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Fujita type theorem for a class of coupled quasilinear convection–diffusion equations
Boundary Value Problems volume 2021, Article number: 36 (2021)
Abstract
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.
1 Introduction
In this paper, we consider the critical Fujita exponent of the following coupled quasilinear convection–diffusion equations:
where \(p, q>m>1\), \(\kappa \in \mathbb{R}\), \(\lambda \ge 0\), \(\mu = \frac{\lambda (q-m)+2(q-p)}{p-m}\ge 0\). In addition, \(B_{1}\) denotes the unit ball in \(\mathbb{R}^{n}\), ν denotes the unit inner normal vector to \(\partial B_{1}\), and \(0\le u_{0}\), \(v_{0}\in C_{0}(\mathbb{R}^{n})\) are nontrivial.
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
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:
where \(q\ge 1\), \(p>1\), \(\boldsymbol{b}_{0}\in \mathbb{R}^{n}\). They demonstrated that the critical Fujita exponent was
Zheng and Wang [29] studied more general nonlinear convection–diffusion systems
where \(p>m\ge 1\), \(\kappa \in \mathbb{R}\), \(-2<\lambda _{1}\le \lambda _{2}\), Ω is the bounded area in \(\mathbb{R}^{n}\) with a smooth boundary ∂Ω and \(B_{R_{1}}\subset \Omega \subset B_{R_{2}}\) for some \(0< R_{1}\le R_{2}\), and \(B_{R}\) denotes the ball in \(\mathbb{R}^{n}\) with radius R and center at the origin, and ν is a unit outer normal vector to \(\partial B_{1}\). 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
In [22], the authors studied the following Newtonian filtration system:
where \(0< m<1\), \(p,q\ge 1\), 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:
and obtained
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 \(( \vert x \vert +1)^{\mu }\). 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.
2 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.
Definition 2.1
Let \(0< T\le +\infty \). A pair of nonnegative functions \((u,v)\) is called a super (sub) solution to problem (1)–(4) in \((0,T)\) if
and the following integral inequalities
are fulfilled for any \(0\le \varphi \), \(\psi \in C^{2,1}(\mathbb{R}^{n}\times [0,T))\) vanishing when t is near T or \(\vert x \vert \) is sufficiently large. \((u,v)\) is called a solution to problem (1)–(4) in \((0,T)\) if it is both a supersolution and a subsolution.
Definition 2.2
A solution \((u,v)\) to problem (1)–(4) is said to blow up in a finite time \(0< T<+\infty \) if
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.
Theorem 2.1
(Local existence)
When \(0\le u_{0}\), \(v_{0}\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n})\cap L^{\infty }(\mathbb{R}^{n})\), the Cauchy problem (1)–(4) admits at least one solution locally in time.
Theorem 2.2
(Comparison principle)
For \(0< T\leq +\infty \), assume that \((u^{*},v^{*})\) and \((u^{**},v^{**})\) are two solutions to system (1) and (2) with nonnegative initial data \(u_{0}^{*}(x)\), \(v_{0}^{*}(x)\) and \(u_{0}^{**}(x)\), \(v_{0}^{**}(x)\) in \((0, T)\), respectively. If \((u_{0}^{*}(x), v_{0}^{*}(x))\le (u_{0}^{**}(x), v_{0}^{**}(x))\) a.e. in \(\mathbb{R}^{n}\), then \((u^{*},v^{*})\le (u^{**},v^{**})\) a.e. in \(\mathbb{R}^{n}\times (0,T)\).
The proofs of Theorem 2.1 and Theorem 2.2 are the same as the one in [23, 25, 28] and are omitted here.
3 Auxiliary lemmas
In order to research the blow-up property of solutions to problem (1)–(4), we need the following auxiliary lemmas.
Lemma 3.1
Assume that \((u,v)\) is a solution to problem (1)–(4). Then there exists \(R_{0}>0\) depending only on n and κ such that, for any \(l>R_{0}\),
where
and
Proof
It follows from Definition 2.1 that
where \(\psi _{l}(r)\in C^{1}([0,+\infty ))\) satisfies \(\psi _{l}'(0)=0\) and
For \(0\le \vert x \vert \le l\), it is easily verified that
While for \(l\le \vert x \vert \le \delta l\), a direct calculation gives
If \(n+\kappa -1\le 0\), one gets
If \(n+\kappa -1>0\), we have
By (9)–(12), we obtain (7). Similarly, one can prove that (8) holds. □
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):
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).
