Existence and blow-up of global solutions for a class of fractional Lane–Emden heat flow system

. In this paper, we consider a class of Lane–Emden heat flow system with the fractional Laplacian


Introduction
The classical Lane-Emden equation −∆u = u p , x ∈ R N , N > 2, p > 1, has been extensively studied, going back to the pioneering work of astronomers and astrophysicists Lane [32] and Emden [15]. It is one of the basic equations in the theory of stellar structure and originally used to compute the pressure, density and temperature on the surface of the Sun. It has been discussed by many scholars, see [13,18,35,43,47] and the references therein. The existence and nonexistence of the global solutions of the equation was once an significant research topic for scholars. For instance, Gidas and Spruck [22] proved that the equation has no positive classical solution in a bounded domain when 1 < p < N+2 N−2 , while the existence of the solution was solved by Caffarelli et al. in [6]. Thereafter, Chen and Li [8] found the form of the positive solution for p = N+2 N−2 (N ≥ 3) in the whole space and obtained that only trivial solutions exist for p < N+2 N−2 by using the method of moving planes. In addition, as for p > N+2 N−2 , Zou [49] proved that the equation has a unique positive radial symmetric solution with polynomial decay at infinity. Meanwhile, scholars also discussed the existence and nonexistence of solutions to nonlinear elliptic equation and system with a more general nonlinearity. In [4], Bernard studied the semilinear elliptic equation −∆u = u p + f (x) in the whole space. He obtained the blow-up of the global solutions for 1 < p ≤ N N−2 , while if p > N N−2 and f ∈ C 0,γ (R N ) with 0 < γ ≤ 1, he showed that the equation has a bounded positive solution. Obviously, the Lane-Emden type system is the natural counterpart of the Lane-Emden equation where N ≥ 3, p, q > 1. When f = 0, Mitidieri [37] proved that there has no nontrivial radial positive solutions of class C 2 (R N ) by contradiction if 1 < p ≤ q and 1 p+1 + 1 q+1 > N−2 N , while if 1 < p ≤ q and 1 p+1 + 1 q+1 ≤ N−2 N , the existence of positive (radial, bounded) classical solution for the system is fully solved by Serrin and Zou in [42]. As for more general cases, when f ∈ L N(pq−1) 2q(p+1) , Ferreira et al. [19] showed the existence of the global solutions in the supercritical case N > max 2q(p+1) pq−1 , 2p(q+1) pq−1 by means of the fixed point theorem, here the range for (p, q) covers the critical and supercritical cases with respect to the hyperbola 1 p+1 + 1 q+1 = N−2 N . In case N ≤ max 2q(p+1) pq−1 , 2p(q+1) pq−1 , the nonexistence results has been pointed out by Mitidieri in [38]. For more researches on elliptic equations, please refer to [2,11,34,40].
The parabolic equation corresponding to the classical Lane-Emden equation, namely the semilinear reaction-diffusion equation x ∈ R N , t > 0, has been studied by many scholars, since the pioneering work [20] of Fujita in 1966, where it was shown that the Cauchy problem of the equation has two cases of the solution: if q > q c = 1 + 2 N , there exist both global and blow-up solutions, corresponding to small and large initial values, respectively; while if q < q c = 1 + 2 N , then the problem does not admit nonnegative global solution. The case of q = q c = 1 + 2 N was decided by Hayakawa [28] for N = 1, 2 and Kobayashi et al. [31] for all N ≥ 1 that the problem does not admit nontrivial nonnegative global solution. Thus, it can be seen that the range of index q plays an important role in the researches of existence and blow-up of the solutions. And q c is called Fujita critical exponent. Since then, there have been a number of extensions to the research of critical exponent in several directions. For instance, Pascucci [39] considered a semilinear Cauchy problem on nilpotent Lie groups and obtained the sharp Fujita critical exponent, which generalized the results in [20,28,31].
