Global Existence of Small Data Solutions to Weakly Coupled Systems of Semi-Linear Fractional σ –Evolution Equations with Mass and Different Nonlinear Memory terms

: We study in this paper the long-term existence of solutions to the system of weakly coupled equations with fractional evolution and various nonlinearities. Our objective is to determine the connection between the regularity assumptions on the initial data, the memory terms, and the permissible range of exponents in a specific equation. Using L p − L q estimates for solutions to the corresponding linear fractional σ –evolution equations with vanishing right-hand sides, and applying a fixed-point argument, the existence of small data solutions is established for some admissible range of powers ( p 1 , p 2 , . . . , p k ) .


Introduction
This paper is devoted to the weakly coupled system of k semi-linear fractional σ-evolution equations.The system incorporates mass terms and different memory terms and our focus is on small data solutions to the corresponding Cauchy problem.
In this discussion, we will illustrate two distinct Cauchy problems: the semi-linear heat equation and the semi-linear wave equation.
Firstly, let us consider the semi-linear heat equation: According to Fujita's results in [1], the critical exponent for this equation is defined as p Fuj := 1 + 2 n .It is noteworthy that for p > p Fuj , small initial data solutions exist globally (in time), while for 1 < p < p Fuj , a blow-up phenomenon occurs.The critical case p = p Fuj was further studied in [2,3], where it was shown that blow-up does indeed occur.
Moving on, let us shift our focus to the semi-linear wave equation: For the specific case when n = 3, ref.[4] proved that the critical exponent can be determined as the positive root of the quadratic equation (n − 1)p 2 − (n + 1)p − 2 = 0.The exponent obtained from the quadratic equation is known as the Strauss exponent, denoted as p S .Based on the Strauss exponent p S , we can conclude that there is the global (in time) existence of small data weak solutions when p ≥ p S .However, for p > 1 and large data, we can only expect the local (in time) existence of solutions.The optimality of the Strauss exponent p S in R 2 was demonstrated in [5,6].After that, the global existence of solutions for n = 2, 3 was treated in [7], while for n ≥ 4, it was addressed in [8,9].The nonexistence of solutions with compactly supported data was studied in [10] for the range 1 < p < n+1 n−1 .For the specific case of n = 3, optimal results were proven in [11] for p = 1 + √ 2.Moreover, in [12], it was proved that for n > 3 and 1 < p < p S , there is a nonexistence result for small data.
In 2017, D'Abbicco et al. [13] studied the semi-linear fractional wave equation, which can be expressed as follows: where λ ∈ (0, 1), which represents the fractional Riemann-Liouville derivative.The authors successfully proved the critical power for the existence of solutions with small initial data in spatial dimensions that are relatively low.The case of non-null Cauchy data and the use of the Caputo fractional order were studied in [13].
In [14], they proved the global (in time) existence of small data solutions for semilinear fractional σ-evolution equations.These equations incorporated either mass or power nonlinearity.Furthermore, a related problem was addressed in [15], where instead of the power nonlinearity, a memory term was considered.
For the weakly coupled system consisting of semi-linear heat equations, we have the following equations: where t ∈ [0, ∞), x ∈ R d , and p, q > 1 with pq > 1.In [16], it was shown that the exponents p and q satisfying d 2 = max{p, q} + , while blow-up occurs for the opposite case.For more details on the system of semi-linear heat equations, please refer to [17][18][19][20].
Considerations are made in several papers regarding weakly coupled systems of semilinear classical damped wave equations with power nonlinearities.The specific problem of interest is: where t ∈ [0, ∞), x ∈ R d .In 2007, Sun and Wang proved in [21] that For the case of d = 1 or d = 3, it has been proven that the solution exists globally in time for small initial data in weakly coupled systems of semi-linear classical damped wave equations with power nonlinearities.However, if λ ≥ d 2 , it has been shown that every solution with a positive average value does not exist globally.
In the paper [22], these results were generalized to the case where d = 1, 2, 3. Additionally, improved time-decay estimates have been provided specifically for the case of d = 2.In 2014, Nishihara and Wakasugi used the weighted energy method to prove the critical exponent for any space dimension in [23].Furthermore, considering time-dependent dissipation terms, the authors in [24][25][26] demonstrated the global (in time) existence of small data solutions under certain conditions that illustrate the interplay between the exponents of the power nonlinearities.
During the last years, many authors have studied the Cauchy problem for weakly coupled systems, see, e.g., [24,27,28], where the derivative introduced in their work is the classical derivative.In [29], the authors studied a weakly coupled system where the fractional derivative involves in the equations with special Cauchy data.
The paper is organized into several sections.First, we provide an overview of the study and present the main results (Section 2).Following that, Section 3 introduces the necessary background information and definitions for the foundation used to prove the results.Then, the proofs of the theorems are presented, utilizing previous estimates of linear equations (Section 5).Finally, Section 6 summarizes the study, highlights its contributions, and suggests potential directions for future research.
In a recent paper [30], the author investigated the following Cauchy problem for weakly coupled systems of semi-linear fractional σ-evolution equations.The system involves mass terms and different power nonlinearities. where D α t ( f ) and I β t f are defined as above.The author proved the following results.

