Local Well-Posedness to the Cauchy Problem for an Equation of the Nagumo Type
In this paper, we show the local well-posedness for the Cauchy problem for the equation of the Nagumo type in this equation (1) in the Sobolev spaces . If , the local well-posedness is given for and for if .
In this paper, we show the local well-posedness for the following Cauchy problem:where is a constant diffusion coefficient, and is a small positive quantity. In , the equation (1) was used to model chemotaxis (see equation (55) in ). Organisms which use chemotaxis to locate food sources include amoebae of the cellular slime mold Dictyostelium discoideum, and the motile bacterium Escherichia coli . Therefore, models the population density, is a positive integer, and is a parameter which determines the minimal required density for a population to be able to survive (for normalized population density, i.e., such that is the maximum sustainable population). Balasuriya and Gottwald  studied the wave speed of travelling waves for the equation (1). Also, they have the numerical evidence for the wave speed of travelling waves for the equation (1). Other results related to the equation (1) can be found in .
When , the equation (1) is called a Nagumo equation or bistable equation [3–7] in which case the model describes an active pulse transmission line simulating a nerve axon.
Also, we can see the equation (1) as a generalized viscous Burgers equation with a source term. Dix  proved local well-posedness of the viscous Burgers equation with a source term using a contraction mapping argument. Moreover, for the classical Burgers equation (without viscosity) is well known that classical solutions cannot exits for all time, but weak global solutions can be established . In addition, the uniqueness of the weak solution depends on some entropy condition. Observe that when , the equation (1) is a generalized Burgers equation (without viscosity) and nonlinear source term. Therefore, from the mathematical viewpoint, the case is very interesting to study the existence and uniqueness of classical solution.
In this paper, we show the local well-posedness for the Cauchy problem to the equation of the Nagumo type (1) in the Sobolev spaces for if , and for if . Our proof of local well-posedness is based on the results given in [10–12]. We use the Banach fixed point in a suitable complete space to guarantee the existence of local solutions to the problem (1) with . The Banach fixed point technique has been widely used to show existence and uniqueness of solutions to differential equations in Banach spaces (for instance, see [10–14] for more details). When , we use the parabolic regularization method to show local well-posedness for the Cauchy problem (1) (e.g., [12,15]).
We will use the following notation: for the real numbers; for the Schwartz’s space usual; denotes the Fourier transform of ; the inverse Fourier transform will be denoted by ; by , , the set of all such that . is called the Sobolev space and it is a Hilbert space with respect to the inner product ; for the space of all continuous functions on an interval into the Banach space ; if is compact, is seen as a Banach space with the sup norm; for the space of all weakly continuous functions on an interval into Banach space ; for the space of all weakly differentiable functions on an interval into Banach space . We also denote by , , the semigroup in generated by the operator where , i.e., is a -semigroup of contractions in , . Moreover, is the unique solution to the linear problem associated with (1), i.e., is the unique solution to the following problem.
Proposition 1. Let , , , and . Then, there exists a constant , depending only on , such that
In particular, for all .
When there is no risk of confusion, we will use the notations for , for , and .
2. Local Well-Posedness of the Problem (1) with
In this section, we use the Banach fixed point in a suitable complete metric space to show the existence of local solutions for integral equation (9) in Sobolev space for . In addition, the uniqueness of the solution and continuous dependence are established.
Proposition 2. Let be fixed. Then, is a continuous map from into and satisfies the estimates as follows:for all , where is a continuous function, nondecreasing with respect to each of their arguments. In particular,
Proof. Observe that . Then, as is a Banach algebra for , we have the following:whereThe following result is to prove the existence of solutions. The proof is based in standard arguments [10,11]. We only present a sketch of proof.
Proposition 3. Let be fixed, , , and is defined by (2). Then, there exists and a unique function satisfying the following integral equation:
Sketch of proof. Let be fixed, but arbitrary. Consider the following:which is a complete metric space with distance . Define on the space the following map:We have the following:(1)If then (2)We can choose sufficiently small such that (3)There exists such that is a contraction on So, has a unique fixed point in which satisfies the integral equation (40) where .