Lemma 3.2
Assume that \(m>1\), \(\kappa >-n\), \(p>p_{c}\) and set
where \(s_{+}= \max \{0,s\}\), \(\eta >0\), and
Then there exists sufficiently small \(\eta >0\) such that \((u,v)\) given by (13), (14), and (17) is a supersolution to system (1) and (2).
Proof
It is clear that \(U^{m}\) and \(V^{m}\) satisfy (15) and (16) when \(r\ge (\eta /A)^{1/2}\). For \(0< r<(\eta /A)^{1/2}\), a simple computation can obtain
and
Due to \(\frac{2Am}{m-1}<\beta \), there exists sufficiently small \(\eta _{1}>0\) such that, for \(0<\eta <\eta _{1}\),
Then, due to \(\lambda , \mu >0\) and the definition of U, V, there exists \(\eta _{2}>0\) such that, for any \(0<\eta <\eta _{2}\),
Combining the above inequations with (18) and (19), we can see that for sufficiently small \(0<\eta _{2}<\eta _{1}\) and \(0<\eta <\eta _{2}<\eta _{1}\), one gets (15) and (16). Thus, \((u,v)\) given by (13), (14), and (17) is a supersolution of system (1) and (2). □
4 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 \(\kappa \le -n\).
Theorem 4.1
Assume that \(p, q>m\), \(\lambda , \mu \ge 0\), \(\kappa \le -n\), and \(0\le u_{0}\), \(v_{0}\in C_{0}(\mathbb{R}^{n}\setminus B_{1})\) are nontrivial, the solution to problem (1)–(4) blows up in a finite time.
Proof
Let \((u,v)\) be the solution to problem (1)–(4). Denote
For any \(l>R_{0}\), Lemma 3.1 shows that
The Hölder inequality leads to
where \(C_{1}>0\) is a positive constant independent of l. Substituting (22) and (23) into (21) shows that
Owing to the Hölder inequality, for any \(t>0\), we have
which imply
where \(C_{2}>0\) is a positive constant independent of l and \(A(q,\mu )=n+\kappa +\mu -q(n+\kappa )\), \(A(p,\lambda )=n+\kappa + \lambda -p(n+\kappa )\). Here, it should be pointed out that the above discussion only requires \(p,q >m\).
Due to \(\kappa \le -n\), it is easy to verify that \(A(q,\mu )>0\), \(A(p,\lambda )>0\). From (24)–(26),
For sufficiently large \(l_{1}>1\), and note that \(-2+n+\kappa -m(n+\kappa +\mu )/q<0\), \(-2+n+\kappa -m(n+\kappa +\lambda )/p<0\), one can get
where \(C_{3}>0\) is a constant depending on \(l_{1}\). Since \(p, q>m>1\), there exists \(0< T<+\infty \) such that
Obviously, \(\operatorname{supp}\psi _{l_{1}}(x)=B_{2{l_{1}}}\). Then one gets
That is to say, \((u,v)\) blows up in a finite time. □
Next, we discuss the case \(\kappa >-n\).
Theorem 4.2
Assume that \(p, q>m>1\), \(\lambda , \mu >0\), \(\kappa >-n\), and \(0\le u_{0}\), \(v_{0}\in C_{0}(\mathbb{R}^{n}\setminus B_{1})\) are nontrivial. Then, for \(p< p_{c}\), any nontrivial solution to problem (1)–(4) blows up in a finite time.
Proof
Let \((u,v)\) be a nontrivial solution to problem (1)–(4). Set
where θ is a constant determined below. According to Lemma 3.1, for any \(l>R_{0}\),
Substituting (22) and (23) into (29) shows that
Let us discuss the classification of symbols of \(A(q,\mu )\) and \(A(p,\lambda )\) in (25), (26).
If \(A(q,\mu )<0\), \(A(p,\lambda )<0\), we substitute (25) and (26) into (30), and this yields that
where \(C_{4}=\max \{C_{2}^{m/p}, C_{2}^{m/q}\}>0\), and
Set
which implies that
namely,
By a simple calculation,
Note that if \(p< p_{c}\), then \(m(\theta )<\Theta \). Further, \(w_{l}(0)\) is nondecreasing with respect to \(l\in (0,+\infty )\) and
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.
