Research Article | Open Access
Luisa Malaguti, Cristina Marcelli, Serena Matucci, "Continuous Dependence in Front Propagation for Convective Reaction-Diffusion Models with Aggregative Movements", Abstract and Applied Analysis, vol. 2011, Article ID 986738, 22 pages, 2011. https://doi.org/10.1155/2011/986738
Continuous Dependence in Front Propagation for Convective Reaction-Diffusion Models with Aggregative Movements
The paper deals with a degenerate reaction-diffusion equation, including aggregative movements and convective terms. The model also incorporates a real parameter causing the change from a purely diffusive to a diffusive-aggregative and to a purely aggregative regime. Existence and qualitative properties of traveling wave solutions are investigated, and estimates of their threshold speeds are furnished. Further, the continuous dependence of the threshold wave speed and of the wave profiles on a real parameter is studied, both when the process maintains its diffusion-aggregation nature and when it switches from it to another regime.
This paper deals with the reaction-diffusion equation: which provides an interesting model in several frameworks such as the population dispersal, ecology, nerve pulses, chemical processes, epidemiology, cancer growth, chemotaxis processes etc. The great variety of contexts where it meets justifies its interest by the scientific community. We refer to the books [1–3] for an updated illustration of the main properties and applications of (1.1).
In many cases the presence of a convective flux, expressed by the function , is a fundamental ingredient of these models. Indeed, it accounts for external forces acting into the process such as, for instance, control strategies. We assume that is a monostable, that is, of Fisher-type, reaction term; that is, it satisfies According to (1.2), and are stationary solutions of (1.1). Since frequently represents a density, of main interest is the investigation of solutions taking values in the interval and connecting these two stationary states. The family of traveling wave solutions (t.w.s.) is an important class of functions which share this property. It was showed in fact that, at least in some sense, t.w.s. are able to capture the main features of all the solutions taking values in (see, e.g., [4, 5]). We recall that a function is said to be a t.w.s. of (1.1) with wave speed if there exist a maximal interval , with , and a function , satisfying such that is a solution of (1.1) for all . The symbol denotes the closure in of , and the quantity is usually called the wave coordinate.
Since we restrict our analysis to those solutions of (1.1) with values in , we assume defined in such an interval and, throughout the paper, we take and .
A population in a hostile environment usually increases its chances to survive if it is able to produce aggregative movements. Therefore, when (1.1) is a model for population dynamics, it has to take into account also this important aspect. As a prototype of equation (1.1) in this framework, we can consider where , , and denote positive constants while the function accounts for the net rate of growth of the individuals and it satisfies (1.2). In this case , and (1.4) is a generalized version of a model proposed by Turchin  in order to describe the aggregative movements of Aphis varians. Under suitable conditions on the parameters, it is straightforward to show the existence of a value such that when while in . This interesting situation is motivated by the remark that it is highly unlikely to find conspecific in the vicinity at low population densities. It seems then more reasonable to expect that the tendency of a population to aggregate, modeled by negative values of , appears after a certain threshold density level. Therefore, in order to include this type of aggregative behavior into the model, we assume the existence of satisfying The extremal cases and respectively correspond to a purely diffusive and a purely aggregative term. In the former case, that is, , the presence of t.w.s. in these models and their main qualitative properties have been investigated since a long time, and we refer to [2, 7, 8] for details. The latter case; that is, , was recently discussed in . As far as we know, the first detailed investigation concerning the existence of t.w.s. of (1.1) when satisfies (1.5) and appeared in , but there was no convective effect included. Under the additional assumption that we complete in Theorem 2.1 the analysis started in  and show that also (1.1), incorporating a convective behavior, is able to support a continuum of t.w.s., parameterized by their wave speeds which satisfy , and we give an estimate of the threshold speed .
The very nature of aggregative processes causes ill posedness of our model (1.1) when and (see, e.g.,  and the references there contained). However, discrete models underlying (1.1) are well posed, and some numerical computations (see ) seem in good agreement with the information obtained in the discrete setting. Moreover, even in this possible ill-posedness context, the t.w.s. above defined are regular solutions for (1.1), and this increases the interest in studying them.
