A Reaction-Diffusion System with Nonlinear Nonlocal Boundary Conditions

Definition 1. A pair of nonnegative functions $$ is called a subsolution of problem (1) in Q_T if

and a pair of nonnegative functions $$ is called a supersolution of problem (1) in Q_T if the reversed inequalities hold in (2).

Definition 2. A pair of functions (u, v) is called a solution of (1) in Q_T if it is both a subsolution and a supersolution of problem (1) in Q_T.

Definition 3. We say that solution (u, v) of (1) is positive in Q_T if u > 0 and v > 0 in Q_T.

Theorem 4. Let $$ and $$ be a nonnegative supersolution and a nonnegative subsolution of problem (1) in Q_T, respectively. Suppose that $$ and $$ or $$ and $$ in $$ if min⁡(p, q, m, n) < 1. If $$ and $$ for $$, then $$ and $$ in $$.

Proof. Let $$ be a nonnegative function such that $$. Then $$, where ν is the unit outward normal to the lateral boundary of Q_T. By the definition of a subsolution we have

If we multiply (3) by ξ and then integrate over Q_t for 0 < t < T, we get

On the other hand, the supersolution $$ satisfies (4) with reversed inequality. Set $$. Then we have

where Θ₁,  Θ₂ are nonnegative continuous functions if min⁡(p, q, m, n) ≥ 1 and positive continuous functions if min⁡(p, q, m, n) < 1 in $$ which satisfy the following equalities:

Obviously there exists a positive constant M₁ such that $$ in $$. Since c₁(x, t), k₁(x, y, t) are nonnegative and continuous functions, then there exists a constant M₂ > 0 such that 0 ≤ c₁(x, t) ≤ M₂, 0 ≤ k₁(x, y, t) ≤ M₂ in $$ and $$, respectively.

Consider the following backward problem in Q_t:

where $$0 ≤ ψ(x) ≤ 1. By the maximum principle for the heat equation 0 ≤ ξ ≤ 1, it is easy to show that −M₃ ≤ ∂ξ/∂ν ≤ 0 on S_t for some M₃ ≥ 0.

Let s₊ = max⁡(s, 0). Then from (5), we get

where K₁ = pM₁M₂,   K₂ = mM₁M₂M₃|∂Ω|, and |∂Ω| is the Lebesgue measure of ∂Ω. Consider the sequence of functions {ϕ_n(x)}, $$, converging in L¹(Ω) to ϕ(x), defined as follows: Replacing ψ(x) by ϕ_n(x) in (8) and passing to the limit as n → ∞, we have Using a similar argument for the inequality $$,  x ∈ Ω,  0 < t < T, we get where K₃, K₄ are positive constants. Adding (10) and (11), we have where K₅ = max⁡(K₁ + K₄, K₂ + K₃).

Applying now Gronwall′s inequality, we conclude that

Since w₊ + z₊ ≥ 0, then $$, $$.

Theorem 5. For small values of T, (14) has a unique solution in Q_T.

Proof. We start the proof with the construction of a supersolution of (14). Let max⁡(sup_Ωu_0ɛ(x),   sup_Ωv_0ɛ(x)) ≤ C,   C > 0. Denote $$. Introduce an auxiliary function   φ(x) with the following properties:

where  $$. Let α, β be positive constants such that αq − β = βp − α and Note that αq − β > 0 for pq > 1 and αq − β ≤ 0 for pq ≤ 1. Obviously the pair of functions (ε, ε) is a subsolution of problem (14). We show that is a supersolution of (14) in Q_T for T ≤ min⁡(1/α, 1/β, 1/(αq − β)) if pq > 1 and T ≤ min⁡(1/α, 1/β) if pq ≤ 1. We have for (x, t) ∈ Q_T. On the other hand, we get that for (x, t) ∈ S_T. Similarly, we can show that To prove the existence of a solution of the problem (14) we introduce the set Clearly B is a nonempty convex subset of $$.

Consider the following problem:

where (s₁, s₂) ∈ B. Problem (22) has a nontrivial positive solution. Let us call A(s₁, s₂) = (u, v). In order to show that A has a fixed point in B we verify that A is a continuous mapping from B into itself such that AB is relatively compact. Thanks to the comparison principle for (22) we have that A maps B into itself.