As for the semilinear parabolic system, Escobedo and Herrero [16] discussed the Cauchy problem of semilinear reaction-diffusion system in the whole space They showed that the system has two significant curves, namely, the global existence curve pq = 1 and the Fujita curve pq = 1 + 2 N max {p + 1, q + 1}. If 0 < pq ≤ 1 or pq > 1 + 2 N max {p + 1, q + 1} with suitably small initial values, then every solution is global by employing the integral equivalent system and Gronwall-type inequalities, respectively; while if pq > 1 + 2 N max {p + 1, q + 1} with large initial values, the system possesses no nontrivial global solution. Meanwhile, the nonexistence of nontrivial global solution is proved based on some heat kernel estimates for 1 < pq ≤ 1 + 2 N max {p + 1, q + 1}. For some problems with boundary conditions or nonlinear terms different from the above, many scholars have also studied the existence and nonexistence of global solutions. For example, Deng and Fila [14] and Bai et al. [5] discussed the Fujita critical exponent of parabolic problems in the upper half space and bounded domain respectively. For more researches on parabolic system, see for example [17,29,30,45].
In mathematical physics, nonlinear evolution equations with the fractional Laplacian are extensively used to describe anomalous diffusion, see [25,26,33] and the references therein. Therefore, it is of theoretical value and practical significance to study the existence of solutions of equations with the fractional Laplacian. Amor and Kenzizi [3] studied the Cauchy problem of the fractional perturbed heat equation on a bounded domain and obtained the necessary conditions for the existence of nonnegative global solution. In [23], Greco et al. concerned the Cauchy problem of the fractional heat equation u t + (−∆) s u = 0 in the whole space. It was showed that the problem has a global solution if the initial value subject to a certain growth condition. In addition, many scholars have also considered the fractional nonhomogeneous parabolic equation When f (t, u) = h(t)u p , Guedda et al. [24] and Tan et al. [44] concerned the Cauchy problem of the equation by means of the integral equivalent equation and the contraction mapping principle, respectively. Their conclusions implied that the Fujita critical exponent is 1 + 2α(1+σ) N . Here, p > 1 and the function h(t) ∈ C ([0, ∞)) satisfied c 0 t σ ≤ h(t) ≤ c 1 t σ with c 0 , c 1 > 0, σ > −1 for t large enough. Besides, the nonexistence of nontrivial nonnegative solutions and the asymptotic symmetry of the solution were obtained in [10] and [9] under suitable assumptions on f (t, u) via narrow region principles and the method of moving planes, respectively. For more works about the fractional parabolic equation, see [1,21,36,41] and the references therein.
Inspired by the above literature, we study the Cauchy problem of the Lane-Emden heat flow system with the fractional Laplacian This problem is used to describe the heat transfer of two mixed combustibles, where u and v represent the temperature of anomalous diffusion at which the two substances interact respectively. We are primarily concerned with the case 0 < α ≤ 2, Q := R N × (0, +∞), N ≥ 3.
loc (R N ) for i = 1, 2 are nonnegative functions. The nonnegative coupling terms for ω 3 , ω 4 > 1, which assumes only nonnegative values. We show the existence of a unique global solution for (1.1) in the supercritical case and the problem does not admit nonnegative global solutions in the critical case. As for the subcritical case, we consider the blow-up of the global solution for the following Cauchy problem of the higher-order nonlinear evolution system For simplicity, throughout the paper, we denote by C a generic positive constant which may vary in value from line to line and even within the same line, but is independent of the terms which will take part in any limit process.
The following Duhamel's integral equivalent system [44] will be used to prove the existence of a global solution for (1.1) in the supercritical case and the blow-up result in the critical case for (1.1).
From [48], we have To facilitate writing, we set Define and In this framework we can write the integral system (1.4)-(1.5) in the abstract form and pq−1 , we denote Below we assume the basic assumptions on the range of p 1 and q 1 : 13) and N αq 1 < Nq (1.14) The range and some basic assumptions of the indexes p ′ 1 and q ′ 1 are 1 (1.16) (1.17) The above assumptions are used in the following statements. Our main results read pq−1 . Let C 0 R N denote the space of all continuous functions decaying to zero at infinity, and let a( where the constant K 1 is as in Lemma 2.3.

