A NOTE ON THE CAUCHY PROBLEM FOR HEAT EQUATIONS WITH COUPLING MOVING REACTIONS OF MIXED TYPE

Abstract

This paper deals with the Cauchy problem for heat equations with coupling moving reactions of mixed type. After obtaining the infinite Fujita blow-up exponent, we classify optimally the simultaneous and non-simultaneous blow-up for two components of the solutions. Moreover, blow-up rates and set are determined. By using the analogous procedures, one can fill in the gaps for the other two systems, which are studied in the paper 'Australian and New Zealand Industrial and Applied Mathematics Journal' 48(2006)37-56.

1. Introduction and main results

Recently, Xiang, Chen and Mu in [1] studied the following two types of parabolic equations

where u0(x), v0(x) ≥ 0, ≢ 0 are continuous bounded functions in ℝN; m, n, p, q ≥ 0 and pq > 0 for the coupling; moving site x0(t) : ℝ+ → ℝN is Hölder continuous. The results for (1) can be summarized below with the help of the figure:

It can be checked that blow-up phenomena are unsettled yet in the exponent regions V, VI, VII , and on the boundary between V and VIII , between IX and VI, between V and III , and between IV and VI. For system (2), the authors obtained that simultaneous blow up occurs for q ≥ m and p ≥ n; If q < m, p ≥ n, or q ≥ m, p < n, non-simultaneous blow-up occurs; The uniform blow-up profiles are obtained for q ≥ m, p ≥ n. Equivalently to VII in Figure 1.1 (i.e., m > q, n > p), blow-up phenomena are unknown.

Figure 1.1.Blow-up classifications for system (1)

Motivated by [1], we study the parabolic equations with coupling moving reactions of mixed type,

with smooth initial data u(x, 0) = u0(x) ≥ 0, v(x, 0) = v0(x) ≥ 0 in ℝN; constants m, n, p, q ≥ 0; x0(t) : ℝ+ → ℝN is Hölder continuous; let T be the maximal existence time of local classical solutions. The existence and uniqueness of local positive classical solutions (u, v) of (3) can be obtained by using the standard methods of [4,5]. Nonlinear parabolic systems like (3) come from population dynamics, chemical reactions, heat transfer, etc., where u and v represent the densities of two biological populations during a migration, the thickness of two kinds of chemical reactants, the temperatures of two different materials during a propagation, etc, in which the nonlinear reactions in such dynamical systems take place only at a single (sometimes several) site(s) and couple through different types of nonlinearities. The interested readers refer to [6,7,8] and the papers therein. We obtain the first theorem about the infinite Fujita blow-up exponent and blow-up set for (3).

Theorem 1.1. Let (u, v) be any positive local classical solution of (3). Then limt→T (u+v) = +∞, x ∈ ℝN if and only if max{m−1, n, pq−n(m−1)} > 0. And the solutions blow up everywhere in ℝN.

In the sequel, we discuss the blow-up solutions only. The second result shows the occurring of only simultaneous or non-simultaneous blow-up.

Theorem 1.2.

Corollary 1.3.

Additionally, we assume that x0 : ℝ+ →ℝ is smooth and

(H) , x ∈ ℝ, t ∈ [0, T); For small ε ∈ (0, 1),

Theorem 1.4. Let assumption (H) be in force.

The key clues on non-simultaneous and simultaneous blow-up are the signals of (m − q − 1) and (n − p). By using the analogous methods in this paper, one can check that the key clues are the signals of (m − q − 1), (n − p − 1) and the signals of (m − q), (n − p) for problems (1) and (2), respectively. The following is the detail:

The following results give the blow-up rates. It can be understood that non-simultaneous blow-up rate is equivalent to the one for the scalar equation (see [9]): If u (v) blows up alone, then for any x ∈ ℝN. By using the analogous methods of [10], simultaneous blow-up rates are obtained.

Theorem 1.5. Let (u, v) be a blow-up solution of (3).

(i) If m < q + 1, n < p, then

(ii) If m < q + 1, n = p, then

(iii) If m = q + 1, n < p, then

(iv) If m = q + 1, n = p, then

(v) If m > q + 1, n > p and there exist initial data such that simultaneous blow-up occurs, then

 

2. Infinite Fujita blow-up exponent

We prove the blow-up criteria and blow-up set for system (3) by the comparison principle.

Proof. (Theorem 1.1) Inspired by Souplet [9], we introduce the following auxiliary functions

It can be checked that

By the comparison principle, we have

Especially, . By a simple calculation, we obtain

One can check from (7) that for every initial data if max{m − 1, n, pq − n(m − 1)} > 0. Hence any positive solution of (3) blows up everywhere in ℝN. On the other hand, it is easy to see that every positive solution is global for m ≤ 1 and n = pq = 0. □

 

3. Any blow-up must be simultaneous or non-simultaneous

By (6), shall the same blow-up time for all x ∈ ℝN, and so do . It suffices to discuss blow-up estimates of

Proof. (Theorem 1.2) By Theorem 1.1, every solution blows up. Combining (4) with (5), we obtain that there exists some positive constant C such that

(i) m ≤ q + 1, n ≤ p.

