Global Attractivity of a Diffusive Nicholson′s Blowflies Equation with Multiple Delays

Lemma 1. (i) The solution N(t, x) of (4)–(6) satisfies N(t, x) ≥ 0 for $$.

(ii) If ψ(θ, x)≢  0 on D_τ, then the solution N(t, x) of (4)–(6) satisfies N(t, x) > 0 for $$.

Definition 2. A lower-upper solution pair for (4)–(6) is a pair of suitably smooth function v and w such that

• (i)

  v ≤ w in $$,

• (ii)

  v and w satisfy

  $$ ()
  for all $$ with $$, and

• (iii)

  v(θ, x) ≤ φ(θ, x) ≤ w(θ, x),   (θ, x) ∈ D_τ.

Lemma 3. Let (v, w) be a lower-upper solution pair for the initial boundary value problem (4)–(6). Then, there exists a unique regular solution N(t, x) of (4)–(6) such that v ≤ N ≤ w on $$.

Lemma 4. (i) The solution N(t, x) of (4)-(6) satisfies

(ii) There exists a constant K = K(ψ) ≥ 0 such that N(t, x) ≤ K on $$.

Proof. Let w(t) be the solution of the following Cauchy problem:

Solving the equation, we have

Taking

By Lemma 3, there is a unique regular solution N(t, x) such that

Note that

Therefore, the formula (8) is correct, and there exists one K(ψ) > 0 such that $$ for any t ∈ (−τ, ∞) and

So we complete Lemma 4.

Theorem 5. Assume that $$, then every solution N(t, x) of (4)–(6) tends to N₀ = 0 (uniformly in x) as t → +∞.

Proof. By Lemma 4, without loss of generality, let $$ for $$. Under the condition $$, we can get

Define m(t) and y(t) to be the solutions of the following two delay equations, respectively:

By using the similar methods to prove Lemma 4, we can get that

Because of N(t, x) < 1/a, for any $$, m(t) ≤ φ ≤ y(t) < 1/a, one can get

Therefore, from Definition 2, (m(t), y(t)) is a lower-upper pair of (4)-(5) with initial condition m(θ) ≤ ψ(θ, x) ≤ y(θ) on D_τ. Consequently, by Lemma 3, we have

By Theorem 1 of Luo and Liu [16], it follows from $$ that the solutions m(t) and y(t) of (17) both satisfy

Hence, we complete the proof of Theorem 5.

Theorem 6. If $$, then every nontrivial solution N(t, x) of (4)–(6) satisfies

Proof. Let f(x) = xe^−ax, then the function f(x) is increasing on (0, (1/a)) and decreasing on ((1/a), +∞), f(1/a) = max _x∈[0,∞)f(x), $$ for $$. Let $$, then it is not difficult to verify that the function g(y) satisfies the following conditions:

•  

   (g₁) the function g(y) is increasing on (0, (1/a)) and decreasing on ((1/a), +∞), $$,

•  

  (g₂) g(y) > δy for y ∈ (0, N^*) and g(y) < δy for y ∈ (N^*, +∞).

There are now two possible cases to consider.

Case  1 (N^* < 1/a). In view of Lemma 4, we may also assume without loss of generality that every solution N(t, x) of (4)–(6) satisfies

Let $$, $$, $$ and $$. By (23), we have

From Lemma 1(ii), let

Let I_∞ = {1,2, …}. Now, we define two sequences {z_k} and {y_k} to satisfy, respectively,

We prove that {z_k} and {y_k} are monotonic and bounded. First of all, we prove that {z_k} is monotonically increasing, and N^* is the least upper bounded. Note (g₁) and (g₂), we have

By induction and direct computation, we have

Similarly, we have

Define v₁(t) and w₁(t) to be the solutions of the following differential equations, respectively:

It follows from (24) and (25) that z₀ ≤ N(t, x) ≤ y₀ for any $$. Consider (30), for any $$, we have

Therefore, from Definition 2, (v₁(t), w₁(t)) is a lower-upper pair of (4)-(5) with initial condition z₀ ≤ N(t, x) ≤ y₀ on $$. Consequently, by Lemma 3, we have

Note that w₁(t) is monotonically decreasing for t ≥ 3τ and lim _t→∞w₁(t) = y₁, while v₁(t) is monotonically increasing for t ≥ 3τ and lim _t→∞v₁(t) = z₁. Hence,

Define v_n(t) and w_n(t) to be the solutions of the following differential equations, respectively:

Repeating the above procedure, we have the following relation:

By (28) and (29), and taking limits on both sides of (35), we have

Case  2 (N^* = y₀). Similarly, let y_k = N^* and z_k be the same as in the proof of Case 1; we can also get (35). Hence, the proof of Theorem 6 is complete.

Remark 7. Our main results are also valid when N does not depend on a spatial variable x ∈ Ω in (4).