A Two-Species Cooperative Lotka-Volterra System of Degenerate Parabolic Equations

Definition 1.1. A nonnegative and continuous vector-valued function (u, v) is said to be a generalized solution of the problem (1.1)–(1.4) if, for any 0 ≤ τ < T and any functions $$ with $$, $$ and

where Q_τ = Ω × (0, τ).

Similarly, we can define a weak supersolution $$ (subsolution $$) if they satisfy the inequalities obtained by replacing “=” with “≤” (“≥”) in (1.3), (1.4), and (1.9) and with an additional assumption φ_i ≥ 0  (i = 1,   2).

Definition 1.2. A vector-valued function (u, v) is said to be a T-periodic solution of the problem (1.1)–(1.3) if it is a solution in [0, T] such that u(·, 0) = u(·, T), v(·, 0) = v(·, T) in Ω. A vector-valued function $$ is said to be a T-periodic supersolution of the problem (1.1)–(1.3) if it is a supersolution in [0, T] such that $$ in Ω. A vector-valued function $$ is said to be a T-periodic subsolution of the problem (1.1)–(1.3), if it is a subsolution in [0, T] such that $$ in Ω.

Lemma 2.1. Let (u_ɛ, v_ɛ) be a solution of the problem (2.1)–(2.4).

• (1)

  If 1 < (m₁ − α)(m₂ − β), then there exist positive constants r and s large enough such that

  $$ ()
  $$ ()
  where C is a positive constant only depending on m₁, m₂, α, β, r, s, |Ω|, and T.

• (2)

  If 1 = (m₁ − α)(m₂ − β), then (2.7) also holds when |Ω| is small enough.

Proof. Multiplying (2.1) by $$ and integrating over Ω, we have that

By Poincaré’s inequality, we have that where K is a constant depending only on |Ω| and N and becomes very large when the measure of the domain Ω becomes small. Since α < m₁, Young′s inequality shows that For convenience, here and below, C denotes a positive constant which is independent of ɛ and may take different values on different occasions. Complying (2.8) with (2.9) and (2.10), we obtain As a similar argument as above, for v_ɛ and positive constant s > 1, we have that Thus we have that

For the case of 1 < (m₁ − α)(m₂ − β), there exist r, s large enough such that

By Young′s inequality, we have that where Together with (2.13), we have that where Furthermore, by Hölder′s and Young′s inequalities, from (2.17) we obtain Then by Gronwall′s inequality, we obtain

Now we consider the case of 1 = (m₁ − α)(m₂ − β). It is easy to see that there exist positive constants r, s large enough such that

Due to the continuous dependence of K upon |Ω| in (2.9), from (2.13) we have that when |Ω| is small enough. Then by Young′s and Gronwall′s inequalities we can also obtain (2.20), and thus we complete the proof of this lemma.

Lemma 2.2. Let (u_ɛ, v_ɛ) be a solution of (2.1)–(2.4); then

where C is a positive constant independent of ɛ.

Lemma 2.3. Let (u_ɛ, v_ɛ) be a solution of (2.1)–(2.4), then

where C is a positive constant independent of ɛ.

Proof. For a positive constant $$, multiplying (2.1) by $$ and integrating the results over Q_T, we have that

where s₊ = max {0, s} and $$ is the characteristic function of [t₁, t₂]  (0 ≤ t₁ < t₂ ≤ T). Let then I_k(t) is absolutely continuous on [0, T]. Denote by σ the point where I_k(t) takes its maximum. Assume that σ > 0, for a sufficient small positive constant ϵ. Taking t₁ = σ − ϵ, t₂ = σ in (2.25), we obtain From we have that Letting ϵ → 0⁺, we have that Denote A_k(t) = {x : u_ɛ(x, t) > k} and μ_k = sup _t∈(0,T) | A_k(t)|; then By Sobolev′s theorem, with we obtain where r > p(m₁ + α)/(p − 2), s > pr/(p(r − m₁ − α) − 2r) and C denotes various positive constants independent of ɛ. By Hölder’s inequality, it yields Then On the other hand, for any h > k and t ∈ [0, T], we have that Combined with (2.35), it yields that is, It is easy to see that Then by the De Giorgi iteration lemma [22], we have that where $$. That is,

It is the same for the second inequality of (2.24). The proof is completed.

Lemma 2.4. The solution (u_ɛ, v_ɛ) of (2.1)–(2.4) satisfies the following:

where C is a positive constant independent of ɛ.

Proof. Multiplying (2.1) by $$ and integrating over Ω, by (2.3), (2.4) and Young′s inequality we have that

which together with the bound of a, b, c, u_ɛ, v_ɛ shows that where C is a positive constant independent of ɛ. Noticing the bound of u_ɛ, we have that It is the same for the second inequality. The proof is completed.

Theorem 2.5. The problem (1.1)–(1.4) admits a generalized solution.

Proof. By Lemmas 2.2, 2.3, and 2.4, we can see that there exist subsequences of {u_ɛ}, {v_ɛ} (denoted by themselves for simplicity) and functions u, v such that

as ɛ → 0. Then a rather standard argument as [23] shows that (u, v) is a generalized solution of (1.1)–(1.4) in the sense of Definition 1.1.

Lemma 2.6. Let $$ be a subsolution of the problem (1.1)–(1.4) with the initial value $$ and $$ a supersolution with a positive lower bound of the problem (1.1)–(1.4) with the initial value $$. If $$, $$, then $$, $$ on Q_T.