As it is clear also from the prototype equation (1.4), very naturally these models include real parameters which frequently cause, on their varying, the transition of the process a diffusive to a diffusive-aggregative regime or from a diffusive-aggregative to a purely aggregative one. The main aim of this paper lies in the analysis of these behaviors, so we consider the more general dynamic: which continuously depends on the real parameter . We take , , and in with and for all . Further, we assume the existence of a continuous function such that We discuss both the case , corresponding to a diffusive-aggregative dynamic, as well as the case or 1, where (1.7) is, respectively, a purely diffusive reaction-convection equation and a purely aggregative one. In Theorem 2.2 we prove that the threshold value is always a lower semicontinuous function (l.s.c.) on the whole interval ; hence, in particular, the lower semicontinuity holds when the process (1.7) switches from a purely diffusive to a diffusive-aggregative and from the latter to a purely aggregative behavior. We provide quite general conditions (see Theorem 2.2 and Proposition 2.4) either guaranteeing that is continuous on or that it fails to be continuous for or . Two simple illustrative examples complete this discussion. In Theorem 2.6 we show the continuous dependence, on the parameter , of any family of profiles corresponding to a continuum of wave speeds . These are the main results of the paper; their statements appear in Section 2 while their proofs can be found in Section 4. Notice that the function can be extended in a continuous way outside the interval , so that the model also includes the purely diffusive or aggregative dynamic. However, the study of the continuity in the former case has been carried on in , while the latter case can be treated by a suitable change of variable, and this discussion is included in Theorem 4.2.
As for the methodology, it is easy to see that every wave profile of (1.1), having wave speed , corresponds to a regular solution of the boundary value problem: where . More precisely, a solution of (1.12) in is a function satisfying the boundary conditions (1.3).
Since, in addition, is strictly monotone in every interval where (see, e.g., Theorem 3.7), the investigation of t.w.s. can be reduced to the study of a first-order singular boundary value problem. In order to investigate this problem, we mainly use comparison-type techniques, that is, suitable upper and lower solutions. In Section 3 we report and complete the original discussion developed in  and show the equivalence between the first-order singular problem and the existence of t.w.s. for (1.1).
2. Statements of the Main Results
Theorem 2.1. Assume that conditions (1.2), (1.5), (1.6) and (2.1) hold. Then, there exists a value such that (1.1) supports t.w.s. if and only if , and the profile is unique, up to shifts. Moreover, the threshold value satisfies the following estimates, depending on the value : and, if ,
As stated in the Introduction section, the main aim of the paper is the study of the continuous dependence, both of the threshold value and of the wave profiles , with respect to the coefficients , , and appearing in (1.1). More precisely, we are mainly interested in studying the continuous dependence when a change in the type of dynamics occurs: from a purely diffusive to a diffusive-aggregative one or from a diffusive-aggregative to a purely aggregative one.
To this aim, we consider a continuous function satisfying conditions (1.9), and we introduce a real parameter varying in , inside the coefficients of (1.1), in such a way that , , and are continuous functions of two variables. We further assume (1.8), (1.10), and (1.11) and suppose that satisfy (2.1) for each .
In this framework, the threshold value is a function of the parameter , say , and, for every , the profile (modulo shifts) of the t.w.s. having speed is a function of , where is the wave variable, say .
As for the function , the following result will be proven.
Theorem 2.2. Let and be continuous functions satisfying (1.8), (1.9), (1.10), and (1.11). For all , assume that condition (2.1) holds for , and let . Then, is l.s.c..
Further, if for some we have then is continuous at .
With regards to the behavior of at the endpoints and , if then is continuous at , and, similarly, if then is continuous at .
Remark 2.3. Assume now that exists and it is continuous on . In this case it is easy to verify that all conditions in (2.5), (2.6), and (2.8) are satisfied as equalities. Furthermore, according to (1.8) and (1.10), we obtain that and also while and for . The continuity of , hence, implies that . Consequently, conditions (2.7) and (2.9), respectively, reduce to Therefore, under conditions (2.10) and (2.11), the function is continuous on the whole interval . Of course, when , then (2.10) and (2.11) trivially hold, and is continuous on .