For \(A(q,\mu )=0\), \(A(p,\lambda )<0\), we set \(\theta =0\). It follows from (25), (26), and (30) that
Here
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 \(\theta =0\). By the similar argument as \(A(q,\mu )=0\), \(A(p,\lambda )<0\), we can also prove that any nontrivial solution blows up in a finite time. □
Theorem 4.3
Assume that \(p, q>m>1\), \(\lambda , \mu >0\), \(\kappa >-n\), and \(0\le u_{0}\), \(v_{0}\in C_{0}(\mathbb{R}^{n}\setminus B_{1})\) are nontrivial. Then, if \(p>p_{c}\), there exist both nontrivial global and blow-up solutions to problem (1)–(4).
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
then (35) leads to
By a similar argument in the proof of Theorem 4.1, one can show that \((u,v)\) blows up in a finite time. □
5 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
Proof
For any sufficiently large \(l>1\), it follows from (32) that
where \(w_{l}\) is defined by (28) with \(\theta =0\). Similar to the end of the proof of Theorem 4.1, there exists some \(l_{3}>1\) such that, for any \(l>l_{3}\),
which implies
Let \(l\to +\infty \) in the above inequality, then we can obtain (38). □
Lemma 5.2
Under the assumption of Lemma 5.1, there exist three positive constants \(M_{1}, M_{2}, M_{3}>0\) independent of l and t such that, for any sufficiently large \(l>1\),
where
Proof
It is easy to verify that
For any sufficiently large \(l>1\), it follows from the Hölder inequality that
where \(C_{8}>0\) is a constant independent of l. Substituting the above two inequalities into (29) with \(\theta =0\), it follows from (25), (26), (37), (40), and (41) that
which yields (39) by choosing
□
Lemma 5.3
Under the assumption of Lemma 5.1, there exists a constant \(M_{4}>0\) independent of l and t such that, for any sufficiently large \(l>1\),
Proof
Owing to the Hölder inequality, one obtains
where \(C_{9}>0\), independent of l. Substitute the above results into (29) and
then it follows from the Young inequality that
where
□
Now we prove the following theorem.
Theorem 5.1
Assume that \(\kappa >-n\). Then any nontrivial solution to problem (1)–(4) with \(p=p_{c}\) blows up in a finite time.
Proof
We prove the theorem by contradiction. Assume that \((u,v)\) is a nontrivial global solution to problem (1)–(4) with \(p=p_{c}\). Set
It follows from (38) and the nontriviality of \((u,v)\) that \(0<\Lambda <+\infty \). Owing to (43) and the monotonicity of \(w_{l}(t)\) with respect to \(l\in (0, +\infty )\), there exist \(l_{0}>1\) and \(t_{0}>0\) such that, for any \(0<\varepsilon <\Lambda \),
From Lemma 5.3, for \(s\ge t_{0}\), we obtain
which yields that
Let \(l=l_{0}\) in (39), from the above inequality, one gets that
Take \(\varepsilon _{0}\in (0,\Lambda )\) and \(M_{5}>0\) to get
where \(\varepsilon _{0}\) and \(M_{5}\) are independent of \(l_{0}\), \(0<\tau <\min \{\frac{p_{c}-m}{p_{c}-1}, \frac{q-m}{q-1} \}\). Then we obtain
where
Integrating (44) over \((t_{0},t_{1})\) with respect to t and using
one gets that
That is to say,
where
is a positive constant independent of \(l_{0}\). It is obviously verified that
Using the same method, one gets
where
Similarly, for any positive integer i, we obtain
where
Letting \(i\to +\infty \) in (45) implies
which contradicts (38). □
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The authors would like to thank the referees for their valuable comments and suggestions which improved the original manuscript.
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This work is supported by the National Natural Science Foundation of China (No. 11871133), by the Department of Science and Technology of Jilin Province (YDZJ202101ZYTS044).
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Zhou, Y., Leng, Y. & Nie, Y. Fujita type theorem for a class of coupled quasilinear convection–diffusion equations. Bound Value Probl 2021, 36 (2021). https://doi.org/10.1186/s13661-021-01513-w
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DOI: https://doi.org/10.1186/s13661-021-01513-w