Let Ω be a bounded domain in ℝ^N (N ≥ 1) with smooth boundary ∂Ω. We consider the following parabolic equation with a localized reaction term

v τ - Δ v m = a ( x ) v q 1 ( x , τ ) v s 1 ( x 0 , τ ) , x ∈ Ω , τ > 0 ,

v ( x , τ ) = 0 , x ∈ ∂ Ω , τ > 0 ,

v ( x , 0 ) = v 0 ( x ) , x ∈ Ω ,

where m ≥ 1, q₁ ≥ 0, s₁ > 0 and x₀ ∈ Ω is a fixed point. Throughout this article, we assume the functions a(x) and v₀(x) satisfy the following conditions:

(A1) a(x) and v₀(x) ∈ C ²(Ω); a(x), v₀(x) > 0 in Ω and a(x) = v₀(x) = 0 on ∂Ω.

When Ω = B = {x ∈ ℝ^N; |x| < R}, we sometimes assume

(A2) a(x) and v₀(x) are radially symmetric; a(r) and v₀(r) are non-increasing for r ∈ [0, R].

Problems (1.1)-(1.3) arise in the study of the flow of a fluid through a porous medium with an internal localized source and in the study of population dynamics (see 123). Porous medium equations (m > 1) with or without local sources have been studied by many authors 456.

Concerning (1.1)-(1.3), to the best of authors knowledge, a number of articles have studied it from the point of the view of blow-up and global existence 78910. Many studies have been devoted to the case m = 1 10111213. The case m = 1, a(x) = 1, q₁ = 0, s₁ ≥ 1 and m = 1, a(x) = 1, q₁, s₁ > 1 were studied by Souple 1011. Souple 10 demonstrated that the positive solution blows up in finite time if the initial value v₀ is large enough. In the case a(x) = 1, q₁ = 0, and s₁ > 1, Souple 11 showed that the solution v(x, τ) blows up globally and the blow-up rate is precisely determined. The case q₁ = 0 and s₁ > 0 was studied by Cannon and Yin 12 and Chandam et al. 13. Cannon and Yin 12 studied its local solvability and Chandam et al. 13 investigated its blow-up properties.

The study of this article is motivated by some recent results of related problems (see 141516. In the case of a(x)(= constant), the global existence and blow-up behavior have been considered by Chen and Xie 15. It turns out that if q₁ + s₁ < m or q₁ + s₁ = m and a(x)(= constant) is sufficiently small, there exists a global solution of problem (1.1)-(1.3); if q₁ + s₁ > m, the solution of problem (1.1)-(1.3) blows up for large initial datum while it admits a global solution for small initial datum. Furthermore, Du and Xiang 16 obtained the blow-up rate estimates under some appropriate hypotheses on initial datum. For some related localized models arising in physical phenomena, we refer the readers to 171819 and the references therein.

For the localized semi-linear parabolic equation of the form

v τ - Δ v = v q 1 ( x , τ ) v s 1 ( x 0 , τ ) , x ∈ Ω , τ > 0 ,

with the Dirichlet boundary condition (1.2) and the initial condition (1.3). In 20, Li and Wang proved that the blow-up set to system (1.2)-(1.4): (a) the system possesses total blow-up when q₁ ≤ 1; (b) the system presents single point blow-up patterns when q₁ > 1.

We now restrict ourselves to the problem of the form

v τ - Δ v m = a ( x ) v q 1 ( x , τ ) v s 1 ( 0 , τ ) , x ∈ B , τ > 0 ,

v ( x , τ ) = 0 , x ∈ ∂ B , τ > 0 ,

v ( x , 0 ) = v 0 ( x ) , x ∈ B ,

where q₁ ≥ 0, s₁ > 0, and q₁ + s₁ > m > 0. When m = 1, it was proved in 14 that

(1) If 0 ≤ q₁ ≤ 1 and q₁ + s₁ > 1, then the solution of (1.5)-(1.7) blows up in a finite time T.

(2) If q₁ > 1, then x = 0 is the only blow-up point for (1.5)-(1.7).