As an immediate consequence of the estimates (2.2), (2.3), and (2.4), if the relation (2.7) or (2.9) in the previous theorem does not hold, then the function could be discontinuous at the endpoints, as the following result states.
Proposition 2.4. Under the same conditions of Theorem 2.2, if then is not continuous at . Similarly, if then is not continuous at .
Example 2.5. Let and , and let Observe that conditions (1.8), (1.9), and (1.10) are trivially satisfied. Moreover, uniformly converges to as , and is continuous in . Furthermore, also condition (2.1) is satisfied. Indeed, is differentiable in , and for every , and finally is differentiable at since for we have , and then is differentiable at . Since we have , whereas then (2.12) holds for every continuous function such that . Therefore, the function is not continuous at ; that is, looses its continuity in the transition from a diffusive-aggregative dynamic to a purely diffusive one.
The last result concerns the continuous dependence of the profiles of the t.w.s..
Theorem 2.6. Let and be continuous functions satisfying (1.8), (1.9), (1.10), and (1.11). For all , assume that condition (2.1) holds for , and let . Take continuous and such that for every , and let be fixed.
Let be the profile of the corresponding travelling wave solution of (1.1) with speed such that , for some fixed . Then converges to uniformly on all the real line.
Remark 2.7. In all the results stated in this section, if convective effects are not present; that is, , then assumptions (1.6) and (1.11) can be removed, as we show in the end of the proof of Theorem 3.7.
3. Reduction to a Singular First-Order Equation
Given continuous functions , with , satisfying conditions (1.2), (1.5), (1.6), and (2.1), let us consider the following singular first-order boundary value problem where is an unknown real constant. By a solution of the b.v.p. (3.1), we mean a differentiable function , satisfying all the conditions in (3.1). Of course, any possible solution of (3.1) is of class in any compact interval not containing the value .
As we will show at the end of this section, the solvability of problem (3.1) is equivalent to the existence of solutions of (1.12). For this reason, we now study the existence of solutions to problem (3.1). To this aim, in  it was proved the existence of solutions to the singular boundary value problem (3.1) in the case when is everywhere positive. In particular, the following Proposition is a consequence of [12, Lemma 2.2] combined to [12, Theorem 1.4].
Proposition 3.1. Let be continuous functions such that , in , and exists finite. Then, there exists a value , satisfying the following estimate: such that the problem admits solutions if and only if . Moreover, the solution is unique.
As far as the regularity of the solution at the endpoints, the following result holds.
Corollary 3.2. Under the same assumptions of Proposition 3.1, if exists finite, then exists finite too, and
Proof. Assume by contradiction that
For a fixed , let be a sequence converging to and such that
we obtain . Replacing this inequality in (3.3) evaluated for , and passing to the limit, we have
Similarly, taking a sequence converging to and such that
arguing as before, we obtain . Combining with (3.8) we get , which is impossible, due to the arbitrariness of . Then,
Since (3.3) can be rewritten as follows:
then would imply that as , a contradiction. Hence, is real, nonnegative.
If , then from (3.11) we get that admits limit as , and this limit must coincide with . Passing to the limit in (3.11) we get that is a root of If , we immediately have as , since . If , since the above trinomial has two discordant zeros, we deduce that Summarizing, if , then (3.13) holds.
Assume now that . If ; from (3.11) we get as , a contradiction. Hence, implies . Moreover, observe that Thus, satisfies (3.13) also in the case .
By a simple change of variable ( and ), it is easy to check the validity of the following results in the case when is everywhere negative in .
Proposition 3.3. Let be continuous functions such that , in , and exists finite. Then, there exists a value , satisfying the following estimate: such that problem (3.3) admits solutions if and only if . Moreover, the solution is unique.
Corollary 3.4. Under the same assumptions of Proposition 3.3, if exists finite, then also exists finite, and
Combining the previous results, we are able to prove an existence theorem for problem (3.1).
Theorem 3.5. Let be continuous functions, with , differentiable at , satisfying condition (2.1) and such that
for a given constant .
