Blow-Up Analysis for a Reaction–Diffusion System Coupled via L α -Norm-Type Sources under Positive Boundary Value Conditions

: This article mainly deals with the blow-up properties of nonnegative solutions for a reaction–diffusion system coupled with norm-type sources under positive boundary value conditions. The local existence of a nonnegative solution and the comparison principle are given. The criteria for the global existence or ﬁnite time blow-up of the solutions are obtained by constructing new functions and utilizing the super-and -sub-solution method. The results reveal a correlation between the blow-up proﬁles of the solutions and the size of the domain, as well as the positive boundary value.

Over the last few decades, extensive research has been conducted on the blow-up properties of solutions to nonlinear parabolic equations with nonlocal sources under homogeneous Dirichlet boundary conditions.For example, in [9], Deng et al. investigated the following degenerate parabolic equation with the initial and boundary conditions x ∈ (−l, l), t > 0, v(±l, t) = 0, t > 0, v(x, 0) = v 0 (x), x ∈ [−l, l], with l > 0, a > 0 and q > m > 1.The authors proved that (4) has a global solution if a is small enough, while the solution of (4) blows up in finite time if a l −l v q 0 dx ≥ 1 and Symmetry 2023, 15, 2074.https://doi.org/10.3390/sym15112074https://www.mdpi.com/journal/symmetryλ < (2al) m q −1 a l −l ϕdx, where λ and ϕ represent the first eigenvalue and the corresponding eigenfunction of the eigenvalue problem −ϕ xx = λϕ, x ∈ (−l, l), ϕ(±l) = 0, respectively.Moreover, the blow-up set is the entire interval [−l, l].Later, Duan et al. [4] extended problem (4) to a parabolic system with a nonlocal source and established the uniform blow-up profiles of solutions.
In [10], Deng et al. studied the following problem with zero homogeneous Dirichlet boundary data.They proved that if pq < mn, the nonnegative solutions are global.For the case pq ≥ mn, there are both global solutions and blow-up solutions, which depend on the initial data and the size of domain Ω.
Recently, Liu et al. [11] considered the following degenerate parabolic equations subject to zero Dirichlet conditions.Based on the studies in [10,12], the authors discussed the criteria for determining whether the solutions of ( 6) would either exist globally or blow up in finite time.Furthermore, they established various forms of uniform blow-up behavior for simultaneous blow-up solutions.
However, there is also a lot of literature concerning the global existence and blow-up properties of solutions for parabolic equations under other boundary conditions, including nonlocal boundary conditions, Neumann conditions, Robin or like-Robin boundary conditions, and so on (see [21][22][23][24][25][26][27][28][29][30] and the references therein).Notably, Ling [21] focused on Equation ( 5) under positive Dirichlet boundary value conditions and pointed out that small diffusion exponents m, n or large coupling exponents p, q may lead to the blow-up of solutions.Simultaneously, the author demonstrated that the boundary values also play a significant role in determining the occurrence of blow-up.
Motivated by the works mentioned above, we would like to study the influence of the condition (3) on determining the behavior of both the global and blow-up solutions.

Theorem 3. Assuming that for every
sufficiently small, then the solution of ( 1)-(3) exists globally; (2) If the domain Ω is sufficiently small, then the nonnegative solution of (1)-( 3) is global; (3) If the domain Ω contains a sufficiently large ball, then the solution of (1)-(3) blows up in finite time provided that u i,0 (x) are positive and continuous in Ω.
The rest of the article is organized as follows.In Section 2, we establish the local existence and the comparison principle which will be used for problems (1)- (3).Then, we give the proof of Theorems 1-4 concerning the global existence and blow-up in finite time in Section 3. Finally, some conclusions are summarized in Section 4.