In the meantime, they obtained the blow-up rate estimate but less precise. Namely,

(i) If 0 ≤ q₁ < 1, then for any x ∈ B

C 1 ( a ( x ) ) 1 ∕ ( 1 - q 1 ) ≤ v ( x , τ ) ( T - τ ) 1 ∕ ( q 1 + s 1 - 1 ) ≤ C 2 , a s τ → T ,

where C 1 = ( ( a ( 0 ) ) s 1 ∕ ( 1 - q 1 ) ( q 1 + s 1 - 1 ) ) 1 ∕ ( 1 - q 1 - s 1 ) , C 2 = ( a ( 0 ) ( p + q - 1 ) ) 1 ∕ ( 1 - q 1 - s 1 ) .

(ii) If q₁ = 1, then for any x ∈ B

a ( x ) a ( 0 ) ln ( T - τ ) - 1 ∕ s 1 ≤ ln v ( x , τ ) ≤ ln ( T - τ ) - 1 ∕ s 1 , a s τ → T .

It seems that the results of 14 can be extended to m ≥ 1 and the blow-up rate can be precisely determined. Motivated by this, in this article, we will extend and improve the results of 14.

The purpose of this article is to determine the blow-up rate of solutions for a nonlinear parabolic equation with a weighted localized source, that is, we investigate how the localized source and the local term affect the blow-up properties of the problem (1.5)-(1.7). Indeed, we find that when q₁ ≤ 1, the solution of (1.5)-(1.7) blows up at the whole domain with a uniformly blow-up profile.

The rest of this article is organized as follows. The results are stated in Section 2. We then prove these results in Section 3.

The following two theorems are our main results.

Theorem 2.1 Assume q₁ + s₁ > m, (A1) and (A2) hold. Let v(x, t) be the solution of problem (1.5)-(1.7), then v(x,t) blows up provided that the initial value v₀(x) is sufficiently large.

The method used in the proof Theorem 2.1 is originally due to 818, and bears much resemblance to that of Theorem 3.2 in 15 and Theorem 1.3 in 16. Therefore, we omitted them here.

For the case q₁ > 1, we do not know how to deal with the uniform blow-up rate of problem (1.5)-(1.7). In the following, we focus only on the case of 0 ≤ q₁ ≤ 1.

Theorem 2.2 Assume (A1) and (A2) hold. Let v(x, t) be the blow-up solution of (1.5)-(1.7), which blows up in finite time T and v(x, t) is non-decreasing in time, then the following limits hold uniformly in all compact subsets of B.

(i) If 0 ≤ q₁ < 1, then

lim τ → T ( T - τ ) 1 ∕ ( q 1 + s 1 - 1 ) v ( x , τ ) = C ( a ( x ) ) 1 ∕ ( 1 - q 1 ) ,

where C = ( ( q 1 + s 1 - 1 ) ( a ( 0 ) ) s 1 ∕ ( 1 - q 1 ) ) 1 ∕ ( 1 - q 1 - s 1 ) .

(ii) If q₁ = 1, then

lim τ → T ln v ( x , τ ) = a ( x ) a ( 0 ) ln ( T - τ ) - 1 ∕ s 1 .

Remark 2.1 The domain we considered here is a ball, it seems that the results of Theorem 2.2 remain valid for the general domain. (It is an open problem in this case.)

To get the blow-up profiles for problem (1.5)-(1.7), we need some transformations. Let u(x, t) = v^m(x, τ), t = mτ, then (1.5)-(1.7) becomes

u t = u p ( Δ u + a ( x ) u q ( x , t ) u s ( 0 , t ) ) , x ∈ B , t > 0 , u ( x , t ) = 0 , x ∈ ∂ B , t > 0 , u ( x , 0 ) = u 0 ( x ) = v 0 m ( x ) , x ∈ B ,

where 0 ≤ p = (m - 1)/m < 1, q = q₁/m, and s = s₁/m.

Under above transformation, assumptions (A1) and (A2) become

(B1) a(x) and u₀(x) ∈ C ²(B); a(x), u₀(x) > 0 in B and a(x) = u₀(x) = 0 on ∂B.

(B2) a(x) and u₀(x) are radially symmetric; a(r) and u₀(r) are non-increasing for r ∈ [0, R].

In our consideration, a crucial role is played by the Dirichlet eigenvalue problem

- Δ φ = λ φ , in B , φ ( x ) = 0 , on ∂ B .