Then, there exists a value , satisfying estimate (2.4), such that problem (3.1) admits solutions if and only if . Moreover, the solution is unique.
Proof. Consider the boundary value problem (3.3) for . Since all the assumptions in Proposition 3.1 are satisfied, there exists a threshold value , satisfying (3.2), such that the b.v.p (3.3), with and , is uniquely solvable if and only if . Similarly, considering the problem (3.3) for , by Proposition 3.3, there is a threshold value , satisfying (3.15), such that this problem is uniquely solvable if and only if . Let . Clearly, satisfies (2.4). Moreover, if problem (3.3), with , is solvable for some , then also problem (3.3) with , and problem (3.3), with and , are solvable for the same value . Hence, .
Conversely, let us fix , and let be the function obtained gluing the unique solution of problem (3.3), with and and the unique solution of problem (3.3) with . In order to prove the assertion, it suffices to observe that, from Corollaries 3.2 and 3.4, the glued function is differentiable at the point .
Remark 3.6. Notice that in the previous theorem the required differentiability of at cannot be dropped. Indeed, in view of the proof, if the right derivative of at does not coincide with the left one, then and is not differentiable at .
Concerning the equivalence between the solvability of problem (3.1) and the existence of t.w.s., the following result holds.
Theorem 3.7. Let the assumptions (1.2), (1.5), (1.6), and (2.1) be satisfied. The existence of solutions to (1.12) for some is equivalent to the solvability of problem (3.1) with the same . If , then condition (1.6) can be removed.
Proof. Let be a t.w.s. of (1.1) with wave speed , that is, a solution of problem (1.12) satisfying condition (1.3). Observe that if with , for some , then since , we have that is a point of proper local maximum for when , while is a point of proper local minimum when . Hence, assume now by contradiction the existence of a value such that and . By the boundary condition , we get the existence of a value such that and . Integrating (1.12) in , we obtain
in contradiction with condition (1.2). Similarly, if with for some , then there exists a value such that and . Integrating (1.12) in , we get again a contradiction. Therefore, whenever , .
Put and . If then is constant in , and we can define if ; if . Of course, since (1.12) is autonomous, is a t.w.s. of (1.1) in its existence interval and satisfies , for every , . Hence, is strictly decreasing and invertible in . Let , . It is easy to see that the function is a solution of (3.1) for the same value of .
Now assume that , and let be a solution of (3.1) for some admissible real value . Let be the unique solution of the Cauchy problem: defined on its maximal existence interval . Since whenever , we have that and . Moreover, according to (1.6) and Corollary 3.2, the limit exists and it is not zero, implying that . Consider now the unique solution of the initial value problem: on its maximal existence interval . Again, for implies and . Moreover, according to (1.6) and Corollary 3.4, we obtain that ; hence, also the former limit is not zero and . Then, as we made above, with a suitable shift, it is possible to glue the two functions , in such a way to have a unique , defined on some interval , with , on the whole , and satisfying (1.12) and (1.3).
When or one considers just one of the Cauchy problems above defined, whose solution is a t.w.s. for (1.1).
Finally, assume . It is well known (see, e.g., ) that is always strictly positive in this case. Moreover, (see, e.g., [2, Lemma 1]), Hence, both limits are finite and nonzero independently on the value of . So condition (1.6) can be removed.
Remark 3.8. If , then . Indeed, , for , implies . Therefore, if for some , we have that Being an admissible value, the wave speed satisfies condition (2.4); hence, there exists such that for , implying . With a similar argument, it can be showed that implies .
4. The Continuous Dependence Results
As we mentioned in the Introduction section, both the dependence on the parameter of the minimal wave speed and of the wave profile corresponding to the speed were already studied in  for the diffusive case, that is, in case , for every . More in detail, the following result holds (see [11, Theorems 4.1 and 4.2]).
Theorem 4.1. Let be continuous functions, with for every and for every . Assume that (1.8) and (2.1) are satisfied for all . Then, the threshold value is a l.s.c. function in . Moreover, if for some then is continuous at .
