Properties of blow-up solutions to a parabolic system with nonlinear localized terms

This paper deals with blow-up properties of the solution to a 
semi-linear parabolic system with nonlinear localized sources 
involved in a product with local terms, subject to the null 
Dirichlet boundary condition. We investigate the influence of 
localized sources and local terms on blow-up properties for this 
system. It will be proved that: (i) when $m, q\leq 1$ this 
system possesses uniform blow-up profiles. In other words, the 
localized terms play a leading role in the blow-up profile for 
this case. (ii) when $m, q>1$, this system presents single 
point blow-up patterns, or say that, in this time, local terms 
dominate localized terms in the blow-up profile. Moreover, the 
blow-up rate estimates in time and space are obtained, 
respectively.

1. Introduction and Main Results. In this paper, we consider the following semi-linear parabolic system with nonlinear localized sources accompanied by local terms x ∈ Ω, t > 0, where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, m, n, p, q are nonnegative constants and satisfy m + n > 0, p + q > 0, and x 0 ∈ Ω is a fixed point. Initial data u 0 (x), v 0 (x) ∈ C 0 (Ω) are non-negative and nontrivial. Using the methods of [8] and [17] we know that (1.1) has a local non-negative solution, and that the Comparison Principle is true. Moreover, if m, n, p, q ≥ 1 then the Uniqueness holds.
The blow-up property of the solution to a single equation of the form      u t − ∆u = u m (x, t)u p (x 0 (t), t)) − µu q (x, t), x ∈ Ω, t > 0, has been discussed by many authors, see [3,4,17,18,19] and the references therein. In the paper [17], Souplet obtained a sharp critical blow-up exponent for system (1.2). Lately, Souplet [18] introduced a new method to investigate the profile of the blow-up solution to system (1.2) with m = µ = 0. He proved that if p > 1, then uniformly on compact subsets of Ω holds In the paper [16], Pedersen and Lin studied the following problem      x ∈ Ω, t > 0, x ∈ Ω, u i = 0, i = 1, · · · , k, u k+1 := u 1 , x ∈ ∂Ω, t > 0 with p i > 1. They first proved that the solution blows up in finite time if the initial data u i0 (x) are large enough, and then derived the blow-up rate of the solution.
The primary purpose of this paper is to explore the influence of localized terms and local terms on the blow-up properties of system (1.1). Our first result is related to some sufficient conditions for that (u, v) blows up in finite time. When m ≤ 1 and q ≤ 1, we have the following uniform blow-up profiles in the interior, which show that the localized terms u p (x 0 , t) and v n (x 0 , t) play a leading role in the blow-up profile.
Theorem 2. Assume that (u, v) is a classical solution of (1.1), which blows up in finite time T . Let m, q ≤ 1, then the following statements hold uniformly on any compact subset of Ω. ( (ii) If m = 1 or q = 1, then For the case m > 1 and q ≤ 1, or the case m ≤ 1 and q > 1, we do not know how to deal with the blow-up properties of system (1.1). In the following, we focus only on the case of m, q > 1. Let us first make some assumptions: (H1) m, q > 1, and Ω = B(0; R), x 0 = 0. (H2) Initial data u 0 (x), v 0 (x) :B(0; R) → R 1 are nonnegative nontrivial, radially symmetric non-increasing continuous functions and vanish on ∂B(0; R). ( Under the above assumption (H2), the solution (u, v) is radially symmetric and non-increasing in x. Therefore, u(  When m > 1 and q > 1, system (1.1) possesses the following single point blow-up patterns, which illustrate that the local terms u m (x, t) and v q (x, t) dominate the localized terms u p (x 0 , t) and v n (x 0 , t) in the blow-up profile. When u and v blow up simultaneously, we may estimate the blow-up rate as follows.
Theorem 6. Under the conditions of Theorem 4, there exist constants 0 < c ≤ C such that the following statements hold for all 0 ≤ t < T .
(ii) If p > m − 1 and n = q − 1, then Remark 1. If m = q, n = p and u 0 (x) = v 0 (x), then system (1.1) turns into a single equation. From above, we draw a complete conclusion on the blow-up profiles. More precisely, the problem possesses uniform blow-up profiles if and only if m, the power of the local term, is less than 1.
Furthermore, for problem (1.1) with suitable initial data, its blow-up rate in space can be evaluated as follows.
2. Proofs of Theorems 1 and 2. In this section, we prove Theorems 1 and 2.
2.1. Proof of Theorem 1. We use the results of [20] and a comparison argument to prove Theorem 1. Without loss of generality we may assume that u 0 (x), v 0 (x) > 0 in Ω. And hence u, v > 0 in Ω × [0, T ), T being the maximal existence time of (u, v).
Let B(x 0 , d) be the ball centered at x 0 with radius d > 0 such that B(x 0 , d) ⊂ Ω, and let (u, v) be the solution of the auxiliary problem where u 0 (x) and v 0 (x) are nonnegative smooth symmetric, radially non-increasing functions which are less than u 0 (x) and v 0 (x) on B(x 0 , d) respectively. Then u(·, t) and v(·, t) are radially symmetric non-increasing. By the comparison principle, u ≥ u, v ≥ v as long as (u, v) and (u, v) exist. Therefore, When the condition (ii) holds, we choose (u 0 (x), v 0 (x)) and (u 0 (x), v 0 (x)) are sufficiently large. By the results of [20], (u, v) blows up in the finite time, and so does (u, v).