Denote λ be the first eigenvalue and by φ the corresponding eigenfunction with φ(x) > 0 in B, normalized by ∫ B a ( x ) φ ( x ) d x = 1 .

For convenience, we denote

g ( t ) = u s ( 0 , t ) a n d G ( t ) = ∫ 0 t g ( s ) d s .

Before proving our result, we would like to give a property of the following problem

w t = w α ( Δ w + a ( x ) g ( t ) ) , x ∈ B , t > 0 , w ( x , t ) = 0 , x ∈ ∂ B , t > 0 , w ( x , 0 ) = w 0 ( x ) = u 0 1 - q ( x ) , x ∈ B ,

where 0 ≤ α ≤ 1 and w = u ^1-q(x, t).

Lemma 3.1 Assume (B1) and (B2) hold. Let w(x, t) be the solution of Equation (3.1), which blows up in a finite time T* and non-decreasing in time t, then the following limits hold uniformly in all compact subsets of B.

(i) If 0 ≤ α < 1, then

lim t → T * w 1 - α ( x , t ) G ( t ) = ( 1 - α ) a ( x ) .

(ii) If α = 1, then

lim t → T * ln w ( x , t ) G ( t ) = a ( x ) .

Proof. (i) Assumption (B2) implies w_r ≤ 0 (r = |x|), it then follows that w ( 0 , t ) = max x ∈ B ̄ w ( x , t ) and Δw(0, t) ≤ 0 for t > 0. From (3.1), we then get

d w 1 - α ( 0 , t ) d t ≤ ( 1 - α ) a ( 0 ) g ( t ) , 0 < t < T * . .

Consequently,

lim t → T * sup w 1 - α ( 0 , t ) G ( t ) ≤ ( 1 - α ) a ( 0 ) ,

which implies

lim t → T * G ( t ) = ∞ a n d lim t → T * g ( t ) = ∞ .

Moreover, it is apparent that lim_t-T* w(0, t)/g(t) = 0, since s > 1 - q.

Set R₁ ∈ (0, R), B₁ = {x ∈ ℝ^N, | x |< R₁} and b(x) = 1/a(x), x ∈ B₁. Since a ^'(r) ≤ 0, we obtain that b'(r) ≥ 0, for 0 ≤ r ≤ R₁.

We now introduce the function

w 1 ( x , t ) = b 1 ∕ ( 1 - α ) ( x ) w ( x , t ) , x ∈ B 1 , 0 < t < T * .

By a simple calculation, and note that ∇w(x, t)∇b(x) = u_r(r, t)b'(r) ≤ 0, then there exist m₁, m₂ > 0 such that

b ( x ) Δ w ( x , t ) ≥ m 1 Δ w 1 ( x , t ) - m 2 w ( x , t ) x ∈ B 1 , 0 < t < T * .

Setting ε(t) = m₂w(0, t)/g(t). From lim_t→T* w(0, t)/g(t) = 0, we infer that there exists t₁ ∈ (0, T*) such that 0 < ε(t) ≤ 1/2 for t₁ ≤ t < T*.

Hence, in view of (3.1), we observe

1 1 - α ( w 1 1 - α ) t = b ( x ) Δ w + g ( t ) ≥ m 1 Δ w 1 + ( 1 - ε ( t ) ) g ( t ) + ε ( t ) g ( t ) - m 2 w ( 0 , t ) = m 1 Δ w 1 + ( 1 - ε ( t ) ) g ( t ) , x ∈ B 1 , t 1 < t < T * .

Set g₁(t) = (1 - ε(t))g(t), G 1 ( t ) = ∫ t 1 t g ( s ) d s , we then obtain

lim t → T * G 1 ( t ) = ∞ and lim t → T * G 1 ( t ) G ( t ) = 1 .

Obviously, w₁(x, t) is a sup-solution of the following equation

{ ( w * ) t = ( w * ) α ( ( m 1 △ w * + g 1 ( t ) ) , x ∈ B 1 ,   t 1 < t < T * , w * ( x , t ) = 0 , x ∈ ∂ B 1 ,   t > t 1 , w * ( x , t 1 ) = b 1 / 1 − α ( x ) w ( x , t 1 ) , x ∈ B 1 .

