Erratum to: “Travelling waves for a system of equations” [Nonlinear Anal. 68(12) (2008) 3909–3912]

Nonlinear Analysis 69 (2008) 2771–2772

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Erratum

Erratum to: ‘‘Travelling waves for a system of equations’’ [Nonlinear Anal. 68(12) (2008) 3909–3912] Shair Ahmad a,∗ , Alan C. Lazer b , Antonio Tineo c a

Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, United States

b

Department of Mathematics, University of Miami, Coral Gables, FL 33124, United States

c

Departamento de Matemáticas, Universidad de Los Andes, Merida, Venezuela

a b s t r a c t We correct an oversight in the proof of Theorem 1.1 in [Shair Ahmad, Alan C. Lazer, Antonio Tineo, Traveling waves for a system of equations, Nonlinear Analysis 68 (12) (2008)], which, in addition to considerably reducing the assumptions made in [M.M. Tang, P.C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Ration. Mech. Anal. 73 (1980). [2]], gives a better derivation and a sharper estimate of the wave speed. © 2008 Elsevier Ltd. All rights reserved.

First of all, we may assume, without loss of generality, that in Lemma 2.2, di = 1. Proof of Theorem 1.1. For each N sufficiently large, there exists an integer kN , 1 ≤ kN ≤ n, such that ykN (τkNN ) = for j 6= kN ,

aj 2

≤ yj (τ ) ≤ aj .

ak

N

2

and

N kN

Let u (t ) be the vector uN (t ) = (y1 (t + τkNN ), . . . , yn (t + τkN )). Then the vector (uN (0), (uN )0 (0), kN ) is bounded in N

2n+1

R

aj 2

. Hence there exists a subsequence of this vector which converges to the vector (α, β, j∗ ) where α = (α1 , . . . , αn ),

≤ αj ≤ aj , 1 ≤ j∗ ≤ n, β ∈ Rn and β ≤ 0. Let u(t ) be the solution of the system u00j (t ) + cu0j (t ) + uj fj (u(t )) = 0,

1≤j≤n

satisfying the initial conditions u(0) = α , u0 (0) = β . Clearly, since uN (t ) converges to u(t ) and uN 0 (t ) converges to u0 (t ) uniformly on compact intervals, then 0 ≤ u(t ) ≤ α and u0 (t ) ≤ 0 for all t in (−∞, ∞) (apply Lemma 2.2 to uN (t )). We claim that 0 < u(t ) ≤ a and u0 (t ) < 0, −∞ < t < ∞. Suppose, on the contrary, that for some j, 1 ≤ j ≤ n, uj (t¯) = 0. Then a u0j (t¯) = 0 since there is a minimum at t¯. Hence, by uniqueness, uj (t ) ≡ 0, contradicting 2j ≤ uj (0) ≤ aj . Next, suppose we have uj (t¯) = aj . Then since we have a max at t¯, u0j (t¯) = 0 and therefore, since u0j (t ) ≤ 0, then u00j (t¯) = 0. It follows from the equation u00j (t ) + cu0j (t ) + uj fj (u(t )) = 0 that fj (u(t¯)) = 0. Therefore, by conditions A1 − A2 , u(t¯) = a. By the same reasoning, all components u0i (t¯) = 0, 1 ≤ i ≤ n. We note that w(t ) ≡ a is a solution. Therefore, by uniqueness, w(t ) = u(t ), contradicting uj∗ (0) =

∗

DOI of original article: 10.1016/j.na.2007.04.029. Corresponding author. E-mail address: [email protected] (S. Ahmad).

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.07.004

aj∗ 2

.

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S. Ahmad et al. / Nonlinear Analysis 69 (2008) 2771–2772

Now we wish to show that u0 (t ) < 0. Suppose that u0j (t¯) = 0. Then, since we have a minimum, u00j (t¯) = 0. But, as shown above, 0 < u(t¯) < a, and hence fj (u(t¯)) > 0. Therefore, it follows from the corresponding differential equation that u00j (t¯) < 0, and hence u0j (t ) > 0 for t < t¯, contradiction. Now, using Lemma 2.4 and an argument similar to that in [1], it follows that limt →∞ u(t ) = 0 and limt →−∞ u(t ) = a. References [1] S. Ahmad, A.C. Lazer, An elementary approach to travelling front solutions to a system of N competition-diffusion equations, Nonlinear Anal. 16 (10) (1991). [2] M.M. Tang, P.C. Fife, Proppagating fronts for competing species equations with diffusion, Arch. Ration. Mech. Anal. 73 (1980).