Let G(x, y; t) denote the Green′s function for the heat equation given by

with zero boundary condition. Then (u, v) is a solution of (22) if and only if

We claim that A is continuous. In fact, let {(s_1k, s_2k)} be a sequence in B converging to (s₁, s₂) ∈ B in $$. Denote (u_k, v_k) = A(s_1k, s_2k). Then we see that

where r = max⁡(Θ₁, Θ₂) and Choosing T so small that r < 1, we conclude that (u_k, v_k)→(u, v) in $$ as k → ∞. The equicontinuity of AB follows from (24), (25) and the properties of the Green′s function (see, e.g. [22]). The Ascoli-Arzela theorem guarantees the relative compactness of AB. Thus, we are able to apply the Schauder-Tychonoff fixed point theorem and conclude that A has a fixed point in B if T is small. Since (u, v) is a fixed point of A, it is a solution of (14). Uniqueness of solution follows from a comparison principle for (14) which can be proved in a similar way as in the previous section.

Theorem 6. For small values of T (1) has a maximal solution in Q_T.

Proof. Let ε₂ > ε₁. It is easy to show that $$ is a supersolution of the problem (14) with ε = ε₁. Then $$,  $$. Using these inequalities and the continuation principle of solutions we deduce that the existence time of (u_ɛ(x, t), v_ɛ(x, t)) does not decrease as ε → 0. Let ε → 0, then

and (u_max⁡(x, t), v_max⁡(x, t)) exist in Q_T for some T > 0.

Moreover, by dominated convergence theorem, (u_max⁡(x, t), v_max⁡(x, t)) satisfies the following equations:

The interior regularity of (u_max⁡(x, t), v_max⁡(x, t)) follows from the continuity of (u_max⁡(x, t), v_max⁡(x, t)) in Q_T and the properties of the Green′s function. Obviously (u_max⁡(x, t), v_max⁡(x, t)) satisfies (1). Let (y₁(x, t), y₂(x, t)) be any other solution of (1). Then by the comparison principle u_ε(x, t) ≥ y₁(x, t), v_ε(x, t) ≥ y₂(x, t). Taking ε → 0, we conclude that u_max⁡ ≥ y₁(x, t), v_max⁡(x, t) ≥ y₂(x, t).

Definition 7. We say that a function g(x, t) has the property (N) if there exist x_k ∈ Ω and t_k > 0,  k ∈ ℕ such that g(x_k, t_k) > 0,  k ∈ ℕ, and t_k → 0 as k → ∞.

Remark 8. Note that if a nonnegative function g(x, t) has no the property (N) then g(x, t) ≡ 0 in Q_τ for some τ > 0.

Theorem 9. Let either u₀(x) or v₀(x) be a nontrivial function in Ω. Supposing that k₁(x, ·, t) and k₂(x, ·, t) are nontrivial functions for any x ∈ ∂Ω and t ∈ (0, T),  c₁(x, t) has the property (N) if u₀(x) ≡ 0,  and  c₂(x, t) has the property (N) if v₀(x) ≡ 0. Let (u(x, t), v(x, t)) be a supersolution of (1) in Q_T. Then (u(x, t), v(x, t)) is positive in $$ for 0 < t < T.

Proof. Suppose for definiteness that u₀(x) is a nontrivial function. We show at first that u(x, t) > 0 in $$ for 0 < t < T. We have

then by strong maximum principle a minimum of u(x, t) should be attained in $$ on a parabolic boundary. Thus, u(x, t) > 0 in Q_T, otherwise, there would be a contradiction with the initial datum. We show that u(x, t) > 0 on ∂Ω × (0, T). Let there exist a point (x₀, t₀) ∈ S_T such that u(x₀, t₀) = 0. But u(x, t₀) > 0 for x ∈ Ω. By boundary conditions (1) and assumption for k₁(x, y, t) we have u(x, t) > 0 for (x, t) ∈ S_T. This contradiction shows that u(x, t) > 0 on S_T, and therefore u(x, t) > 0 in $$ for 0 < t < T.

Now we show the positiveness of v(x, t). If v₀(x) is a nontrivial function, then v(x, t) > 0 in $$ for 0 < t < T by previous arguments. If v₀(x) ≡ 0 we suppose that there exists a constant τ > 0 such that v(x, t) ≡ 0 in Q_τ since otherwise we can use the arguments from the beginning of the proof again. But this is a contradiction with the second equation in (1) since u(x, t) > 0 in Q_τ and c₂(x, t) has the property (N). Hence, we conclude that v(x, t) > 0 in $$ for 0 < t < T.

Theorem 10. Let problem (1) have a solution in Q_T with nonnegative initial datum for min⁡(p, q, m, n) ≥ 1 and with positive initial datum under conditions min⁡(p, q, m, n) < 1 and k₁(x, ·, t),   k₂(x, ·, t) are nontrivial functions for any x ∈ ∂Ω and t ∈ (0, T). Then solution of (1) is unique in Q_T.