By the maximum principle, w₁(x, t) ≥ w*(x, t) and w r * ≤ 0 . Similar to the proof of (4.15) in 15 that

lim t → T * ( w * ) 1 - α ( x , t ) G ( t ) = ( 1 - α ) ,

uniformly in all compact subsets of B₁,

Therefore, by the arbitrariness of B₁, we obtain that the following inequatlity holds uniformly in all compact subsets of B

lim t → T * inf w ( 1 - α ) ( x , t ) G ( t ) ≥ ( 1 - α ) a ( x ) .

In particular,

lim t → T * inf w ( 1 - α ) ( 0 , t ) G ( t ) ≥ ( 1 - α ) a ( 0 ) .

From (3.2) and (3.4), we deduce

lim t → T * w ( 1 - α ) ( 0 , t ) G ( t ) = ( 1 - α ) a ( 0 ) .

Multiplying both sides of (3.1) by φ and integrating over B × (0, t), 0 < t < T*

1 1 - α ∫ B w 1 - α φ d x - ∫ B w 0 1 - α φ d x = - λ ∫ 0 t ∫ B w φ d x d s + G ( t ) .

Since ∫ 0 t ∫ B w φ d x d s ≤ ∫ B φ d x ∫ 0 t w ( 0 , s ) d s , so we have

lim t → T * ∫ 0 t ∫ B w φ d x d s G ( t ) = 0 .

It then follows that

lim t → T * ∫ B w 1 - α φ d x G ( t ) = ( 1 - α ) .

Note that w_r ≤ 0, (3.3) and (3.6), it is sufficient to prove

lim t → T * sup w 1 - α ( x , t ) G ( t ) ≤ ( 1 - α ) a ( x ) , ∀ x ∈ B .

Assume on the contrary that there exists a point x₁ ∈ B, x₁ ≠ 0 such that

lim t → T * sup w 1 - α ( x 1 , t ) ∕ G ( t ) = c > ( 1 - α ) a ( x 1 ) .

Then there exists a sequence {t_n} such that t_n → T*

lim t n → T * sup w 1 - α ( x 1 , t n ) ∕ G ( t n ) = c > ( 1 - α ) a ( x 1 ) .

By the continuity of a(x), we deduce that there exists x₂ ∈ B such that (1 - α)a(x) < c for B₁ = {x ∈ ℝⁿ : |x₂| ≤ |x| ≤ |x₁|}. Using w_r ≤ 0, (3.3) and (3.6), it is easy to check that

lim t n → T * ∫ B w 1 - α ( x , t n ) φ ( x ) d x G ( t n ) = lim t n → T * ∫ B \ B 1 w 1 - α ( x , t n ) φ ( x ) d x + ∫ B 1 w 1 - α ( x , t n ) φ ( x ) d x G ( t n ) ≥ ∫ B \ B 1 ( 1 - α ) a ( x ) φ ( x ) d x + lim t n → T * c ∫ B φ ( x ) d x > ( 1 - α ) ,

which is a contradiction to (3.6). Combining (3.3) and (3.7), Lemma 3.1 (i) is proved. Case (ii) can be treated similarly.

The key step in establishing the result of Theorem 2.2 is the following lemma.

Lemma 3.2 Under the assumption of Lemma 3.1, let u(x, t) be the blow-up solution of (2.3), which blows up in a finite time T* and non-decreasing in time t, then the following statements hold uniformly in all compact subsets of B:

(i) If p + q < 1, then

lim t → T * u 1 - q - p ( x , t ) G ( t ) = ( 1 - q - p ) a ( x ) .

(ii) If p + q = 1, then

lim t → T * ln u ( x , t ) G ( t ) = a ( x ) .

Proof. (i) Since u_r ≤ 0 and u_t ≥ 0, it then follows that u ( 0 , t ) = max x ∈ B ̄ u ( x , t ) and Δu(0, t) ≤ 0 for t > 0, which imply lim_t→T* u(0, t) = ∞. Obviously,

u t ( 0 , t ) ≤ a ( 0 ) u p + q ( 0 , t ) g ( t ) , 0 < t < T * ,

which implies

lim t → T * sup u 1 - p - q ( 0 , t ) G ( t ) ≤ ( 1 - p - q ) a ( 0 ) .