Local Existence and Comparison Principle
Since equations of (1) are degenerate, there are usually no classical solutions.Therefore, we may give a definition of a weak solution for problem (1)- (3).For convenience, we denote Q T = Ω × (0, T), S T = ∂Ω × (0, T) and define a class of test functions as (2) u i ≤ (≥)ε 0 , (x, t) ∈ S T , and u i (x, 0) ≤ (≥)u i,0 (x), x ∈ Ω; (3) For every t ∈ (0, T) and Accordingly, we can say that the vector function 3) if it is both a sub-solution and a super-solution.Moreover, we say Next, we give a maximum principle, which is important for proving the local existence of a solution to (1)- (3).
Then, w i (x, t) ≥ 0 on Q T .
Proof.It can be proven by the similar method in [25,26], so we omit it here.
The local existence of solutions may be proven by the regularization procedure, so we adopt a similar method (see [15]) and consider the following regularized system, for where u j i,0 is a smooth approximation of u i,0 (x) with suppu j i,0 ⊂ Ω, and By using a discussion similar to that of Theorems A.1-A.4 in [13], it is known that the problem of ( 8) has a unique classical solution with the corresponding initial and boundary conditions.Obviously, passing to the limit j → ∞ , it follows that and with the corresponding initial and boundary conditions on (0, T(n)), where is the maximal existence time.Here, a weak solution of ( 10) is defined similarly to that for problem ( 1)-( 3), only the equalities for u i , ( 7) may be replaced with Due to u i,n ≥ ε 0 + 1 n and Lemma 1, we have Lemmas 2 and 3 which will be used to prove the local existence of solutions.The proof is standard (see [15]).
Hence, the limit T * ≡ lim n→∞ T(n) exists and there are the pointwise limits for any (x, t) ∈ Ω × [0, T * ).In addition, as the convergence of the sequence u i,n is monotone, passage to the limit n → ∞ in identities (11) for ψ i ∈ Φ and t ∈ [0, T * ), the following theorem can be established by monotone and dominated convergence theorems.

Theorem 5. (Local existence). Assuming that for every
or hold.Then, (u Although the proof is quite standard and similar to that in [8,15], the comparison principle is very important in proving the existence or blow-up of the solution of ( 1)- (3).Therefore, we sketch the outline for the reader's convenience.
Proof.Subtracting the first inequalities of (7) for u i and u i yields where Since u i and u i are bounded Q T , it follows from m i > 1, α i , β i ≥ 1 that Φ i (x, s), H i (x, s) are bounded.Clearly, if p i /α i , q i /β i ≥ 1, then D i (s) and E i (s) are also bounded.On the other hand, if p i /α i , q i /β i < 1, we have D i (s) ≤ δ p i /α i −1 , E i (s) ≤ δ q i /β i −1 according to the assumptions (12) or (13).Therefore, we may choose some suitable test function ψ i ∈ Φ as in [15] (pp.118-123) to obtain where U + ≡ max{U(x, t), 0} and C i are bounded constants.Thus, where By (15), it follows from the Gronwall lemma that Based on the above argument, it can be observed that the solutions of (1)-( 3) are unique provided that p i ≥ α i , q i ≥ β i .As a result, we have the following corollary.
) be a sub-solution and a super-solution of (1)-(3), respectively.Then, By the theory of linear system, we give the following lemma, which will be used later.

Lemma 4.
Assuming that m i > p i and D = 0, then there exist positive constants l 1 , Proof.Due to D = 0, then rank(A) < k, where rank(A) represents the rank of matrix A.
Without loss of generality, by taking l 1 = θ > 0, and the lemma has been proven directly.

