Roles of Weight Functions to a Nonlocal Porous Medium Equation with Inner Absorption and Nonlocal Boundary Condition

Theorem 1.1. Suppose that p + q > r. If ∫_Ωf(x, y)dy ≥ 1 for x ∈ ∂Ω, and the initial data u₀(x)>(k/(|Ω| − k)) ^1/(p+q)  (|Ω | > k), then the solution of problem (1.1)–(1.3) blows up in finite time.

Remark 1.2. There may exist a global solution of problem (1.1)–(1.3) for small enough initial data under the condition of Theorem 1.1. Unfortunately, since the weight function satisfies the condition ∫_Ωf(x, y)dy ≥ 1 on the boundary, we cannot construct a suitable supersolution of problem (1.1)–(1.3).

Theorem 1.3. Suppose that p + q > r, if ∫_Ωf(x, y)dy ≤ 1 for x ∈ ∂Ω, then the solution of problem (1.1)–(1.3) exists globally for the initial data u₀(x)<(k/|Ω|) ^1/(p+q−r). If p + q ≥ max  {m, r}, then the solution of problem (1.1)–(1.3) blows up in finite time for large enough initial data and arbitrary f(x, y) > 0.

Theorem 1.4. Suppose that p + q < r,then the solution of problem (1.1)–(1.3) exists globally for arbitrary f(x, y) > 0.

Theorem 1.5. Suppose that p + q = r.  ∫_Ωf(x, y) ≤ ρ < 1 for x ∈ ∂Ω, where ρ is a positive constant and ρ < 1.

• (1)

  If m > p + q, then every nonnegative solution of problem (1.1)–(1.3) exists globally.

• (2)

  If m = p + q, then for |Ω | < k + (δ/M₁)(1 − ρ^m), the solution of problem (1.1)–(1.3) exists globally, where δ, M₁ > 0 are as defined in (3.13).

• (3)

  If m < p + q, the solution of problem (1.1)–(1.3) exists globally for sufficient small initial data while it blows up in finite time for large enough initial data.

Theorem 1.6. Suppose that p + q > max  {m, r},   ∫_Ωf(x, y)dy ≤ 1 for x ∈ ∂Ω, and the initial data satisfies the conditions (C₁)-(C₂), then

Definition 2.1. Suppose that $$ is nonnegative and satisfies

Proposition 2.2 (comparison principle). Suppose that $$ and $$ are the nonnegative subsolution and supersolution of problem (1.1)–(1.3), respectively. Moreover $$, and $$ in $$, then $$ in $$.

Proof. Let ψ(x, t) ∈ C^2,1(Q_T) be a nonnegative function with ψ = 0 on S_T. Multiplying the inequality in (2.1) by ψ(x, t) and integrating it on Q_T, we get

For x ∈ ∂Ω,   y ∈ Ω,   t > 0,

Theorem 2.3 (local existence and uniqueness). Suppose that the nonnegative initial data $$ satisfies the compatibility condition. Then, there exists a constant T^* > 0 such that the problem (1.1)–(1.3) admits nonnegative solution $$ for each T < T^*. Furthermore, either T^* = ∞ or

Remark 2.4. The existence of local nonnegative solutions in time to problem (1.1)–(1.3) can be obtained by using the fixed point theorem (see [18]) or the regular theory to get the suitable estimate in a standard limiting process (see [19, 20]). By the previous comparison principle, we can get the uniqueness of solution to the problem (1.1)–(1.3) in the case of p + q ≥ 1,   r ≥ 1.

Proof of Theorem 1.1. Consider the following problem:

It is obvious that the solution of problem (3.1) is a lower solution of problem (1.1)–(1.3) when ∫_Ωf(x, y)dy ≥ 1 and u₀(x) > v₀. By Proposition 2.2, u(x, t) is a blow-up solution of problem (1.1)–(1.3).

Proof of Theorem 1.3. (1) The case of p + q > r. Let $$. It is easy to show that if ∫_Ωf(x, y)dy < 1 and $$ is the upper solution of problem (1.1)–(1.3), then we can draw the conclusion.

(2) The case of p + q ≥ max  {m, r}. We need to establish a self-similar blow-up solution in order to prove the blow-up result. We first suppose that $$, and ω(x) is not identically zero, and ω(x)|_∂Ω = 0. Without loss of generality, we assume that 0 ∈ Ω and ω(0) > 0.

Let $$, and ξ = |x | (T − t) ^−μ, where A > 1,   0 < T < 1,   γ and μ > 0. We know that

It is easy to see that V^′(ξ) ≤ 0 and V < 1, and

If x ∈ ∂Ω,   ω(0) > 0, and ω is continuous, it is known that there exist positive ε and ρ such that ω ≥ ε for x ∈ B(0, ρ). We can get B(0, RT^σ) ⊂ B(0, ρ) ⊂ Ω if T is small enough. Then, $$ on ∂Ω × (0, T). It follows from (3.3) that $$ for sufficiently large K₀. Therefore, one can observe that the solution to (1.1)–(1.3) exists no later than t = T provided that u₀ ≥ K₀ω(x). This implies that the solution blows up in finite time for large enough initial data.

Proof of Theorem 1.4. Suppose that λ₁ > 0 is the first eigenvalue of −Δ with homogeneous Dirichlet boundary condition, and ϕ(x) is the corresponding eigenfunction. Let M∫_Ω(1/(ϕ(y) + ε))dy ≤ 1 for some 0 < ε < 1, where $$

Now, we assume that u^m = v, then (1.1)–(1.3) becomes that

Set $$, where C is determined later, then it follows that

For x ∈ ∂Ω,   t > 0,

Proof of Theorem 1.5. Suppose that Φ(x) solves the following problem:

Set ω(x, t) = A(ρ^m/(1 − ρ^m) + Φ(x)/M₁) ^1/m, here A > 0 will be determined later and 0 < ρ < 1,

• (1)

  If m > p + q, choosing A = max  {max  |u₀(x)|, (|Ω | − k)(M/δ)(1/(1 − ρ^m)) ^{(p + q)/m}}, we have

  $$ (3.16)

• (2)

  If m = p + q, selecting |Ω | < k + (δ/M₁)(1 − ρ^m), and A = max  u₀(x), we get

  $$ (3.17)

• (3)

  If m < p + q, we choose max  u₀(x) ≤ A ≤ (|Ω | − k)(M₁/δ)(1/(1 − ρ^m)) ^{−(p + q)/m} such that

  $$ (3.18)
  For x ∈ ∂Ω,
  $$ (3.19)

For the blow-up case of m < p + q, it holds clearly from the second part of the proof of Theorem 1.3.

Proof of Theorem 1.6. (1) We can easily know that V(t) is Lip continuous and differential almost everywhere,

(2) Next, we set $$, where δ > 0, then

Since p₁ + q₁ > r₁, we get p₁ + 2q₁ − n > q₁ + 1 − n and p₁ + q₁ − r₁ > 0, then by (4.7)-(4.8),

For (x, t) ∈ S_T, because of $$, we then have

Integrating it over (t, T), we have