Proof of Theorems 2.
In what follows we intend to verify Theorem 2. For convenience, denote Before we prove Theorem 2, we claim that if (u, v) is a classical solution of system (1.1) which blows up in finite time T , that is, then u and v blow up simultaneously. In fact, we have Moreover, u and v blow up simultaneously.
Proof. Since (u, v) blows up in finite time T , it can be deduced that Without loss of generality we may assume that u(t) ∞ → ∞ as t → T . Suppose on the contrary that lim t→T g(t) < ∞. So, from the equation of u in system (1.1), we know that u exists globally, since 0 < m ≤ 1. This is a contradiction. Therefore, lim t→T g(t) = ∞.
Combining lim t→T g(t) = ∞ and g(t) = v n (x 0 , t) yields that v(x 0 , t) → ∞ as t → T . Namely, u and v blow up simultaneously.
Next, we infer that lim t T G(t) = ∞. Set U (t) = max x∈Ω u(x, t), then U (t) is Lipschitz continuous and Furthermore, because of lim t→T v(t) ∞ = ∞ which was showed above, applying similar arguments as above to the equation of To illustrate Theorem 2, We start by presenting two lemmas, which show the relationships among u, v, F (t) and G(t).

Lemma 2.
Under the conditions of Theorem 2, the following statements hold uniformly on any compact subset of Ω.
(i) If m < 1 and q < 1 then (ii) If m = 1 and q < 1 then (iv) If m < 1 and q = 1 then Proof. (i) When m < 1 and q < 1. Direct computations demonstrate uniformly on any compact subset of Ω. By comparison methods, we obtain that Hence, from (2.4) it follows that, uniformly on any compact subset of Ω holds On the other hand, we know that So, (2.5) and (2.6) guarantee that, uniformly on any compact subset of Ω, (ii) When m = 1 and q < 1. Analogous to case (i), we find that (ln u, Proceeding as case (i) we arrive at the corresponding conclusion.
Cases (iii) and (iv) can be treated similarly.
(i) If m < 1 and q < 1 then . By (i) of Lemma 2, we know that for chosen positive constants δ < 1 < τ , there exists t 0 < T such that And thus, In view of the right-hand side of (2.7), Integrating the above yields that for t 0 ≤ t < T , Due to lim t T F (t) = ∞ and q < 1, for given constant 0 < ε < 1, there exists Application of similar analysis as above to the left-hand side of (2.7) guarantees that there exists t * (2.9) and (2.10) ensure (i) of Lemma 3. Analogous to case (i), we can draw the other conclusions of Lemma 3. Lemma 3, we getT i < T such that the corresponding (i)-(iv) of Lemma 3 hold for allT i ≤ t < T .

Proof of Theorem 2. Choose {δ
(i) m < 1 and q < 1. From (i) of Lemma 2 it follows that for such sequences Integrating (2.12) and (2.13) from t from T and using of lim t T F (t) = ∞, we obtain that, for T * i ≤ t < T , where Since c i → 1 and C i → 1 on account of δ i , ε i , τ i → 1, and T * i → T as i → ∞, by letting i → ∞ in (2.14) we find (2.15) Similar as above, it can be inferred that uniformly on any compact subset of Ω. That is, uniformly on any compact subset of Ω holds lim (ii) m = 1 or q = 1. We divide this case into three subcases: (1) m = 1, q < 1; (2) m = q = 1; and (3) m < 1, q = 1. We first discuss subcase (1) m = 1 and q < 1.
Analogous as the beginning of the proof of case (i), it follows from (ii) of Lemma 2 and (ii) of Lemma 3 that for T * i ≤ T < T ,

Consequently, (2.18) and (2.19) guarantee that for T
(2.20) (2.21) As v 1−q (x, t) ∼ (1 − q)F (t) uniformly on compact subsets of Ω, we claim that, uniformly on compact subsets of Ω there holds Therefore, it can be deduced from (ii) of Lemma 2, (2.21) and (2.22) that uniformly on compact subsets of Ω, And thereby, uniformly on any compact subset of Ω, Finally, we can verify subcase (2) and (3) by similar means of subcase (1) and case (i). So, we complete the proof of Theorem 2.

