Weakly Coupled Systems of Semilinear Damped Waves with Different Scale-Invariant Time-Dependent Dissipation Terms

where (t, x) ∈ [0,∞) × R, was investigated by a lot of researchers. For f(t, u) |u|, p> 1, the critical exponent describes the threshold between global (in time) existence of small data weak solutions and blow-up of local (in time) small data weak solutions (for more details, the readers can see [1–16]). Let us turn to the study of semilinear damped wave equation with power nonlinearity utt − Δu + ut f(t, u), u(0, x) u0(x), ut(0, x) u1(x), (3)

Many papers treated the limit where r � 1 which means the following semilinear Cauchy problem: In [34][35][36], the authors showed that the situation depends strongly on the value of μ; in other words, the transition of μ from 1 to ∞ describes the change from a hyperbolic to a parabolic-like model from the point of decay estimates for solutions. Furthermore, they proved that the decay rate of solutions for large μ is the same as that obtained for solutions of the classical damped wave equation. A particular case of the Cauchy problem (5) with μ � 2 was studied in [37].
Some papers are devoted to the weakly coupled systems of semilinear classical damped wave equations with power nonlinearities. e model we have in mind is where (t, x) ∈ [0, ∞) × R n . In 2007, Sun and Wang proved in [38] that if for n � 1 or 3, then the solution exists globally in time for small initial data, while if (max p; q + 1)/(pq − 1) ≥ n/2, then every solution having positive average value does not exist globally. In [39], the author generalized the previous results to the case where n � 1, 2, 3 and improved time decay estimates for n � 2. In 2014, using the weighted energy method, Nishihara determined in [32] the critical exponent for any space dimension. In [40][41][42][43], the authors studied the system of weakly coupled semilinear damped waves with time-dependent coefficients in the dissipation terms Recently, in [43], the author treated problem (8) with modified nonlinearities.
Let us now turn to our main problem described in (1). In this paper, we will show how the interaction between parameters μ 1 and μ 2 influence the results obtained for the effective case, that is, the change from effective to noneffective. e paper is organized as follows. We start by some background and previous results for single equation. After that, we will show our main results of global (in time) existence; moreover, we summarize all cases appearing in Section 2 in a table. Next, in Section 3, we prove existence of solution by applying Banach's fixed point. Section 4 concludes the paper.
We introduce for s > 0 and m ∈ [1, 2) the function space with the norm In [41], the estimates for the solution to the Cauchy problem were proved for different classes of regularity of the data, low regular data, data from energy space, data from Sobolev spaces with suitable regularity, and large regular data. We summarize these results in the following theorems.

Theorem 1.
Let us assume the data (u 0 , u 1 ) ∈ A m,s with s > 0. en, the solution u to the Cauchy problem (11) satisfies for μ > 1 the following decay estimates. For s ≥ 0, In order to use Duhamel's principle in the next sections, we consider the family of parameter-dependent Cauchy problems en, the solution v to the Cauchy problem (14) satisfies for μ > 1 the following decay estimates for s ≥ 0: and for s ≥ 1: where _ H s is the homogeneous Sobolev space.

Low Regular Data.
In this section, we are interested in Cauchy problem (1), where the initial data are supposed to have low regularity; or in other words, the data belong to the Sobolev space H s (R n ) for s ∈ (0, 1), with additional regularity L m (R n ) for m ∈ [1, 2). From the estimates of eorem 1 and further considerations, we remark the existence of five cases corresponding to the value of μ. ese cases are as follows: ese cases generate for the system (1) a lot of cases corresponding to the values of μ 1 and μ 2 . In eorem 3, we restrict ourselves to three cases which are from our point of view are more interesting and important. e remaining cases will be treated in Remark 1.

Lemma 1. Let p satisfy the conditions
en, the following statements are valid: then ese estimates imply then ese estimates imply e proof of Lemma 1 is basically concluded after using the Gagliardo-Nirenberg inequality from Proposition A.1 and the definition of the space M(τ, u).

Data from Energy
Space. If the data are in the energy space, then we get for s � 1 a similar case to the case of the previous section because the estimates for ‖|D| s�1 u(t, ·)‖ L 2 (R n ) and ‖u t (t, ·)‖ L 2 (R n ) coincide with those of the previous section. Moreover, we obtain the global existence in time of energy solutions. Consequently, we have the following result. 2), and min μ 1 ; μ 2 > 1. Moreover, let the exponents p and q of power nonlinearities satisfy condition (18) and en, there exists a small constant ϵ 0 such that if then there exists a uniquely determined global (in time) energy solution to (1) in C 0, ∞), H 1 R n ∩ C 1 0, ∞), tL 2 n R n 2 .
Journal of Mathematics Furthermore, the solution satisfies the following decay estimates: Example 2. If we consider system (1) for n � 2 and m � 2, then for given μ 1 � μ 2 � 21/10, we get after using the last case the following admissible ranges for p and q:

Data from Sobolev Spaces with Suitable Regularity.
is section is devoted to the case where the data are from Sobolev spaces with suitable regularity. We will treat the same cases of the previous sections corresponding to the values of μ 1 and μ 2 . In the following lemma, we will provide some estimates which are important tools in the proofs of our main results.

Lemma 2.
Let p > ⌈ s ⌉ satisfy the following condition: en, the following statements are valid. then Journal of Mathematics ese estimates imply ese estimates imply Using the fractional chain rule from the Appendix and the definition of space M(τ, u), one can prove the desired statements.
Theorem 5. Let n ≥ 4. e regularity parameters s 1 and s 2 satisfy the following conditions: e data (u 0 , u 1 ) and (v 0 , v 1 ) are supposed to belong to A m,s 1 × A m,s 2 with m ∈ [1, 2). Furthermore, we assume for the exponents p and q the following conditions: Furthermore, the solution satisfies the following decay estimates: then where we assume s * < n/2. We remark that ese estimates imply the estimate then ese estimates imply the estimate (55) Theorem 6. Let n ≥ 4. e data (u 0 , u 1 ) and (v 0 , v 1 ) are supposed to belong to A m,s 1 × A m,s 2 with m ∈ [1, 2) and s 2 > s 1 > (n/2) + 1. Moreover, we assume where s 2 ∈ [s 1 , s 1 + 1]s and s 2 ≤ s 2 . Furthermore, we assume for the exponents p and q the following conditions: en, there exists a uniquely determined global (in time) energy solution to (1) in Furthermore, the solution satisfies the following decay estimates: Journal of Mathematics 11

Proof of eorem 3.
Let us define the space of solutions X(t) as follows: with the norm where M 1 (τ, u) and M 2 (τ, u) will be defined in the treatment of each case. Let N be the mapping on X(t) which is defined by We denote by E 1,0 � E 1,0 (t, 0, x) and E 1,1 � E 1,1 (t, 0, x) the fundamental solutions to the Cauchy problem and by E 2,0 � E 2,0 (t, 0, x) and E 2,1 � E 2,1 (t, 0, x) the fundamental solutions to the Cauchy problem From Proposition E.1, the goal is to prove the following estimates: From the definition of the norm of the solution space X(t), which we will define in each case in correspondence with the main goals, we can immediately obtain We complete the proof of all results separately by showing the inequality which leads to (66).