Proof. Without loss of generality, we might assume that $$, $$, $$, where M is a positive constant. By the definitions of subsolution and supersolution, we have that

Take the test function as where H_ɛ(s) is a monotone increasing smooth approximation of the function H(s) defined as follows: It is easy to see that H_ɛ′(s) → δ(s) as ɛ → 0. Since $$, the test function φ(x, t) is suitable. By the positivity of a, b, c we have that where C is a positive constant depending on $$. Letting ɛ → 0 and noticing that we arrive at Let $$ be a supsolution with a positive lower bound σ. Noticing that with x, y > 0, we have that where C is a positive constant depending upon α, σ, M.

Similarly, we also have that

Combining the above two inequalities, we obtain By Gronwall′s lemma, we see that $$. The proof is completed.

Corollary 2.7. If b_lf_l > c_Me_M, then the problem (1.1)–(1.4) admits at most one global solution which is uniformly bounded in $$.

Proof. The uniqueness comes from the comparison principle immediately. In order to prove that the solution is global, we just need to construct a bounded positive supersolution of (1.1)–(1.4).

Let ρ₁ = (a_Mf_l + d_Mc_M)/(b_lf_l − c_Me_M) and ρ₂ = (a_Me_M + d_Mb_l)/(b_lf_l − c_Me_M), since b_lf_l > c_Me_M; then ρ₁, ρ₂ > 0 and satisfy

Let $$, where η > 1 is a constant such that (u₀, v₀)≤(ηρ₁, ηρ₂); then we have that That is, $$ is a positive supersolution of (1.1)–(1.4). Since $$ are global and uniformly bounded, so are u and v.

Lemma 3.1. In case of b_lf_l > c_Me_M, there exists a pair of T-periodic supersolution and T-periodic subsolution of the problem (1.1)–(1.3).

Proof. We first construct a T-periodic subsolution of (1.1)–(1.3). Let λ be the first eigenvalue and ϕ be the uniqueness solution of the following elliptic problem:

then we have that Let where ɛ > 0 is a small constant to be determined. We will show that $$ is a (time independent, hence T-periodic) subsolution of (1.1)–(1.3).

Taking the nonnegative function $$ as the test function, we have that

Similarly, for any nonnegative test function $$, we have that We just need to prove the nonnegativity of the right-hand side of (3.4) and (3.5).

Since ϕ₁ = ϕ₂ = 0,    | ∇ϕ₁ | , |∇ϕ₂ | > 0 on ∂Ω, then there exists δ > 0 such that

where $$. Choosing then we have that which shows that $$ is a positive (time independent, hence T-periodic) subsolution of (1.1)–(1.3) on $$.

Moreover, we can see that, for some σ > 0,

Choosing then on Q_T, that is These relations show that $$ is a positive (time independent, hence T-periodic) subsolution of (1.1)–(1.3).

Letting $$, where η, ρ₁, ρ₂ are taken as those in Corollary 2.7, it is easy to see that $$ is a positive (time independent, hence T-periodic) subsolution of (1.1)–(1.3).

Obviously, we may assume that $$, $$ by changing η, ɛ appropriately.

Lemma 3.2 (see [24], [25].)Let u be the solution of the following Dirichlet boundary value problem

where f ∈ L^∞(Ω × (0, T)); then there exist positive constants K and α ∈ (0,1) depending only upon τ ∈ (0, T) and $$, such that, for any (x_i, t_i) ∈ Ω × [τ, T]    (i = 1,2),

Lemma 3.3 (see [26].)Define a Poincaré mapping

where (u(x, t), v(x, t)) is the solution of (1.1)–(1.4) with initial value (u₀(x), v₀(x)). According to Lemmas 2.6 and 3.2 and Theorem 2.5, the map P_t has the following properties:

• (i)

  P_t is defined for any t > 0 and order preserving;

• (ii)

  P_t is order preserving;

• (iii)

  P_t is compact.

Theorem 3.4. Assume that b_lf_l > c_Me_M and there exists a pair of nontrivial nonnegative T-periodic subsolution $$ and T-periodic supersolution $$ of the problem (1.1)–(1.3) with $$; then the problem (1.1)–(1.3) admits a pair of nontrivial nonnegative periodic solutions (u_*(x, t), v_*(x, t)),   (u^*(x, t), v^*(x, t)) such that

Proof. Taking $$, $$ as those in Lemma 3.1 and choosing suitable B(x₀, δ), B(x₀, δ′), Ω′, k₁, k₂, and K, we can obtain $$. By Lemma 2.6, we have that $$. Hence by Definition 1.2 we get $$, which implies $$ for any k ∈ ℕ. Similarly we have that $$, and hence $$ for any k ∈ ℕ. By Lemma 2.6, we have that $$ for any k ∈ ℕ. Then

exist for almost every x ∈ Ω. Since the operator P_T is compact (see Lemma 3.3), the above limits exist in L^∞(Ω), too. Moreover, both u_*(x, 0) and u^*(x, 0) are fixed points of P_T. With the similar method as [26], it is easy to show that the even extension of the function u_*(x, t), which is the solution of the problem (1.1)–(1.4) with the initial value u_*(x, 0), is indeed a nontrivial nonnegative periodic solution of the problem (1.1)–(1.3). It is the same for the existence of u^*(x, t). Furthermore, by Lemma 2.6, we obtain (3.16) immediately, and thus we complete the proof.

Remark 3.5. If b_lf_l > c_Me_M, the problem (1.1)–(1.3) admits a maximal periodic solution (U, V). Moreover, if (u, v) is the solution of the initial boundary value problem (1.1)–(1.4) with nonnegative initial value (u₀, v₀), then, for any ɛ > 0, there exists t depending on u₀, v₀, and ɛ, such that