Remark 1.4.
It is worth noting that, compared with the semilinear reaction-diffusion system of the classical Laplacian in [16], the influence of the fractional operator and the nonlinear terms for (1.1) we consider on the estimates are more complicated. Hence, when we prove Theorem 1.2, we argue by contradiction, the integral related to the initial value is estimated skillfully, which reduces a large number of calculations generated by using the method in [16], and the method here is more convenient. Remark 1.5. From Theorem 1.3, if α = β and k = 1, then hypothetical condition will correspondingly change to N < max α(p+1) pq−1 , pq−1 , which is consistent with the indexes in Theorems 1.1 and 1.2. So we can get that the critical curve for (1. pq−1 . Next, we give some comments about the critical curve(exponent) for (1.1). ( , which is the critical curve for semilinear reaction-diffusion system in [16].
( [24,44]. We conclude this introduction by describing the plan of the paper. Section 2 recalls some lemmas and some properties of the fundamental solution Γ(x, t) which we shall use in the sequel. In Section 3, we use the contraction mapping principle to prove the existence of a unique global solution for (1.1) in the supercritical case, and further obtain some relevant properties of the global solution. The blow-up of the global solutions in the critical case is discussed via Duhamel's integral equivalent equations and combined with proof by contradiction, which is gathered in Section 4. As for the blow-up result for a more general higher-order system (1.3) in the subcritical case, we utilize the test function method to obtain and make up the content of Section 5. Section 6 is an appendix, in which we prove some lemmas given in Section 2 in detail.

Preliminaries
In this section, we mainly introduce some lemmas, as well as some properties and estimates related to the kernel function Γ(x, t), which will be utilized in the following proofs. For general k, we first give the definition of weak solutions for (1.3).
(5) For all x ∈ R N and t, s > 0, the following Chapman-Kolmogorov equation holds: is a bounded map. Furthermore, for any T > 0 and h(x, t) ∈ L m (R N ), there are positive constants K 1 and K 2 depending only on m, n and l, such that for all t ∈ (0, T] and any l > 0, where 1 + 1 Here and hereafter, " * " stands for the convolution in the space variable.
See Appendix for detailed proof of Lemma 2.3.

Lemma 2.4 (See [7]
). Let a ∧ b := min {a, b} for a, b ∈ R. Then there exist positive constants C α, N and C ′ α,N , depending only on N and α, such that

5)
for q > 1 and all x ∈ R N .
Similar estimates can be found in [24, Lemma 3.2]. To make the paper self-contained, we give the proof of Lemma 2.5 in Appendix.

Existence of the global solution for (1.1) in the supercritical case
In this section, we utilize (1.10) and the contraction mapping principle to prove the existence of a global solution for (1.1) in the supercritical case. To achieve this, we first derive a key lemma, which provides estimates for the integrals in (1.10). Define For each δ > 0 fixed we consider the space D defined by where constants b 1 , b 2 are given by formulas (3.9)-(3.10).
On the space D, we show the following lemma: Lemma 3.1. Let p 1 , q 1 be as in (1.12)-(1.14) and p ′ 1 , q ′ 1 be as in (1.15)-(1.16), (u, v) ∈ D. For all v 1 , v 2 ∈ L q 1 R N and u 1 , Proof. We will only prove the estimates in (3.1) and (3.3) because the ones in (3.2) and (3.4) can be obtained analogously.
Proof of Theorem 1.
Combining (1.15)-(1.17) and (3.24)-(3.26), we conclude that We consider the space It is easy to see that E 1 ⊂ E. Applying Lemma 2.3, similar to (1), we have and It follows that , and then integrating on Q, we obtain One can invert the order of integration and utilize the self-adjointness of (−∆) α 2 to obtain that (3.36) Furthermore, A 2 is estimated as follows where we have used the fact that φ ∈ C ∞ c (Q) and then repeating the fixed point argument, it is easily conclude that On the metric space (B δ , d B ), using (3.14) and (3.18), we can get Similarly, we obtain As a result, Hence, Next, from (1.14) we can get Thus, there exists p 2 > p 1 , q 2 > q 1 such that For t ≥ T, using (3.8)-(3.10), we can get Taking into account that the fixed point is in B δ , employing Lemma 2.3, Hölder's inequality and (3.43), we have Combining (3.9) and (3.12), (3.14) and (3.16), we have and Using (3.44)-(3.47), we can calculate Iterating this procedure a finite number of times, we deduce that This completes the proof.