By means of a change of variable, in the aggregative case, that is, in case for every , it can be proved that the analogous result holds.
Theorem 4.2. Let be continuous functions, with for every and for every . Assume that (1.8) and (2.1) are satisfied for all . Then, the threshold value is a l.s.c. function in . Moreover, if for some then is continuous at .
Proof of Theorem 2.2. In view of the equivalence proved in Theorem 3.7, we analyze the continuous dependence of the threshold value for the problem:
where is an unknown real constant.
For every we have ; hence, we can consider the boundary value problem:
Put and , problem (3.22) is equivalent to the normalized one: Indeed, a function is a solution of (3.22) if and only if the function is a solution of (). Hence, if and denote, respectively, the minimal values of the parameter for which (3.22) and () are solvable, we have Since is continuous and positive in , from Theorem 4.1 we have that is lower semicontinuous at for every .
Similarly, for every , consider the problem: Using an argument analogous to the previous one and taking into account that, in this case, , we deduce that the threshold value for problem () is a lower semicontinuous function at every .
Let and let be the threshold value for problem (4.3). Since (see the proof of Theorem 3.5), with lower semicontinuous functions, then is lower semicontinuous at every .
If , then it results in , and The case can be treated in a similar way, and the semicontinuity of in is proved.
Concerning the study of the continuity of under condition (2.5), let us first consider a value . Analogously to the arguments above developed, we can consider the problem (3.22) and the normalized one (). Notice that, if satisfies condition (4.1), then also the function does. Indeed, we have by the positivity and the continuity of and the differentiability of at 0. Hence, Theorem 4.1 assures that the function is continuous at , and in force of (4.4) we conclude that is continuous too. Analogously, considering the problem (), we can prove that the threshold value is continuous at every . Since (see the proof of Theorem 3.5), then is continuous at every .
It remains to study the continuity of at the values and , where the dynamic change its nature. We limit ourselves in considering the value , since the proof for is analogous.
Let be fixed, and observe that, by assumption (2.6), real values and exist, such that Let be such that for every . Then Taking estimate (3.15) into account, with and , we get implying By the arbitrariness of , assumption (2.7), and estimate (3.2) with and , we have Therefore, since as , we conclude that and the function is upper semicontinuous in . Taking into account the lower semicontinuity of in the whole interval , it follows the continuity in . The proof for the value is similar.
In order to prove Theorem 2.6 about the convergence of the profiles, we need some preliminary results.
Let denote the unique solution of problem (4.3), emphasizing the dependence on the parameters and . The following result holds.
Lemma 4.3. Let and be continuous functions satisfying conditions (1.8), (1.9), (1.10), and (2.1) for every . Let be a continuous function such that for every . Finally, let be the unique solution of problem (4.3).
Then, for every , there exists a real value such that the function is continuous in .
Proof. Let be fixed. We divide the proof in two cases. Case 1 (). Assume (the proof in the other case is analogous), and let be a compact interval containing the value . Put and let denote the solution of problem () (see the proof of Theorem 2.2). By applying [11, Theorem 5.1] to the problem () in , we get the convergence of to as , uniformly in ; that is, for a fixed ,
Taking account of the continuity of and of the uniform continuity of , there exists a value such that for every we have and
Since whenever and , we deduce that
for every and , that is, the uniform convergence of to in . Taking account of the uniform continuity of in , we get as . Finally, since , by the continuity of we also get
and the continuity of the function at is proved.
Notice that, for the opposite case , the usual change of variable allows to apply [11, Theorem 5.1].Case 2 (). For every , let Notice that the function is well defined since by assumption (1.6) we have . Moreover, is continuous and negative for every . Since from Corollary 3.2 it follows that Let us fix a value and define We have From the definition of , we get Therefore, Hence, there exists a value such that that is, Take now such that and whenever . Observe that, by (4.20), we have . Since , then for every , there exists a value such that Assume, by contradiction, the existence of a value and of a point such that Then, in contradiction with the definition of . Therefore, we deduce that Similarly, we can prove that The previous conditions (4.30) and (4.31) imply that Let us now take and consider the functions We have and, from (4.23), we get