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 Hs(ℝ). If D > 0, the local well-posedness is given for s > 1/2 and for s > 3/2 if D=0.


Introduction
In this paper, we show the local well-posedness for the following Cauchy problem: where D > 0 is a constant diffusion coefficient, α ∈ (0, 1/2) and ϵ > 0 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]. erefore, u � u(x, t) models the population density, n is a positive integer, and α 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 u � 1 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].
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 D � 0, the equation (1) is a generalized Burgers equation (without viscosity) and nonlinear source term. erefore, from the mathematical viewpoint, the case D � 0 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 H s (R) for s > 1/2 if D > 0, and for s > 3/2 if D � 0. Our proof of local well-posedness is based on the results given in [10][11][12]. We use the Banach fixed point in a suitable complete space to guarantee the existence of local solutions to the problem (1) with D > 0. e 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][11][12][13][14] for more details). When D � 0, 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: R for the real numbers; S(R) for the Schwartz's space usual; f denotes the Fourier transform of f; the inverse Fourier transform will be denoted by ∨; by H s (R), s ∈ R, the set of all f ∈ S ′ (R) such that (1 + ξ 2 ) s/2 f ∈ L 2 (R). H s (R) is called the Sobolev space and it is a Hilbert space with respect to the inner product (f, g) s � R (1 + ξ 2 ) s f(ξ)g(ξ)dξ; C(I; X) for the space of all continuous functions on an interval I into the Banach space X; if I is compact, C(I; X) is seen as a Banach space with the sup norm; C w (I; X) for the space of all weakly continuous functions on an interval I into Banach space X; C 1 w (I; X) for the space of all weakly differentiable functions on an interval I into Banach space X. We also denote by is the unique solution to the following problem.

Local Well-Posedness of the Problem
(1) with D > 0 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) for all u, v ∈ H s (R), where L s (·, ·) is a continuous function, nondecreasing with respect to each of their arguments. In particular, Proof. Observe that en, as H s (R) is a Banach algebra for s > 1/2, we have the following: where e following result is to prove the existence of solutions. e proof is based in standard arguments [10,11]. We only present a sketch of proof.
Sketch of proof. Let M, T > 0 be fixed, but arbitrary. Consider the following: which is a complete metric space with distance Define on the space X(M, T, ψ) the following map: We have the following:   (1). However, and the right hand side of (57) is a integrable function of τ in [0, t]. us, using the dominated convergence theorem, we have as follows: e Scientific World Journal 3 Now, from the mean value theorem for integrals, there exists a value c on the interval (t, t + h) such that and therefore, where z + t is the right derivative. In similar way, we can conclude that the left derivative is i.e., in (0, T), then e proof of this lemma is given in Lemma 7.1.2 in [16].
be the corresponding solutions of equation (9). If s > 1/2, then proof. Let ψ, ϕ, u and v as in the statement of the proposition. Let s > 1/2. From (9) we have as follows: By Propositions 1 and 2, we obtain the following: As for all we have that 1 < e θ(x) < e and e (m/6)θ(m/2)− (6/m+1)θ(m+1/2) is bounded for m ≥ 1. From (21), we obtain as follows: Now, similar to the proof of the previous proposition, we have as follows: us, E 1/2,1 ((bΓ(1/2)) 2 T) is finite (where E 1/2,1 is given in Lemma 1) and we have as follows: is finishes the proof. Finally, from Propositions 3, 5 and 6, we can summarize in the following theorem:

Local Well-Posedness of the Problem
(1) with D = 0 In this section, we show the local well-posedness of the problem (1) with D � 0 using a priori estimate and the parabolic regularization method, the so-called vanishing viscosity method (for more details see [12]).

Lemma 2. Let η(t), a(t) and b(t) be real valued positive continuous functions defined on
and H(r) be positive continuous functions for r ≥ 0, with G strictly increasing and H nondecreasing. Define implies the inequality is is a particular case of the theorem given in [ x k y n− 1− k + (ϵn/2)y n + x 2 + xy + y 2 + (α + 1)(x + y).
proof. We define q(u, w) � n− 1 k�0 u k w n− 1− k . As s > 3/2 thus H s (R) and H s− 1 (R) are Banach algebras. Moreover, we us, using the Cauchy-Schwartz inequality, we have as follows: e Scientific World Journal In particular, |(v, hz x v) s | ≤ C‖z x h‖ s− 1 ‖v‖ 2 s .
proof. Let T s � T s (ψ) be as in eorem 2. Now, we will split the proof into four steps:

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Step 1. First we will show that (u D (t)) D > 0 is a net which converges to a function u 0 ∈ C([0, T s ]; L 2 (R)) in the L 2 − norm, uniformly over [0, T s ].
Let D 1 , D 2 ∈ (0, +∞). en, e Scientific World Journal 7 where ρ is the function defined in the proof of eorem 2. We bound separately each term on the right-hand side of (40) as follows: In order to bound the first term, we have as follows: We can bound the second term by the following: e Scientific World Journal (where q is defined in the proof of Proposition 7). From eorem 2, we obtain as follows: for all t ∈ [0, T s ]. erefore, from the above bounds, we have as follows: Applying Gronwall's inequality to the last relation, we show that there is a constant C > 0 satisfying and so u 0 ∈ C([0, T s ]; L 2 (R)).
Step 2. Now we show that We obtain by Fatou's Lemma as follows: Step 3. We must show that u D ⇀u 0 in H s (R) for all t ∈ [0, T s ] as D ⟶ 0+. First of all, we will show that (u D (t)) D > 0 is a weak Cauchy net in H s (R), uniformly with respect to t ∈ [0, T s ].
In fact, given φ ∈ H s (R) and ϵ > 0, choosing φ ϵ ∈ H s (R) such that ‖φ − φ ϵ ‖ s ≤ ϵ, then and therefore, we have lim D 1 , us, we have that u D ⇀u 0 for all t ∈ [0, T], i.e., e Scientific World Journal for all φ ∈ H s (R). Moreover, since the convergence is uniform for all φ ∈ H s (R), we can conclude that u 0 ∈ C w ([0, T s ]; H s (R)).
Step 4. Finally, we show that u 0 ∈ C 1 w ([0, T s ]; H s− 2 (R)). Let φ ∈ H s− 2 (R). en, Observe that if r > 1/2, f n ⇀f in H r (R) and g n ⇀g in H r (R) then f n g n ⇀fg in H r (R). After, we have uniformly on [0, T s ]. ereby, taking the limit as D ⟶ 0+ in (51), we obtain as follows: Corollary 1. Let u 0 be as in the preceding theorem, then proof. Since t ∈ [0, T s (ψ)]↦u(u − α)(u − 1) + ϵu n u x is weakly continuous in H s− 2 (R) and the Sobolev space is separable, then applying the Bochner-Pettis theorem, it is a strongly measurable function in H s− 2 (R). erefore, exists as a Bochner integral. So, from (53) we conclude that and therefore, u 0 ∈ AC([0, T s ]; H s− 2 (R)).  (1) with where L 0 is as in the Proposition 7 and where t ∈ [0, T] is fixed, h is such that t + h ∈ [0, T], and 〈·| · |〉 s is the H s duality bracket. As t ∈ [0, T]↦ w(t) ∈ H s (R) is bounded and exists in the norm of H s− 2 (R) ⟶ H − s (R), from (58) and (59) we have as follows: Observe that on the right-hand side of (70) is well-defined for t ∈ [0, T s (ψ)] and therefore we can extend (if necessary) u � u(t) to [0, T s (ψ)] as a solution in H s+1 (R).
us, we conclude that Observe that the last inequality is independent of D > 0 and since u D weakly converges and uniformly to u 0 in H s+1 (R), then we have u 0 ∈ C([0, T s (ψ)]; H s+1 (R)).
First, we will estimate (80a). Applying the Cauchy-Schwartz inequality to (80a) we have e Scientific World Journal 13 where L(x, y) � x 2 + xy + y 2 + (α + 1)(x + y) + α and C � L(M, M). Now, we will estimate (80b). Observe that Finally, we estimate (80c). As s > 3/2, there is s 0 such that 3/2 < s 0 + 1 < s. From the Cauchy-Schwartz inequality, we obtain Now, we will estimate each term on the right-hand side of the last inequality. First, observe that where q is defined in the proof of Proposition 7. We also estimate ‖(u τ 0 ) n − (u θ 0 ) n ‖ s 0 ‖u τ 0 ‖ s+1 . From Lemma 4 and the inequality (71), we have for all τ ≤ τ 0 . From Lemma 4, we have where ϱ � (s 0 /s). To estimate the term ‖u τ 0 − u θ 0 ‖ 0 , observe that where q(u, v) � n j�0 u j v n− j , and from Gronwall inequality we have as follows: From Lemma 4,