Modified different nonlinearities for weakly coupled systems of semilinear effectively damped waves with different time-dependent coefficients in the dissipation terms

We prove the global existence of small data solution in all space dimension for weakly coupled systems of semi-linear effectively damped wave, with different time-dependent coefficients in the dissipation terms. Moreover, nonlinearity terms $ f(t,u) $ and $ g(t,v) $ satisfying some properties of the parabolic equation. We study the problem in several classes of regularity.


Introduction
Let us consider the Cauchy problem for the semilinear classical damped wave equation with power nonlinearity where t ∈ [0, ∞), x ∈ R n , and exponent means that we have the global (in time) existence of small data weak solutions for p > p Fuj (n), whereas the local (in time) existence for p > 1 and large data can be only expected.
Assuming a time-dependent coefficient in the dissipation term, we first consider the Cauchy problem Among other classifications of the dissipation term b(t)u t introduced in [28] and [29], we are interested in the effective case, where b = b(t) satisfies the following properties: • b is a positive and monotonic function with tb(t) → ∞ as t → ∞, (1+t) k for k = 1, 2, 3, , and there exists a constant a ∈ [0, 1) such that tb (t) ≤ ab(t).
Examples of functions belonging to this class are the followings with r ∈ (-1, 1): • b(t) = μ (1+t) r for some μ > 0, b(t) = μ (1+t) r (log(e + t)) γ for some μ > 0 and γ > 0, and b(t) = μ (1+t) r (log(e+t)) γ for some μ > 0 and γ > 0. In [5] the authors derived such estimates for solutions to the family of parameterdependent Cauchy problems Using theses estimates together with Duhamel's principle, in the same paper the authors proved the global existence of small data solutions to the following semilinear Cauchy problem: where f (u) satisfies condition (2). In 2013, D' Abbicco [3] proved the global existence of small data solution for low space dimensions and derived decay estimates for solutions to the Cauchy problem where f (t, 0) = 0 and f (t, v)f (t,ṽ) 1 + Weakly coupled systems can be an interesting problem, treated and improved in [16] and [1]. In this paper, we study in all space dimensions the Cauchy problem of weakly coupled system of semilinear effectively damped waves for If we take γ 1 < -1 or γ 2 < -1, then we will get an empty admissible range for p or q (see the table in  Remark 2.3).
Recently, Nishihara and Wakasugi [23] studied the particular case of (3), where b 1 (t) = b 2 (t) = 1, f (t, v) = |v| p , and g(t, u) = |u| q . Using the weighted energy method, they proved the global (in time) existence if the inequality is satisfied. Using an additional regularity L m (R n ) for data, we conclude the so-called modified Fujita exponent p Fuj,m := 1 + 2m n ; this new exponent implies a modified condition corresponding to (7), max{p;q}+1 In [20] and [18] the authors studied the above system with the same nonlinearities assumed in [23] by taking the equivalent coefficients b 1 = b 1 (t) and b 2 = b 2 (t) or, in other words, α = β = 1. The global (in time) existence of small initial data solutions was proved assuming different classes of regularity of data and for all space dimensions. Considering (3) in [21], the authors proved a global existence result for a particular case from the set of effective dissipation terms b 1 (t) = μ (1+t) r 1 , r 1 , r 2 ∈ (-1, 1), and b 2 (t) = μ (1+t) r 2 with the nonlinearities f (t, v) = |v| p and f (t, u) = |u| q .

Notations
For s > 0 and m ∈ [1, 2), we introduce the function space We denote byp andq the modified exponents of the exponents p and q in the power nonlinearities appearing in (5) and (6). Theñ Remark 1.1 If α = β = 1, then (1+B 1 (t, 0)) ≈ (1+B 2 (t, 0)). This case was studied in previous papers. In this work, we restrict ourselves to the remaining cases.

Main results
We study the Cauchy problem (3) in several cases with respect to the regularity of the data to cover all space dimensions and the modified exponents of power nonlinearitiesp,q and parameters α, β, γ 1 , γ 2 . Therefore we introduce the following classification of regularity: Data from energy space s = 1, data from Sobolev spaces with suitable regularity s ∈ (1, n 2 + 1], and, finally, large regular data s > n 2 + 1.