Theorem 11. Let min⁡(pq, m, n) < 1, u₀(x) = v₀(x) ≡ 0. Suppose that the maximal solution of problem (1) exists in Q_T. Assume that at least one of the following conditions is fulfilled:

Then the maximal solution of problem (1) is nontrivial in Q_T.

Proof. In the local existence theorem we constructed a maximal solution (u_max⁡(x, t), v_max⁡(x, t)) of (1) in the following way: u_max⁡(x, t) = lim_ε→0u_ε(x, t), v_max⁡(x, t) = lim_ε→0v_ε(x, t), where (u_ε(x, t), v_ε(x, t)) is some positive supersolution of (1). To prove the theorem we construct a nontrivial nonnegative subsolution $$ of some problem with trivial initial datum. By the comparison principle we conclude that $$$$, and therefore maximal solution is a nontrivial solution.

Consider at first the case when pq < 1 and c₁(x₀, t₀) > 0,   c₂(x₀, t₀) > 0 for some x₀ ∈ Ω and 0 ≤ t₀ < T. Then there exists a neighborhood U(x₀) of x₀ in Ω and $$ such that c₁(x, t) ≥ c₀,   c₂(x, t) ≥ c₀,   c₀ > 0 for $$.

Consider the following problem:

where w₀(x) is a bounded nontrivial nonnegative continuous function which satisfies a boundary condition. By the strong maximum principle 0 < w(x, t) < M₀ for $$, where $$.

Let α = (1 + p)/(1 − pq) > 1,   β = (1 + q)(1 − pq) > 1. Denote that

where C = min⁡(c₀/(αM₀), c₀/(βM₀)). Note that $$ if t = t₀ or x ∈ ∂U(x₀). After simple calculations we obtain Similarly we can get that Then $$ and (u_ε(x, t), v_ε(x, t)) are subsolution and supersolution, respectively, of the following problem: By comparison principle for (38) we conclude that $$, $$ and hence $$,  $$, for $$.

Now consider the case when m < 1 and k₁(x, y₁, t₁) > 0 for any x ∈ ∂Ω and some y₁ ∈ ∂Ω,  0 ≤ t₁ < T. We will consider the following problem:

where T₀ > t₁ will be defined later. Construct a subsolution of (39) using the change of variables in a neighborhood of ∂Ω as in [23]. Let $$. We denote by $$ the inner unit normal to ∂Ω at the point $$. Since ∂Ω is smooth it is well known that there exists δ > 0 such that the mapping ψ : ∂Ω × [0, δ] → ℝ^N given by $$ defines new coordinates $$ in a neighborhood of ∂Ω in $$. A straightforward computation shows that, in these coordinates, Δ applied to a function $$ which is independent of the variable $$, evaluated at a point $$ is given by where $$ denote the principal curvatures of ∂Ω at $$.

Under the made assumption there exists $$ such that k₁(x, y, t)>0 for $$, where V(y₁) is some neighborhood of y₁ in $$. Let α > 1/(2(1 − m)) and assume that 0 < ξ₀ ≤ 1 and $$. For points in ∂Ω × [0, δ]×(t₁, T₀] of coordinates $$ define

and extend $$ as zero to the whole $$. Using (40), we get that for sufficiently small values of ξ₀. It is clear that It remains to verify the validity of the inequality for x ∈ ∂Ω and t₁ < t < T₀. Here $$ is Jacobian of the change of variables. Estimating the integral I in the right-hand side of (44) we get where constant C does not depend on t; we obtain that (44) is true if T₀ − t₁ is small enough.

By comparison principle for (39) we conclude that $$, $$ and, respectively, $$,  $$ for (x, t) ∈ Ω × (t₁, T₀).

The proof in the case n < 1 and k₂(x, y₂, t₂) > 0 for any x ∈ ∂Ω and some y₂ ∈ ∂Ω,   0 ≤ t₂ < T is similar.

Corollary 12. Let the conditions of Theorem 11 hold with t_i = 0,   i = 0,1, 2. Suppose also that k₁(x, ·, t) and k₂(x, ·, t) are nontrivial functions for any x ∈ ∂Ω and t ∈ (0, T),   c₂(x, t) has the property (N) if (32) is realized, and c₁(x, t) has the property (N) if (33) is realized. Then maximal solution is positive in $$ for 0 < t < T.