Blow-up of nonnegative solutions for (1.1) in the critical case
Throughout this section, we shall assume 1 < p ≤ q for definiteness. The following estimate of the solution for (1.1) is the key step in proving the blow-up theorem for (1.1) in the critical case.
Lemma 4.1. Assume u, v ∈ C 1 (Q) ∩ L ∞ (Q), and let (u, v) is a nonnegative global solution for (1.1) and satisfies (−∆) , v 0 (x, t) be as in (1.6)-(1.7), then there exists a constant C, depending on only p and q, such that Proof. We will only show the first estimate in (4.1) because the proof of the second one is similar. Arguing as in Lemma 2.5 one has v(x, t) ≥ Ct We now substitute (4.2) into (1.4), drop the first and third terms on the right there, and use (1.2a), Jensen's inequality and Tonelli's theorem to obtain We next substitute (4.3) into (1.5). Ignoring again the first and third terms, we have Plugging (4.4) into (1.4) we obtain in turn Iterating the previous procedure, it follows that for any integer k where here constant C in A k is changed one by one according to the different k. We next note that for any positive integers k and l, the following equalities hold Now set λ = pq, then (4.7) can be written as (4.9) We note that (p + 1) 1 + λ + λ 2 + · · · + λ i > 1 for any integer i > 1, then Substitution of (4.9) and (4.10) into (4.6) yields Since ∥u(x, t)∥ ∞ < +∞ for any t ∈ [0, ∞), letting k → ∞ in (4.11) and recalling that λ = pq, we finally arrive at for some constant C that only depends on p and q.
In the critical case, applying Lemma 4.1, the blow-up theorem (Theorem 1.2) of the nonnegative solutions for (1.1) is proved as follows.
Proof of Theorem 1.2.
We suppose by contradiction that there exists a nonnegative global solution u . Using Lemma 4.1, there exists a constant C which depends only on p and q such that By employing Fatou's lemma and Lemma 2.4, we can derive Estimates (4.12) and (4.13) yield ∥b(x)∥ 1 ≤ C, (4.14) where C > 0 depends only on α and N. Regarding v(·, t) as initial value, by (4.14) we get Let t 0 > 0 be as in Lemma 2.5. For t > 1, we setũ(·, t) = u(·, t + t 0 ),ṽ(·, t) = v(·, t + t 0 ). Obviously, (ũ,ṽ) is also a solution for (1.1). Applying (1.4) and Lemma 2.5, it follows thatũ In the proceeding estimate, we consider the integral R N Γ q (y, 1)dy as a constant.
In addition, estimate (4.15) also holds for the functionṽ, which conflicts with (4.18) as t large enough.

Blow-up of nonnegative solutions for (1.3) in the subcritical case
Next, we prove the nonexistence of nonnegative solutions for (1.3) in the subcritical case.
When σ ≥ max α k , β k , the powers in (5.6) and (5.8) satisfy the following inequality: Thus, we eventually arrive at for sufficiently large R. Analogously, we next substitute φ(x, t) into (2.2), use Hölder's inequality and the definition of the global weak solution to obtain Therefore u(x, t) ≡ 0 or v(x, t) ≡ 0 in Q. This is a contradiction with the assumption that f i (x) ̸ ≡ 0 for i = 1, 2, which ends the proof.

A Appendix
Below we give the complete proof of Lemma 2.3.
Its proof is similar to the proof of Lemma 2.5. Therefore, in the case of p ≥ q > 1, through the homologous proof of Theorem 1.2, we can still get the conclusion of Theorem 1.2.