According to the above four subcases, we have, if m ≤ q + 1 and n ≤ p, only simultaneous blow-up occurs.

(ii) If m > q +1 and p > n, then . Hence there must be the case for u blowing up alone for every initial data. If m > q+1 and p = n, then . Then u blows up alone for every initial data.

Case (iii) can be obtained by the similar method of (ii). □

 

4. Existence of simultaneous and non-simultaneous blow-up

We use two lemmas to prove Theorem 1.4 (i)-(ii). Define the set of the initial data = {(u0, v0) | (u0, v0) satisfies (H)}. Without loss of generality, we consider the case: , x ∈ ℝ.

Lemma 4.1. For any (u0, v0) ∈ , there are

Proof. By calculations, we have utx − uxxx = vtx − vxxx = 0 in ℝ × (0, T), and in ℝ. Then ux, vx ≥ 0 in ℝ × [0, T) by the comparison principle. One can also check that

By the comparison principle, ut(x, t), vt(x, t) ≥ 0 in ℝ × [0, T). Then (9) is obtained. In order to prove (10), construct auxiliary functions

For constant ε ∈ (0, 1), one can check

By the comparison principle, J(x, t), K(x, t) ≥ 0 in ℝ × [0, T). □

By Lemma 4.1, we obtain the following important estimates

In fact, by using (10),

By integrations, estimates (11) and (12) can be obtained.

We will use (11) and (12) to prove the existence of non-simultaneous blow-up.

Lemma 4.2. Assume the initial data satisfies (H).

(i) There exist suitable initial data such that u blows up alone if m > q + 1.

(ii) There exist suitable initial data such that v blows up alone if n > p.

Proof. At first, we prove the phenomena for u blowing up alone under suitable initial data. Assume that (ũ0, ṽ0) be a pair of initial data such that the positive solution of (3) blows up. Fix v0(≥ ṽ0) and take M0 > ║v0║∞. Let u0(≥ ũ0) be large such that T satisfies

Consider the auxiliary problem

For m > q + 1 and by Green’s identity [4,11], we have

where Γ is the fundamental solution of the heat equation. Hence w(x0(t), t) ≤ M0. So w satisfies

It follows from (11) that v satisfies

By the comparison principle, v ≤ w ≤ M0 in ℝ × (0, T). Since (u0, v0) ≥ (ũ0, ṽ0), (u, v) blows up. And hence only u blows up at time T.

Secondly, we prove the phenomena for v blowing up alone under suitable initial data. Assume that (ũ0, ṽ0) be a pair of initial data such that the positive solution of (3) blows up. Fix u0(≥ ũ0) and take M1 > ║u0║∞. Let v0(≥ ṽ0) be large such that T satisfies

Consider the auxiliary problem

For n > p and by Green’s identity, we have . So z satisfies in ℝ × (0, T). It follows from (12) that in ℝ × (0, T). By the comparison principle, u(x, t) ≤ z(x, t) ≤ M1 in ℝ × (0, T). Since (u0, v0) ≥ (ũ0, ṽ0), (u, v) blows up. And hence only v blows up. □

We use two lemmas to prove Theorem 1.4 (iii).

Lemma 4.3. The set of (u0, v0) in such that v (u) blows up alone is open in L∞-norm.

Proof. Without loss of generality, we only prove the case for v blowing up with u remaining bounded. Let (u, v) be a solution of (3) with initial data (u0, v0) ∈ such that v blows up while u remains bounded up to blow-up time T, say 0 < u(x, t) ≤ M. By Corollary 1.3, there must be n > p. It suffices to find a L1-neighborhood of (u0, v0) in such that any solution of (3) coming from this neighborhood maintains the property that blows up while û remains bounded. Take M2 > M + ║u0║∞/2. Let (ũ, ṽ) be the solution of the following problem

where (ũ0, ṽ0) and ε0 are to be determined. Denote

Since v blows up at time T, there exists some small constant ε0 > 0 such that (ũ, ṽ) blows up and T0 satisfies , provided (ũ0, ṽ0) ∈ ℕ(u0, v0). Consider the auxiliary system

By Green’s identity, we have

Hence there is

On the other hand, we have

By the comparison principle, ũ ≤ U ≤ M2 in ℝ × (T − ε0, T − ε0 + T0), then ṽ blow up.

According to the continuity about initial data for bounded solutions, there exists a neighborhood of (u0, v0) in such that every solution starting from the neighborhood will enter ℕ(u0, v0) at time T−ε0, and keeps the property that blows up while û remains bounded. □

Lemma 4.4. Assume m > q + 1 and n > p. Then both non-simultaneous blow-up and simultaneous blow-up may occur.

Proof. Assume (u0, v0) ∈ such that the solution of (3) blows up. Then the positive solution with initial data (u0/l, v0/(1 − l)) ∈ for any l ∈ (0, 1) also blows up. By Lemma 4.2, we know there exist some l1 near 0 such that u blows up while v remains bounded if l = l1, and some l2 near 1 such that v blows up while u remains bounded if l = l2, respectively. By Lemma 4.3, such sets of initial data are open and connected. Then there exists some l ∈ (l1, l2) such that simultaneous blow-up happens. □

Till now, the proof of Theorem 1.4 is finished.