Theorem 13. Let the conditions of Corollary 12 hold. Suppose also that there exists t₀ > 0 such that for 0 ≤ t ≤ t₀ the functions c₁(x, t), c₂(x, t),  k₁(x, y, t),   and  k₂(x, y, t) are nondecreasing with respect to t.

Then there exists exactly one solution of problem (1) which is positive in $$ for 0 < t < T.

Proof. Suppose that there exists different from (u_max⁡(x, t), v_max⁡(x, t)) solution (u(x, t), v(x, t)) of (1) with trivial initial datum which is a positive in $$ for 0 < t < T. Denote t_* = min⁡(t₀, T). Due to the conditions of the theorem it is easy to see that u(x, t + τ), v(x, t + τ) is positive supersolution of (1) with trivial initial datum in $$ for any τ ∈ (0, t_*). By Theorem 4 we have u_max⁡(x, t) ≤ u(x, t + τ), v_max⁡(x, t) ≤ v(x, t + τ) for every 0 ≤ t ≤ t_* − τ. Passing to the limit as τ → 0 we get u_max⁡(x, t) ≤ u(x, t),   v_max⁡(x, t) ≤ v(x, t) for 0 ≤ t ≤ t_*. Hence u_max⁡(x, t) = u(x, t),   v_max⁡(x, t) = v(x, t) in Q_T.

Theorem 14. Let the conditions of Corollary 12 fulfill only u₀(x)≢0 or v₀(x)≢0. Then the solution of (1) is unique.

Proof. To prove the uniqueness of the solution if min⁡(pq, m, n) < 1, it suffices to show that if (u(x, t), v(x, t)) is any solution of (1), then

where (u_max⁡(x, t), v_max⁡(x, t)) is a maximal solution of (1).

First, consider the case when 0 < p < 1,   0 < q ≤ 1,   0 < m < 1,   and  0 < n ≤ 1. Let

Then (w₁, w₂) satisfies for some T₁ > 0. By Theorem 13, there exists a unique solution (h₁(x, t), h₂(x, t)) of the following problem: such that h₁(x, t) > 0,   h₂(x, t) > 0 for $$ for some T₂ > 0. Let T₀ = min⁡(T₁, T₂). In a similar way as in Theorems 13 and 4 we can prove that u_max⁡(x, t) ≥ h₁(x, t) ≥ w₁(x, t),   v_max⁡(x, t) ≥ h₂(x, t) ≥ w₂(x, t). Put a₁ = h₁ − w₁,   a₂ = h₂ − w₂. Now, use an elementary inequality, which is recalled for instance in [20], where 0 < k ≤ 1,  min⁡(a, b, c) > 0, and max⁡(a, c) ≤ b ≤ a + c. Then we obtain We show that a₁(x, t) > 0,   a₂(x, t) > 0 in $$. In fact, otherwise, by Theorem 9 there exists $$ such that a₁(x, t) = a₂(x, t) ≡ 0 in $$. Suppose that c₁(x₀, 0) > 0 for some x₀ ∈ Ω. Then we get for x ∈ Ω and $$. This is a contradiction since h₂(x, t) > 0,   v(x, t) > 0 in $$c₁(x₀, t₀) > 0 for some $$, and 0 < p < 1.

If k₁(x, y₀, 0) > 0 for any x ∈ ∂Ω and some y₀ ∈ ∂Ω we can obtain a contradiction by another way. Indeed,

for x ∈ ∂Ω and $$ and we get again a contradiction since h₁(x, t) > 0,  u(x, t) > 0 in $$, and 0 < m < 1.

Since a₁ > 0,   a₂ > 0 in $$ by comparison principle with arguments of Theorems 13 and 4 we conclude that a₁ ≥ h₁, a₂ ≥ h₂ in $$. This implies (46) and completes the proof for the first case.

Now suppose that max⁡(p, q, m, n) > 1. Assume, for example, that 0 < p ≤ 1,   q > 1,   0 < m < 1,   n > 1. Then, as in the first case, we introduce the functions w₁(x, t), w₂(x, t). We use the following relations:

where θ₁, θ₂ are nonnegative continuous functions which are between u_max⁡(x, t),   u(x, t) and v_max⁡(x, t),  v(x, t), respectively, $$ in Q_T. Then we have that a pair of functions (w₁(x, t), w₂(x, t)) satisfies relations

Further develop the arguments, as in the first case, only for q = 1,  n = 1. Using the linearization of terms with powers greater than 1 in the equations and boundary conditions of (1) as above we can prove the theorem for the remaining cases in a similar way.