0) = 0. Observe that from (15) we get ġ0 (0) = 0; moreover by Corollary 1 we deduce c * 0 = h 0 (0).If the strict inequality holds in formula (21), then we get g n (u) < g 0 (u) for n large enough and u in a right neighborhood of 0. Hence, in this case (22) holds and the proof proceeds as above.So, let us now consider the case lim sup u→0, n→∞ g n (u) − g 0 (u) u = 0.
Remark 1. Theorem 4.2 improves the analogous convergence result proved in [1], even in the case of absence of convective effects.Indeed, assumption (21) is weaker than the uniform convergence of the sequence ( gn(u) u ) n in (0, 1), as the following example shows.with m > 1 and h(u) ≥ 0 for every u ∈ [0, 1].Taking d(u) := mu m−1 , this equation can be seen as a particular case of (1).Supposing that f is differentiable at 0 when 1 < m < 2, according to Theorem 4.2 we are able to state that the minimal admissible speed c * (m) is a continuous function of the parameter m (see also the discussion in Introduction about the extension from the discrete to the continuous point of view).Observe that in [9] the exact value of c * (m) was determined in the particular case h(u) = bu m−1 , f (u) = cu(1 − u m−1 ), with b ∈ R, c > 0.
5. Convergence of the profiles.In this section we will prove Theorem 1.1 about the convergence of the wave profiles.According to the approach used in this paper, we first prove the continuous dependence of the solutions of the singular problem (8).
Theorem 5.1.Let (h n ) n and (g n ) n be sequences of continuous functions uniformly convergent to functions h 0 and g 0 respectively, and let (c n ) n be a sequence in R convergent to c 0 , satisfying c n ≥ c * n for every n ∈ N. Assume that condition (9) holds for every n ≥ 0, and let z n (u) := z(u; c n , h n , g n ), n ≥ 0. Then the sequence (z n ) n converges to z 0 , uniformly in each compact interval [a, b] ⊂ (0, 1).