Notice that p + q < 1 and (3.8), hence lim_t→T* G(t) = ∞ and lim_t→T* g(t) = ∞.

A simple calculation yields

1 1 - r Δ u 1 - r = - r u - ( 1 + r ) | ∇ u | 2 + u - r Δ u ( if 0 < r < 1 ) .

In view of (2.3), we have, for x ∈ Ω, 0 < t < T*

1 - q 1 - p - q d u 1 - p - q d t = Δ u 1 - q + q ( 1 - q ) u - q - 1 | ∇ u | 2 + ( 1 - q ) a ( x ) g ( t ) .

Multiplying both sides of Equation (3.9) by φ and integrating over B × (0, t), it follows that

1 1 - p - q ∫ B u 1 - p - q φ d x - ∫ B u 0 1 - p - q φ d x = - λ 1 - q ∫ 0 t ∫ B u 1 - q φ d x d s + G ( t ) + ∫ 0 t ∫ B q u - q - 1 | ∇ u | 2 φ d x d s ,

for 0 < t < T*. Clearly,

∫ 0 t ∫ B u 1 - q φ d x d s ≤ ∫ 0 t u 1 - q ( 0 , t ) d s ∫ B φ ( x ) d x ,

which yields

lim t → T * ∫ 0 t ∫ B u 1 - q φ d x d s G ( t ) = 0 .

Setting u₁(r, t) = u ^(1-q)/2(r, t)(r = |x|). We may claim that

lim t → T * ( u 1 ( r , t ) ) r ( g ( t ) ) 1 ∕ 2 = 0 , a . e . r ∈ ( 0 , R ) .

Indeed, due to lim_t→T* g(t) = lim_t→T* u^s(0, t) = ∞, u_r ≤ 0, and s > 1 - q, we then have

lim t → T * ∫ 0 R ( u 1 ( r , t ) ) r d r ( g ( t ) ) 1 ∕ 2 = lim t → T * u 1 ( R , t ) - u 1 ( 0 , t ) ( g ( t ) ) 1 ∕ 2 = 0 .

Therefore, by Lebesgue's dominated convergence theorem, we infer that

lim t → T * ∫ B q u - q - 1 | ∇ u | 2 φ ( x ) d x g ( t ) = q ω n lim t → T * ∫ 0 R u - q - 1 ( r , t ) u r 2 φ ( r ) r n - 1 d r g ( t ) ≤ q ω n R n - 1 lim t → T * ∫ 0 R u - q - 1 ( r , t ) u r 2 φ ( r ) d r g ( t ) = 4 q ( 1 - q ) 2 R n - 1 ω n lim t → T * ∫ 0 R ( ( u ( 1 - q ) ∕ 2 ) r ) 2 φ ( r ) d r g ( t ) ≤ C ∫ 0 R lim t → T * ( ( u ( 1 - q ) ∕ 2 ) r ( g ( t ) ) 1 ∕ 2 ) 2 d r = 0 ,

where w_n is the surface area of unit ball in ℝ^N.

Now according to (3.10)-(3.12), we obtain

lim t → T * ∫ B u 1 - p - q φ d x G ( t ) = ( 1 - p - q ) .

On the other hand, By (3.9), we find

d u 1 - q d t ≥ u p ( Δ u 1 - q + ( 1 - q ) a ( x ) g ( t ) ) , x ∈ B , 0 < t < T * ,

where γ = p/(1 - q). Consequently, u ^1-q is a sup-solution of the problem

d v d t = v γ ( Δ v + ( 1 - q ) a ( x ) g ( t ) ) , x ∈ B , 0 < t < T * , v ( x , t ) = 0 , x ∈ ∂ B , t > 0 , v ( x , 0 ) = u 0 1 - q ( x ) , x ∈ B .

By the maximum principle, u ^1-q ≥ v in B × (0, T*). Note that 0 ≤ γ < 1, we know from Lemma 3.1 (i) that

lim t → T * v ( 1 - p - q ) ∕ ( 1 - q ) ( x , t ) G ( t ) = ( 1 - p - q ) a ( x ) ,

uniformly in all compact subsets of B.