Proposition 1. Let us assume
and Then, there exists a positive constant ε, such that for any data ≤ ε, we have a uniquely determined global (in time) Sobolev solution to the Cauchy problem (6).Moreover, for all s ≥ 0, the solution satisfies the following decay estimates: Then, there exists a positive constant ε, such that for any data we have a uniquely determined global (in time) Sobolev solution to the Cauchy problem (6).Moreover, for all s ≥ 0, the solution satisfies the following decay estimates: Assume that δ > 0 is small enough for all Then, there exists a positive constant ε, such that for any data to the Cauchy problem (6).Moreover, for all s ≥ 0, the solution satisfies the following decay estimates: In the subsequent sections, we will utilize the notation f ≲ g, indicating the existence of a non-negative constant C, such that f ≤ Cg.Our main findings concerning the global (in time) existence of small data Sobolev solutions will be presented in the following section.

Theorem 1. Let us assume
Then, there exists a positive constant ε, such that for any data (u 01 , .., to the Cauchy problem (1).Moreover, for all s ≥ 0 and l = 1, ..., k, the solution satisfies the following decay estimates:

Theorem 2 (Loss of decay). Let us assume
, , Then, there exists a positive constant ε, such that for any data to the Cauchy problem (1).Moreover, for all s ≥ 0 and l = 2, • • • , k − 1, the solution satisfies the decay estimate We suppose m 1 = m 2 = 1 in the following result.

Theorem 3 (Loss of decay). Let us assume
Then, there exists a positive constant ε, such that for any data to the Cauchy problem (1).Moreover, for l = 1, • • • , k − 1 and for all s ≥ 0, the solution satisfies the following decay estimate: Remark 1.The nonlinear term F µ,p (t, w) in ( 2) may be written as where Γ is the Euler Gamma function, and I . Therefore, it is reasonable to expect that the relations with the power nonlinearities introduced in Proposition 1, Proposition 2, and Proposition 3 as µ l tend to 1, for all l = 1, • • • , k and k = 2.

Preliminaries
Let us consider the Cauchy problem With parameters α ∈ (0, 1), σ ≥ 1, and m > 0, and under the data condition v t (0, x) = 0, the problem can be formally transformed into an integral equation.The solution of the problem is then given by: with where the semigroup of operators G m σ,α (t, •) t≥0 is defined through the Fourier transform as follows: Here, Γ(βk+1) denotes the Mittag-Leffler function (see [31]).According to [14], a representation of solutions to the linear problem associated with Equation (9) (without the term F(t, x)) can be given as v(t, x) = G m σ,α (t, x) ⋆ v 0 (t, x).This representation involves convolving the initial data v 0 (t, x) with the semigroup of operators G m σ,α (t, x).In [14], the authors proved the following result.Proposition 4 (see [14]).Let us assume that α ∈ (0, 1), r ≥ 1, σ ≥ 1, and v 0 ∈ L r ∩ L ∞ .Then, the solution of the linear Cauchy problem for all t ≥ 0 and 1 ≤ r ≤ q ≤ ∞, satisfies the following L r − L q estimates:

Analysis of Weakly Coupled Linear Systems
We will use the decay estimates for solutions to: In order to establish the global existence (over time) of Sobolev solutions with small initial data for the weakly coupled systems of semi-linear models (1), we express their solutions in the following form: Then, the solution of the linear Cauchy problem (15) satisfies the following L m l − L q estimates: By applying Duhamel's principle and some fixed-point argument, we can derive the formal integral representation of solutions to (1) as follows: for all l = 2, • • • , k.
Here, u nl .

Proof of Main Results
Before showing our results, we recall the following lemma from [32].

