Local Well-Posedness to the Cauchy Problem for an Equation of the Nagumo Type

Lizarazo, Vladimir and De la cruz, Richard and Lizarazo, Julio and Endoh, Daiji (2022) Local Well-Posedness to the Cauchy Problem for an Equation of the Nagumo Type. The Scientific World Journal, 2022. pp. 1-16. ISSN 2356-6140

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Abstract

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 .

1. Introduction
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 [1], the equation (1) was used to model chemotaxis (see equation (55) in [1]). Organisms which use chemotaxis to locate food sources include amoebae of the cellular slime mold Dictyostelium discoideum, and the motile bacterium Escherichia coli [1]. 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 [1] 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 [2].

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 [8] 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 [9]. 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

Item Type: Article
Subjects: Q Science > Q Science (General)
Depositing User: APLOS Library
Date Deposited: 02 Jul 2022 08:11
Last Modified: 02 Jul 2022 08:11
URI: http://eprints.asianrepository.com/id/eprint/558

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