Thus,

lim t → T * inf u ( 1 - p - q ) ( x , t ) G ( t ) ≥ ( 1 - p - q ) a ( x ) ,

uniformly in all compact subsets of B.

Next, we prove that

lim t → T * sup u ( 1 - p - q ) ( x , t ) G ( t ) ≤ ( 1 - p - q ) a ( x ) ,

uniformly in all compact subsets of B.

We can verify (3.15) by similar means of (3.7). Therefore, we conclude the proof of case (i).

(ii) Proceeding as (3.8), we have

lim t → T * sup ln u ( 0 , t ) G ( t ) ≤ a ( 0 ) .

For any compact subset B₁ ∈ B, there exists t₁ ∈ (0, T*) such that u(x, t₁) ≥ 1 for all x ∈ B ̄ 1 , and thus ln u(x, t) ≥ 0 in B ̄ 1 × ( t 1 , T * ) .

Direct calculation shows

d ln u d t = 1 1 - q Δ u 1 - q + q u - q - 1 | ∇ u | 2 + a ( x ) g ( t ) .

Let λ₁ be the first eigenvalue of -Δ in H 0 1 ( B 1 ) and by φ₁ > 0 the corresponding eigenfunction, normalized by ∫ B 1 a ( x ) φ 1 ( x ) d x = 1 . Set G 1 ( t ) = ∫ t 1 t g ( s ) d s . Clearly, lim_t→T* G(t)/G₁(t) = 1.

Multiplying both sides of Equation (3.16) by φ₁ and integrating over B₁ × (t₁, t), we get

∫ B 1 ( ln u ) φ 1 d x - ∫ B 1 ( ln u ( x , t 1 ) ) φ 1 d x = - λ ∫ t 1 t ∫ B 1 u 1 - q φ d x d s + G ( t ) + ∫ t 1 t ∫ B 1 q u - q - 1 | ∇ u | 2 φ d x d s , t 1 < t < T * .

The result of case (ii) follows by analogy with the argument used in the proof of case (i).

Proof of Theorem 2.2

(i) By Lemma 3.2 (i), we infer that

u ( 0 , t ) ~ ( ( 1 - q - p ) a ( 0 ) ) 1 ∕ ( 1 - q - p ) G ( t ) , as  t → T * ,

hence

G ′ ( t ) = g ( t ) = u q ( 0 , t ) ~ ( ( 1 - q - p ) a ( 0 ) ) q ∕ ( 1 - q - p ) G ( t ) q ∕ ( 1 - q - p ) .

Integrating equivalence (3.18) between t and T*, we obtain

G ( t ) ~ ( a ( 0 ) ( 1 - q - p ) ) - 1 ( a ( 0 ) ( p + q + s - 1 ) ( T * - t ) ) ( 1 - q - p ) ∕ ( 1 - p - q - s ) .

Using Lemma 3.2 (i) and substituting p = (m - 1)/m, q = q₁/m, s = s₁/m, t* = mτ, and u(x, t) = v^m(x, τ) into (3.19), we complete the proof of Theorem 2.2 (i).

(ii) To obtain the blow-up rate of the exponent type, we need to be more careful in this case, since exponentiation of equivalents is not permitted. Similar to the proof of Theorem 3 in 14 and Lemma 2.3 in 16, we get

lim t → T * G ( t ) = a - 1 ( 0 ) ln ( T * - t ) ( - 1 ∕ s ) .

Thanks to Lemma 3.2(ii) and (3.20), we then get the desired result.

This article deals with the porous medium equation with local and localized source terms, represented by two factors v q 1 ( x , τ ) and v s 1 ( 0 , τ ) , respectively. As we all know that, in the absence of weight function, the solutions of model (1.5)-(1.7) have a global blow-up and the rate of blow-up is uniform in all compact subsets of the domain. A natural question is what happens in the model (1.5)-(1.7), where the source term is the product of localized source, local source, and weight function. It is shown by Theorem 2.2 that if 0 ≤ q₁ ≤ 1, this equation possesses uniform blow-up profiles. In other words, the localized term plays a leading role in the blow-up profile for this case. Moreover, the blow-up rate estimates in time and space is obtained.

The authors declare that they have no competing interests.

All the authors typed, read, and approved the final manuscript.