Proofs of Theorems 3-7.
In this section, we pay attention to system (1.1) with m, q > 1. By the results of [8] and [20], applying standard methods we find from assumptions (H2)-(H3) that the following results are true. ( To prove Theorems 3-6, we begin with giving an elementary lemma, which will play an important part in the following. Proof. By the result (2) listed above, Hence, ∆u(0, t) ≤ 0, ∆v(0, t) ≤ 0 for any 0 < t < T . And thus, which is just the assertion (3.2). Next, we infer the assertion (3.1) and proceed our discussion as that in [22]. Since (u, v) blows up in finite time T , and u t , v t ≥ 0 for all (x, t) ∈ B(0; R) × (0, T ), it can be deduced that for any t 0 : The maximal principle shows that where η is the unit outward normal. By the standard method it follows that for any t 1 : t 0 < t 1 < T , there exists 0 < ε ≤ 1 such that i.e., for t = t 1 and x ∈B(0; R), t). By using the ideas of [9,21], we are sure that w( The maximum principle implies that w ≥ 0, z ≥ 0. Therefore, which means the assertion (3.1) is true.

Proofs of Theorems 3 and 4. In this subsection, we prove Theorems 3 and 4.
Proof of Theorem 3. Assume on the contrary that u blows up in finite time T and v is bounded in B(0; R) × (0, T ). By (3.1) and (3.2) in Lemma 4, we have As v is nonnegative and bounded in B(0; R) × (0, T ), we claim that v(0, t) ≥ c > 0, where c is a constant. Indeed, let w be the solution of the heat equation w t = ∆w with null Dirichlet boundary condition and w(x, 0) = v 0 (x), then the comparison principle asserts that v ≥ w in B(0; R) × (0, T ). Since 0 ∈ B(0; R) and v 0 (x) ≥ 0, ≡ 0, for any fixed t 1 ∈ (0, T ), there exists some constant c = c(t 1 , T ) > 0 such that w(0, t) ≥ c for all t 1 ≤ t ≤ T , and so does v. Without loss of generality, we assume that t 1 = 0, thus v(0, t) ≥ c > 0 for all t ∈ [0, T ). Thereafter, there exist positive constants C 1 ≥ C 2 and C 3 ≥ C 4 , such that for t ∈ [t 1 , T ), Due to m > 1 and lim t T u(0, t) = ∞, integrating the first inequality of (3.3) yields Consequently, for t ∈ [t 1 , T ), As p ≥ m − 1 > 0, it can be deduced that lim t→T v(0, t) = ∞. This is a contradiction. Therefore, Theorem 3 is completed.
Proof of Theorem 4. By (3.1) and (3.2) in Lemma 4, we have In view of the right-hand side of (3.4), When p ≥ m − 1, suppose on the contrary that n < q − 1. By integrating (3.5), we see that Since lim t→T u(0, t) = ∞, taking t → T in the above leads to a contradiction. Consequently, n ≥ q − 1.
When n ≥ q − 1, by using of analogous arguments, we can show p ≥ m − 1.
Similarly, we may conclude (b) of Theorem 4.

3.2.
Proof of Theorem 5. We adopt the ideas of [9] to verify Theorem 5.
Proof of Theorem 5. Assume on the contrary that (u, v) blows up at another point x * = 0. Furthermore, we may think without loss of generality that u blows up at the point x * as t → T , i.e., lim sup t→T u(x * , t) = ∞. Set r * = |x * |, then r * > 0. Since u(x, t) = u(r, t) is non-increasing in r, lim sup t→T u(r, t) = ∞ for any r ∈ [0, r * ] with r = |x|. Let a be fixed number satisfying a = r * /3 and c(x 1 ) = ε(x 1 − a) 2 with ε > 0 is a small constant to be determined.

3.3.
Proof of Theorem 6. In this subsection, we deduce Theorem 6 by introducing a lemma first, which shows the relationship between u(0, t) and v(0, t).
Lemma 5. Under the conditions of Theorem 4, for any given 0 < δ < 1, there exists t 1 ≤ T 0 < T such that the following statement hold for all t ∈ [T 0 , T ).
Analogously, we can demonstrate other cases.
Proof of Theorem 6. (i) We need only prove the case (a) p > m − 1 and n > q − 1, since the case (b) p < m − 1 and n < q − 1 can be treated similarly. Combining the first inequality of (3.2) with (i) of Lemma 5, we see that and lim t→T u(0, t) = ∞, by integrating (3.15) we find that there exists a constant c 1 > 0 such that (3.16) Applying the above arguments to the first inequality of (3.1) and using (i) of Lemma 5 show that there exists a constant C 1 > 0 such that (3.17) Consequently, it follows from (3.16), (3.17) and (i) of Lemma 5 that there exist positive constants 0 < c 2 ≤ C 2 such that (3.18) Let c = min{c 1 , c 2 } and C = max{C 1 , C 2 }, then (3.16)-(3.18) imply the desired conclusion (i) of Theorem 6.
(iii) p = m − 1 and n > q − 1, the conclusion (iii) of Theorem 6 can be derived in the similar way as case (ii).
3.4. Proof of Theorem 7. Similar as in subsection 3.2, we still apply the ideas of [9] to proceed our discussion for Theorem 7.