Proposition 4. The problem (1) is equivalent to the integral equation (40). More precisely, if and is a solution of (1), then satisfies the integral equation (40). Conversely, if and is a solution of (40) then and satisfies (1).
proof. Assume that is a solution of (1). Then, , . So, satisfies the integral equation (40). Conversely, assume that is a solution of (40). For , let . Then, for arbitrary,However,and the right hand side of (57) is a integrable function of in . Thus, using the dominated convergence theorem, we have as follows:Now, from the mean value theorem for integrals, there exists a value on the interval such thatand therefore, .
After, in . where is the right derivative. In similar way, we can conclude that the left derivative is in . So, and . As is the solution of the linear problem (3), we conclude that and satisfies (1).
Lemma 1. Suppose , , , , , is nonnegative and is locally integrable on . Ifi.e., in , thenwhere , with and for .
The proof of this lemma is given in Lemma 7.1.2 in .
Proposition 5. Let and be the corresponding solutions of equation (9). If , thenwhere , and (here is given by previous lemma).
proof. Let and as in the statement of the proposition. Let . From (9) we have as follows:By Propositions 1 and 2, we obtain the following:where . Let . Observe that is finite. In fact, where . Therefore, and from Lemma 1 we have thatAs for all , with , we have that and is bounded for . From (21), we obtain as follows:So, and , with , is convergent.
Proposition 6. Let . Then, the map is continuous in the following sense: if in and , where , are the corresponding solutions (of the problem (1) with ). Let . Then, there exists a positive integer such that for all and
proof. As is a continuous function a , then there exists such that for all . Let . Therefore, is defined on for all . It follows that for all and satisfies where . Therefore, for all and . Now, similar to the proof of the previous proposition, we have as follows:Let . Thus, is finite (where is given in Lemma 1) and we have as follows:This finishes the proof.
Finally, from Propositions 3, 5 and 6, we can summarize in the following theorem:
Theorem 1. Let . The problem (1) is locally well-posed in .
3. Local Well-Posedness of the Problem (1) with D = 0
In this section, we show the local well-posedness of the problem (1) with using a priori estimate and the parabolic regularization method, the so-called vanishing viscosity method (for more details see ).
Lemma 2. Let , and be real valued positive continuous functions defined on . Let and be positive continuous functions for , with strictly increasing and nondecreasing. Define and Then, the inequalityimplies the inequalitywhere , , and .
proof. This is a particular case of the theorem given in  [pp. 78].
Proposition 7. Let be fixed. Then, satisfies the estimatefor all , where .
proof. We define . As thus and are Banach algebras. Moreover, we have that and . Thus, using the Cauchy–Schwartz inequality, we have as follows:
Lemma 3. (T. Kato). Let and be fixed and are real valued functions. Then, there exists a constant such that
In particular, .
proof. See Lemma A.5. in .
Theorem 2. Let be fixed. For , consider the initial value problem (1) with initial data and let be the corresponding solution of (1) for some . Then, there exists a , depending on , such that can be extended to the interval , and there is a function such that and .
proof. Using the inner product in and Lemma 3 we have thatThen, for all , where is the maximally extended solution of the following problem.For , from the problem (32) we obtain as follows:and integrating from 0 to we have as follows:From Lemma 2 with , , and we have the following bound:for , where . Observe that , since the function is strictly decreasing for , and there exists an unique such that . Therefore, we can choose such that . Moreover, the function is increasing on , and therefore we have thatfor all .
For the case , from (32) we have thatand from Lemma 2 we have , for where . So, we can choose such that , and therefore, we conclude thatfor all .
As, for all , and since and do not depend on , the usual extension method shows that we must have for all , where is any positive number satisfying .
Theorem 3. Let be fixed. If , then there exists a and a function such that , and satisfies (1) with , in the weak sense, i.e.,for all and .
Moreover, for all , where is as in Theorem 2.
proof. Let be as in Theorem 2. Now, we will split the proof into four steps:
Step 1. First we will show that is a net which converges to a function in the norm, uniformly over .
Let . Then,