Fujita type theorem for a class of coupled quasilinear convection–diffusion equations

In this paper, we establish the Fujita type theorem for a homogeneous Neumann outer problem of the coupled quasilinear convection–diffusion equations and formulate the critical Fujita exponent. Besides, the influence of diffusion term, reaction term, and convection term on the global existence and the blow-up property of the problem is revealed. Finally, we discuss the large time behavior of the solution to the outer problem in the critical case and describe the asymptotic behavior of the solution.

In 1966, the first result of the exponent of the quasilinear diffusion equation was introduced by Fujita [5]. Precisely, he investigated the Cauchy problem of the semilinear equation and showed that the problem does not have any nontrivial global nonnegative solution if 1 < p < p c = 1 + 2/n, whereas there exist both nontrivial global (with small initial data) and nonglobal nonnegative (with large initial data) solutions when p > p c = 1 + 2/n. Among many other results, it proved p = p c belonged to the blow-up case by Hayakawa [12], Kobayashi et al. [13], and Weissler [27]. This is what we know as the blow-up theorem of Fujita type, and p c is called the critical Fujita exponent. Early results of the Fujita type theorem can be seen in the review articles of Deng [2], Levine [15], and relevant references. In recent years, there are a lot of Fujita's results, such as [2, 3, 8, 10, 11, 14, 16-21, 24-26, 30, 31] and the references therein. Among those works, Galaktionov et al. [6,7] considered the critical Fujita exponent of the Cauchy problem ∂u ∂t = u m + u p , x ∈ R n , t > 0 (p, m > 1) and proved that the critical Fujita exponent is p c = m + 2/n. Aguirre and Escobedo [1] demonstrated the Fujita type theorem of the following convective-diffusion equation: They demonstrated that the critical Fujita exponent was p c = min 1 + 2 n , 1 + 2q n + 1 .
Zheng and Wang [29] studied more general nonlinear convection-diffusion systems where p > m ≥ 1, κ ∈ R, -2 < λ 1 ≤ λ 2 , is the bounded area in R n with a smooth boundary ∂ and B R 1 ⊂ ⊂ B R 2 for some 0 < R 1 ≤ R 2 , and B R denotes the ball in R n with radius R and center at the origin, and ν is a unit outer normal vector to ∂B 1 . It displayed that the critical Fujita exponent is Following from a lot of results, it shows that critical Fujita exponent of a single equation is usually a constant, while the critical Fujita exponent of the coupled equations is usually a curve which is called the critical Fujita curve. In 1991, Escobedo and Herrero [4] investigated the following coupled systems: where p, q > 0, and showed that the Fujita curve is (pq) c = 1 + 2 n max{p + 1, q + 1}.
In [22], the authors studied the following Newtonian filtration system: where 0 < m < 1, p, q ≥ 1, and pq > 1. It was proved that the critical Fujita curve is In [9], the authors studied the Fujita type theorem for the outer problem of the following coupled nonlinear diffusion equations with convective terms: In this paper, we prove that the critical Fujita exponent is The main attention of this paper is to prove the global existence and blow-up properties of solutions. For the global existence of the problem solution, we use the method of constructing the self-similar solution and the comparison principle to prove our conclusion. For the blow-up properties of solutions, we adopt the integral estimation method. It is noted that when discussing the global existence of solutions, we construct the self-similar upper solution to the system. In order to let the self-similar solutions have the same compact supported set, we introduce the perturbation term (|x| + 1) μ . But the disturbance term has a negative impact on our results, which is the problem we need to solve. The paper is organized as follows. In Sect. 2, we state some definitions and some theorems. Then, several useful auxiliary lemmas are given. In Sect. 4, we derive a Fujita type theorem for problem (1)- (4). At last, we study the asymptotic behavior of the solution to problem (1)-(4) in the critical case.

Preliminaries
In this section, we introduce the definition of the solutions to problem (1)-(4) that will be useful for the rest of the paper.
and the following integral inequalities is both a supersolution and a subsolution.
which T is called the blow-up time. Otherwise, (u, v) is said to be global.
The following existence theorem and the comparison principle to problem (1)-(4) play an important role in proving our main results.
, the Cauchy problem (1)-(4) admits at least one solution locally in time.
The proofs of Theorem 2.1 and Theorem 2.2 are the same as the one in [23,25,28] and are omitted here.
To prove the existence of a nontrivial global solution to problem (1)-(4), we introduce the following form of self-similar supersolutions to system (1) and (2): v(x, t) where By a simple calculation, we show that for any r > 0. Then the self-similar function (u, v) with the structure (13)- (14) is a supersolution to (1) and (2).

Blow-up theorems of Fujita type
In this section, we establish the blow-up theorems of Fujita type for problem (1)-(4). First, we consider the case κ ≤ -n.
Then there exists sufficiently large l 2 > 1 such that Combining (32) with (33), we get Just like the proof of Theorem 4.1, we can obtain that (u, v) blows up in a finite time.
Then there exists sufficiently large l 3 such that where C 5 > 0 is a positive constant depending only on l 3 . Therefore, we can obtain that (u, v) blows up in a finite time by a similar proof process of Theorem 4.1.
For other cases, select θ = 0. By the similar argument as A(q, μ) = 0, A(p, λ) < 0, we can also prove that any nontrivial solution blows up in a finite time. Proof The comparison principle and Lemma 3.1 can prove the existence of the nontrivial global solution to problem (1)-(4) with sufficiently small initial value. Next, we study the blow-up solution to problem (1)-(4) when the initial value is sufficiently large.
For l > 1 and (u, v) is the solution to problem (1)-(4), set According to the Hölder inequality and (30), we have where If (u 0 , v 0 ) is so large that By a similar argument in the proof of Theorem 4.1, one can show that (u, v) blows up in a finite time.

The critical case
In this section, we consider the critical case Obviously, we can prove that (29), (32) still hold, and The result of the critical case is based on the following three lemmas.

Lemma 5.1
Assume that (u, v) is a nontrivial global solution to problem (1)-(4) with p = p c , then there exists M 0 > 0 independent of t such that where Proof It is easy to verify that For any sufficiently large l > 1, it follows from the Hölder inequality that ≥ -m q R n \B 1 |x| μ u q (x, t)ψ l |x| dx + R n \B 1 |x| μ u q (x, t)ψ l |x| dx qm q (C 0 C 9 ) q/(q-m) l (q(n+κ-2)-m(n+κ+μ))/(q-m) -m p c R n \B 1 |x| λ v p (x, t)ψ l |x| dx + R n \B 1 |x| λ v p (x, t)ψ l |x| dx