Data from the energy space
In this section, we are interested in system (3), where the data are taken from the function space A m,1 . In Theorem 2.1, we treat the case where both modified exponents powerp andq are above the modified Fujita exponents p Fuj,m,γ 1 := 1 + 2m(γ 1 + 1) n and q Fuj,m,γ 2 := 1 + 2m(γ 2 + 1) n , respectively. 2). Moreover, let the modified exponents satisfỹ and let the exponents p and q of the power nonlinearities satisfy Then there exists a constant 0 such that if then there exists a uniquely determined global (in time) energy solution to (3) in Furthermore, the solution satisfies the following decay estimates: where j + l = 0, 1.
Remark 2.2 We remark that for γ 1 = γ 2 = 0, system (3) behaves in this case like one single equation because the modified power nonlinearitiesp andq are influenced separately only by the modified Fujita exponent p Fuj,m (n) = 2m n + 1. Then we cannot feel in an optimal way the interplay between the powers of nonlinearities in the existence conditions. Remark 2.3 The final admissible ranges for the exponents p and q of power nonlinearities can be fixed using several parameters such as α, β, the exponents γ 1 , γ 2 , the space dimension n, and the parameter of additional regularity m. As an example for the dimension n = 1, if we take 0 < β < 1, thenp < p. We distinguish two cases: , then the solution exists for The general case for the admissible ranges from below can be summarized as follows: β Nonlinearity parameter γ 1 Admissible range for p In the same way, we can get the admissible range for q with respect to the parameters α and γ 2 .
Example 2.4 Let us choose the space dimension n = 2, the parameters γ 1 = -1, γ 2 = - 1 3 , and the coefficients of the dissipation terms b 1 (t) = (1 + t) -1 2 and b 2 (t) = (1 + t) 1 2 , which implies β = 1 α = 3. Using (10) from the previous theorem for m = 2, we getp > 1,q > 7 3 . Theses conditions together with (11) after applying (8) and (9) imply the following admissible range for the exponents of power nonlinearities: The case where one exponentp orq is below the modified Fujita exponent, we distinguish four cases with respect to the values of α and β: 1 . Moreover, let the modified exponents satisfỹ Moreover, we assume that and the exponents p and q of the power nonlinearities satisfy Then there exists a constant 0 such that if then there exists a uniquely determined global (in time) energy solution to (3) in Furthermore, the solution satisfies the following decay estimates: where j + l = 0, 1, and represents the loss of decay in comparison with the corresponding decay estimates for the solution u of the linear Cauchy problem with vanishing right-hand side.
We summarize the remaining results for all cases with respect to α, β,p, andq as follows: • If we assume in the statement of the previous theorem that α < 1 and β ≥ 1, then, instead of (13), we get the condition .

Data from Sobolev spaces with suitable regularity
In this section the regularity of data has a strong influence on the admissible range of the modified exponents or the exponents of power nonlinearities, respectively. For this reason, we assume that the data have a different suitable regularity, that is, with an additional regularity L m (R n ), m ∈ [1, 2). In this section, we use a generalized (fractional) Gagliardo-Nirenberg inequality used in [11] and [25]. Furthermore, we use a fractional Leibniz rule and a fractional chain rule, which are explained in the Appendix.

Large regular data
This case has been classified to benefit from the embedding in L ∞ (R n ), where the data are supposed to have a high regularity, which means that andp > 2m n s 1 + 1 + 2γ 1 2 + 1,q > 2m n s 2 + 1 + 2γ 2 2 + 1.
Then there exists a constant 0 such that if

then there exists a uniquely determined globally (in time) energy solution to (3) in
Furthermore, the solution satisfies for l = 0, 1 the estimates

Some tools
First, we recall the following result from [5].

Lemma 3.1
The primitive B = B(t, τ ) of 1 b satisfies the following properties: To use Duhamel's principle, we need the following results in the proofs of our main results.

Theorem 3.2 The Sobolev solutions to the Cauchy problem
satisfy the following estimates for t > 0: For data from the energy space (s = 1), where j + l = 0, 1; for high regular data (s > 1), The proof of this theorem follows from [28] and [29].

Theorem 3.3 The Sobolev solutions to the parameter-dependent family of Cauchy problems
satisfy the following estimates for t > τ , τ ≥ 0: For data from the energy space (s = 1), where j + l = 0, 1; For high regular data (s > 1), The proof of this theorem follows from [5] and [19].

Proofs
We define the norm of the solution space X(t) by where we will choose M 1 (τ , u) and M 2 (τ , v) with respect to the goals of each theorem. Let N be the mapping on X(t) 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 the Cauchy problem Our aim is to prove the estimates We can immediately obtain from the introduced norm of the solution space X(t), which will be fixed for each case, the following inequality: We complete the proof of all results separately by showing (22) with the inequality which leads to (21).
Proof of Theorem 2.1 We choose the space of energy solutions with the following norms for τ ∈ (0, t]: To prove (23), we need to estimate all terms appearing in (u nl , v nl ) X(t) . Let us begin to estimate u nl t (t, ·) L 2 . Using (19) with m = 2 for τ ∈ [ t 2 , t], we get By a fractional version of the Gagliardo-Nirenberg inequality (see Proposition 4.1) and (5) we obtain where we use condition (11). Plugging the last estimates into (24) and using (4), (17), and (18), we get The last integral can be obtained from the definition of B 1 (t, τ ); indeed, where ν sufficiently small.
For the proof, see [24]. For the proof, see [26]. From Proposition 4.4 we can derive the following corollary. For the proof, see [6] and [25].