Proof of Global Existence and Blow-Up
In this section, we will employ the super-and sub-solution method to prove the Theorems 1-4.By the comparison principle, we only need to construct appropriate solutions or sub-solutions for problem (1)-( 3).Firstly, we give the proof of Theorem 1, which is a necessary condition for global solutions.
Proof of Theorem 1.If the inference is not true, we may assume without loss of generality that p 1 > m 1 and consider the following problem By [17], the solution w of (16) blows up in finite time.Since , it implies that (w, ε 0 , • • • , ε 0 ) is the sub-solution of (1)-(3).Thus, by the comparison principle, the solution (u 1 , u 2 , • • • , u k ) of ( 1)-(3) blows up in finite time.
Proof of Theorem 2. Now, let ϕ(x) be the unique positive solution of the following elliptic problem: Taking as the following: where l i (i = 1, 2, • • • , k) and A > 0 are positive constants to be fixed later.Clearly, where l k+1 = l 1 , m k+1 = m 1 .Denote Due to m i > p i , D > 0, one may find two positive constants l 1 and l k such that Furthermore, there exist constants Thus, By ( 20) and ( 22), we choose A sufficiently large to satisfy Due to On the other hand, for (x, t) ∈ S T , and It follows from ( 24)-( 26) that (u Proof of Theorem 3. Case (1).According to Lemma 4, there exist positive constants Clearly, if a i (i = 1, 2, • • • , k) are sufficiently small such that defined by ( 18) is a positive super-solution of ( 1)-( 3).This means that the solution (u 1 , • • • , u k ) of ( 1)-( 3) exists globally.Case (2).Next, we consider the case where the domain Ω is small enough.Denote by λ 1 the first eigenvalue of the following eigenvalue problem, with the corresponding eigenfunction φ 1 (x).Then, λ 1 > 0 and φ 1 (x) > 0 in Ω.It is well known that λ 1 continuously depends on the domain Ω.Then, for an arbitrary constant λ 2 ∈ (0, λ 1 ), we can find a bounded domain Ω 1 ⊃ Ω such that λ 2 is the first eigenvalue of the following eigenvalue problem Let φ 2 (x) be the corresponding eigenfunction of (30) with λ 2 .Here, φ 2 (x) can be normalized as φ 2 (x) ∞ = 1 and φ 2 (x) > 0 in Ω 1 .So, there exists some constant δ 0 > 0 such that φ 2 (x) > δ 0 on Ω.Now, we define the functions u i (x, t), (i = 1, 2, • • • , k) as follows: where l i satisfies ( 27), and M > 0 will be determined later.Note u it = 0.After a direct calculation, we get where By the relationship between the domain and the corresponding first eigenvalues of problem (30) (see [31]), if the domain Ω is sufficiently small such that λ By taking M appropriately large to satisfy M ≥ max 1≤i≤k and Thus, we prove that (u 1 , u 2 ,• • • , u k ) is a super-solution of (1)- (3).By the comparison principle, the nonnegative solution (u that are appropriately large such that Thus, To complete the proof, without loss of generality, we can assume that 0 ∈ Ω, and B R ⊂⊂ Ω is an open ball with center at 0 and radius R.
Denoted by λ B R > 0 the first eigenvalue of the following eigenvalue problem Utilizing the scaling property (let τ = r R ) of eigenvalues and eigenfunctions, we obtain that , where λ B 1 and φ 1 (τ) are the first eigenvalue and the corresponding normalized eigenfunction of the eigenvalue problem in the unit ball B 1 .Thus, max Constructing functions u i (x, t) , (i = 1, 2, • • • , k) as follows: where c, d > 0 are determined later.Clearly,(u 1 , u 2 , • • • , u k ) blows up in finite time T ≤ 1+d cd .For i = 1, 2, • • • , k, by directly calculating Then, for (x, t) ∈ Ω × (0, 1+d cd ), we have Letting , we may assume that the ball B R is sufficiently large, i.e., the radius R is large enough such that Taking c ≤ min 1≤i≤k l i d(1+d) m i l i −l i −1 .Since u i,0 are a positive and continuous in Ω, we choose d > 0 large enough to satisfy On the other hand, By ( 40)-( 43), (u 1 , u 2 , • • • , u k ) is a positive sub-solution of ( 1)-( 3), which blows up in finite time in the ball B R .It implies that the solution of (1)-(3) blows up in the larger domain Ω.

Conclusions
In this article, we discuss the existence and blow-up properties of a diffusion system with positive boundary value conditions.As the above arguments, the positive boundary value conditions guarantee the local existence of solutions of (1)- (3), which also influence the occurrence of the blow-up phenomenon.In the meantime, the blow-up profiles are also affected by the interactions among the multi-nonlinearities m i , p i , q i .We derive a critical case k ∏ i=1 (m i − p i )= k ∏ i=1 q i , which belongs to the scenario of global existence (or blow-up) under other assumptions, such as small a i or small (or large) size of the domain Ω.Comparing the results of this article with those under homogeneous Dirichlet boundary conditions, as in [11] (with k = 2 in Equation ( 1)), we observe differences in the blow-up profiles of the solutions when considering positive value boundary conditions.In this case, the application of specialized processing skills and the construction of new auxiliary functions become necessary to overcome the difficulties caused by boundary values.It is worth noting that the global existence of the two situations appears to be similar, as stated in Theorems 2 and 3 (1).The global solutions of ( 1)-(3) may be independent of the boundary value ε 0 .However, Theorems 3(3) and 4 show that the blow-up situation is slightly different, where ε 0 plays a significant role.
In fact, it follows from the proof of Theorem 3(3) that the global nonexistence of solutions mainly depends on the relation between the boundary value ε 0 and λ B R , where λ B R is the characteristic of the size of B R .That is to say, if we fix the boundary value ε 0 , that λ B R should be properly small (i.e., B R is large enough) such that Correspondingly, for fixed Ω and p i + q i > m i , ε 0 should be suitably large to satisfy

}
Similarly, we can observe the relation according to the inequality (49).As a special case, when the domain Ω is symmetric, particularly when it takes the form of a ball (or an interval), the results of this paper also hold.However, the following problem is still unsolved: how to construct blow-up solutions in an explicit form supporting the theoretical results obtained above for system (1)-(3).Symmetry-based methods (see, e.g., [32,33]) can be useful for its solving.