Instability of traveling waves for a generalized diffusion model in population problems

In this paper, we study the instability of the traveling waves of a gen- eralized diusion model in population problems. We prove that some traveling wave solutions are nonlinear unstable under H 2 perturbations. These traveling wave solu-


Introduction
In this paper we consider the following equation The equation (1.1) arises naturally as a continuum model for growth and dispersal in a population, see [1].Here u(x, t) denotes the concentration of population, the term g(u) is nonlinear function, denotes reaction term or power with typical example as g(u) = a(1 − u 2 ), a > 0. During the past years, many authors have paid much attention to the equation (1.1), see [2,3,4].Liu and Pao [2] based on the fixed point principle, proved the existence of classical solutions for periodic boundary problem.Chen and Lü [3] proved the existence, asymptotic behavior and blow-up of classical solutions for initial boundary value problem.Chen [4] proved existence of solutions for Cauchy problem.
In this paper we study instability of the traveling waves of the equation (1.1) for g(u) = a(1 − u 2 ), a > 0 .The stability and instability of special solutions for the equation (1.1) are very important in the applied fields.E. A. Carlen, M. C. Carvalho and E. Orlandi [8] proved the nonlinear stability of fronts for the equation (1.1) with g(u) = 0, under L 1 perturbations.We prove that it is nonlinearly unstable under H 2 perturbations, for some traveling wave solution that is asymptotic to a constant as x → ∞.Our proof is based on the principle of linearization.We invoke a general theorem that asserts that linearized instability implies nonlinear instability.
Our main result is as follows This paper is organized as follows.We first find a exact traveling wave solution for the equation(1.1) in Section 2, and then give the proof of our main theorem in Section 3.

Exact Traveling Wave Solution
In this section, we construct an exact traveling wave which satisfies all conditions of theorem 1.1.
If ϕ(x − ct) = ϕ(z) is a traveling wave solution of (1.1), then ϕ satisfies the ordinary differential equation (2.1) Substituting above equation into (2.1),we have Then comparing the order of ϕ, we get We easily proved that and ϕ(z) satisfies the conditions of the theorem.
EJQTDE, 2004 No. 18, p. 2 To prove the theorem 1.1, we first consider an evolution equation where L is a linear operator that generates a strongly continuous semigroup e tL on a Banach space X, and F is a strongly continuous operator such that F (0) = 0.In [9] the authors considered the whole problem only on space X, that is to say, the nonlinear operator maps X into X.However, many equations possess nonlinear terms that include derivatives and therefore F maps into a large Banach space Z.Hence, they again got the following lemma.
Lemma 3.1 [5] Assume the following (i) X, Z are two Banach spaces with X ⊂ Z and u Z ≤ C 1 u X for u ∈ X.
(ii) L generates a strongly continuous semigroup e tL on the space Z, and the semigroup e tL maps Z into X for t > 0, and Then the zero solution of (3.1) is nonlinearly unstable in the space X.
In this paper, we are going to use Lemma 3.1 for proof of the theorem.Definition 3.1 A traveling wave solution ϕ(x − ct) of the equation (1.1) is said to be nonlinearly unstable in the space X, if there exist positive ε 0 and C 0 , a sequence {u n } of solutions of the equation (1.1), and a sequence of time R) is a traveling wave solution of the equation (1.1), then letting w(x, t) = u(x, t) − ϕ(x − ct), we have that is i.e. where So the stability of traveling wave solutions of (1.1) is translated into the stability of the zero solution of (3.2).In order to prove Theorem, taking Z = L 2 (R), X = H 2 (R), we need to prove that the four conditions of Lemma 3.1 are satisfied by the associated equation (3.2).The condition (i) is satisfied, by our choice of Z and X.
Denote the linear partial differential operator in (3.2) by 2) may be rewritten in the form (3.1) So, the condition (iv) is satisfied.
To prove condition (ii) in Lemma 3.1, we need the following two lemmas.
. On the other hand, letting s = ξ 2 , we have Proof.Consider the initial value problem where a(t) is defined in Lemma 3.2 and we use u(t) to denote u(•, t).

EJQTDE, 2004
No. 18, p. 5 By iteration, (3.10) The second term on the right of (3.10) is where By exchanging the order of integration, we get from the third term on the right side of (3.10), (3.12) Therefore (3.9)-(3.12)imply Multiplying both sides of the above inequality by e −25C3M 2 t , we have EJQTDE, 2004 No. 18, p. 6 Integrating the above inequality with respect to t over (0, t), we obtain Observing that v(t) = t 0 u(s) H 2 ds, and substituting above inequality into (3.13),we get Similarly iterating and computing as above, we obtain We now prove that the same curve belongs to the essential spectrum of L.

Lemma 3.4
The essential spectrum of L on H 2 (R) contains that of L 0 .
.14) Thus (3.7) has been proven.To prove (3.8), replacing the first term on the right side of (3.9) by e tL0 H 2 →H 2 u 0 H 2 and using (3.5), we have u(t) H 2 ≤ e t/4 u 0 H 2 + M [6]) also satisfies the above equation.By uniqueness of solution, we know that L generates a strongly continuous semigroup on the Banach space H 2 (R) (see[6]p.344).By Fourier transformation, the essential spectrum of L 0 } does not have a strongly convergent subsequence in H 2 (R).(here we use the definition: λ ∈ σ(L) if and only if L − λ is Fredholm with index zero.)