Proof of Theorem 1
Let T > 0. We introduce the space X k (T) as follows: where and the operator P by In order to prove the global (in time) existence and uniqueness of Sobolev solutions in X k (T), we will demonstrate that the operator P satisfies the following two inequalities: Using the definition of the norm in X k (T) and Proposition 5, we may conclude: Hence, in order to complete the proof of (19), it is reasonable to show the following inequality: On the other hand, we also have where We are interested in estimating the right-hand side of (21).For this we need Lemma 1.We put By using Lemma 1, we obtain ω(s) ≲ (1 + s) −µ 1 , if we assume that p 1 > 1 Once more, we apply Lemma 1 to obtain where To estimate the right-hand side of ( 22), we require the use of Lemma 1.Let By using Lemma 1, we obtain ω(s) ≲ (1 + s) −µ ℓ , if we assume that p ℓ > On the other hand, the conditions q ∈ [m l , ∞] and Once more, we apply Lemma 1 to obtain In order to prove (24), let us consider two vector-functions u and ũ belonging to X k (T).Then, we have Using Hölder's inequality implies the inequality By using the definition of the norm of the solution space X k (T), for p 1 ≥ m k m 1 and 0 ⩽ s ⩽ t, we obtain the following estimates: Hence, we obtain By the same argument, for l = 2, ..., k and 0 ⩽ s ⩽ t, we obtain the following estimate: Remark 2. All estimates (19) and ( 20) are uniform with respect to T ∈ (0, ∞).
From (19), we can see that P maps X k (T) into itself for all T and for small data.By using standard contraction arguments, the estimates (19) and (20) lead to the existence of a unique solution to u = P(u) and, consequently, to (1).This implies that the solution of (1) satisfies the desired decay estimate.Since all constants are independent of T, we can let T tend to ∞, which yields a global (in time) existence result for small data solutions to (1).This concludes the proof.

Proof of Theorem 2
Let T > 0. We introduce the space X k (T) as follows:

and the operator P by
We will prove that, for u = (u 1 , u 2 , ..., u k ); ũ = ( ũ1 , ũ2 , ..., ũk ) in X k (T), the operator P satisfies the following two inequalities: Using the definition of the norm in X k (T) and Proposition 5, we may conclude: Hence, in order to complete the proof of (19), it is reasonable to show the following inequality: On the other hand, for q ∈ [m 1 , ∞], we have where We are interested in estimating the right-hand side of (25).For this we need Lemma 1.We put Once more, we apply Lemma 1 to obtain On the other hand, the conditions q ∈ [m 1 , ∞] and where We are interested in estimating the right-hand side of (26).For this, we need Lemma 1.We put Thanks to Lemma 1, we obtain Once more, we apply Lemma 1 to obtain On the other hand, the conditions q ∈ [m 2 , ∞] and where dη ds dϱ.
On the other hand, we are interested in estimating the right-hand side of (27).For this, we need Lemma 1.We put dη.
We are interested in estimating the right-hand side of (28).For this we need Lemma 1.We put dη.
Once more, we apply Lemma 1 to obtain The proof of ( 24) is similar to the proof of (20) of Theorem 1.This completes the proof.

Proof of Theorem 3
Let T > 0. We introduce the space X k (T) as follows: The operator P is defined by We will prove that, for u = (u 1 , u 2 , ..., u k ); ũ = ( ũ1 , ũ2 , ..., ũk ) in X k (T), the operator P satisfies the following two inequalities: Using the definition of the norm in X k (T) and Proposition 5, we may conclude: Hence, in order to complete the proof of ( 29), it is reasonable to show the following inequality: On the other hand, for q ∈ [1, ∞], we have where We are interested in estimating the right-hand side of (31).For this, we need Lemma 1.We put Once more, we apply Lemma 1 to obtain On the other hand, we are interested in estimating the right-hand side of (32).For this, we need Lemma 1.We put ω(s) = s 0 (s − η) −µ ℓ (1 + η) −p ℓ (µ ℓ−1 −α ℓ−1 ) dη.
Once more, we apply Lemma 1 to obtain The proof of ( 30) is similar to the proof of (20) of Theorem 1.This completes the proof.

Conclusions
In the present paper, we proved the global (in time) existence of small data Sobolev solutions to the weakly coupled system of k semi-linear fractional σ-evolution equations with mass and different memory terms.We studied the relationship between the regularity assumptions for the data, the memory terms, and the admissible range of exponents (p 1 , p 2 , . . ., p k ) in Equation (1).In a forthcoming paper, we will study the blow-up of solutions to (1).