Abstract
We prove the global existence of small data solution in all spaces of all dimensions \(n\geq 1\) for weakly coupled systems of semilinear effectively damped wave, with different time-dependent coefficients in the dissipation terms. Moreover, we assume that the nonlinearity terms \(f(t,u) \) and \(g(t,v) \) satisfy some properties of parabolic equations. We study the problem in several classes of regularity.
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1 Introduction
Let us consider the Cauchy problem for the semilinear classical damped wave equation with power nonlinearity
where \(t\in [0,\infty ), x\in \mathbb{R}^{n}\), and
Having the estimates proved in [17] for the corresponding homogeneous problem, for given compactly supported initial data \((u_{0},u_{1}) \in H^{1}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})\) and for \(p\leq p_{GN}(n):=\frac{n}{n-2}\) if \(n\geq 3\), the authors in [22] proved the local (in time) existence of energy solutions \(u\in \mathcal{C}([0,T), H^{1}(\mathbb{R}^{n}))\cap \mathcal{C}^{1}([0,T), L^{2}(\mathbb{R}^{n}))\). Moreover, they proved the global (in time) existence of small data solutions by using the technique of “potential well” and “modified potential well”. The Cauchy problem (1) was also studied in [7, 12, 13, 27, 30], where the Fujita exponent \(p_{Fuj}(n):=1+\frac{2}{n}\) has an important role as the critical exponent. The critical 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 \(t b(t)\rightarrow \infty \) as \(t\rightarrow \infty \),
-
\(((1+t)^{2}b(t))^{-1} \in L^{1}(0,\infty )\),
-
\(b\in \mathcal{C}^{3}[0,\infty )\) and \(|b^{(k)}(t)| \lesssim \frac{b(t)}{(1+t)^{k}}\) for \(k=1,2,3\),
-
\(\frac{1}{b} \notin L^{1}(0,\infty )\), and there exists a constant \(a\in [0,1)\) such that \(tb'(t)\leq ab(t)\).
Examples of functions belonging to this class are the followings with \(r\in (-1,1)\):
-
\(b(t)=\frac{\mu }{(1+t)^{r}}\) for some \(\mu >0\), \(b(t)=\frac{\mu }{(1+t)^{r}}(\log (e+t))^{\gamma }\) for some \(\mu >0\) and \(\gamma >0\), and \(b(t)=\frac{\mu }{(1+t)^{r}(\log (e+t))^{\gamma }}\) for some \(\mu >0\) and \(\gamma >0\).
In [5] the authors derived such estimates for solutions to the family of parameter-dependent 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
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
where
for \(B_{1}(t,\tau )=\int _{\tau }^{t}\frac{1}{b_{1}(r)}\,dr, B_{2}(t,\tau )= \int _{\tau }^{t}\frac{1}{b_{2}(r)}\,dr, \alpha,\beta \in \mathbb{R}_{+}^{*}\), and \(\gamma _{1},\gamma _{2}\in [-1,\infty )\). If we take \(\gamma _{1}<-1 \) or \(\gamma _{2}<-1 \), 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 \(L^{m}(\mathbb{R}^{n}) \) for data, we conclude the so-called modified Fujita exponent \(p_{Fuj,m }:=1+\frac{2m}{n} \); this new exponent implies a modified condition corresponding to (7), \(\frac{\max \{p;q\}+1}{pq-1}<\frac{n}{2m} \). 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, \(\alpha =\beta =1\). 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 \(b_{1}(t)=\frac{\mu }{(1+t)^{r_{1}}}, r_{1},r_{2}\in (-1,1)\), and \(b_{2}(t)=\frac{\mu }{(1+t)^{r_{2}}}\) with the nonlinearities \(f(t,v)=|v|^{p}\) and \(f(t,u)=|u|^{q}\).
1.1 Notations
For \(s>0\) and \(m \in [1,2)\), we introduce the function space
with the norm
We denote by p̃ and q̃ the modified exponents of the exponents p and q in the power nonlinearities appearing in (5) and (6). Then
and
Remark 1.1
If \(\alpha =\beta =1 \), then \((1+B_{1}(t,0)) \approx (1+B_{2}(t,0))\). This case was studied in previous papers. In this work, we restrict ourselves to the remaining cases.
2 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 nonlinearities \(\tilde{p}, \tilde{q}\) and parameters \(\alpha,\beta,\gamma _{1},\gamma _{2}\). Therefore we introduce the following classification of regularity: Data from energy space \(s=1\), data from Sobolev spaces with suitable regularity \(s\in (1,\frac{n}{2}+1]\), and, finally, large regular data \(s>\frac{n}{2}+1\).
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 \(\mathcal{A}_{m,1}\). In Theorem 2.1, we treat the case where both modified exponents power p̃ and q̃ are above the modified Fujita exponents
respectively.
Theorem 2.1
Let the data \((u_{0},u_{1}),(v_{0},v_{1})\) belong to \(\mathcal{A}_{m,1} \times \mathcal{A}_{m,1}\) for \(m\in [1,2)\). Moreover, let the modified exponents satisfy
and let the exponents p and q of the power nonlinearities satisfy
Then there exists a constant \(\epsilon _{0}\) 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 \(\gamma _{1}=\gamma _{2}=0\), system (3) behaves in this case like one single equation because the modified power nonlinearities p̃ and q̃ are influenced separately only by the modified Fujita exponent \(p_{Fuj,m}(n)=\frac{2m}{n}+1\). 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 \(\alpha,\beta \), the exponents \(\gamma _{1},\gamma _{2}\), the space dimension n, and the parameter of additional regularity m. As an example for the dimension \(n=1\), if we take \(0<\beta <1\), then \(\tilde{p}< p\). We distinguish two cases:
-
If \(\gamma _{1}\geq -\frac{1}{2}\), then \(p\geq \frac{2}{m}\) for \(\tilde{p}>p_{Fuj,m,\gamma _{1} }\) which is equivalent to \(p>\frac{1}{\beta } (2m(\gamma _{1}+1)-\frac{m}{2} +1 ) + \frac{m}{2}\).
-
If \(\gamma _{1}\in [-1,-\frac{1}{2})\), then the solution exists for
$$ p>\max \biggl\lbrace \frac{1}{\beta } \biggl(2m(\gamma _{1}+1)- \frac{m}{2}+1 \biggr)+\frac{m}{2};\frac{2}{m} \biggr\rbrace . $$
The general case for the admissible ranges from below can be summarized as follows:
β | Nonlinearity parameter \(\gamma _{1}\) | Admissible range for p |
---|---|---|
0<β<1 | \(\gamma _{1}\geq -1+\frac{n}{2} \) | \(p>\frac{1}{\beta }+\frac{2m(\gamma _{1}+1)}{n\beta }-\frac{m}{2\beta }+\frac{m}{2} \) |
\(\gamma _{1}\in [-1,-1+\frac{n}{2})\) | \(p>\max \biggl\lbrace \frac{1}{\beta }+\frac{2m(\gamma _{1}+1)}{n\beta }-\frac{m}{2\beta }+ \frac{m}{2}; \frac{2}{m} \biggr\rbrace \) | |
β ≥ 1 | \(\gamma _{1}\geq -1+\frac{n\beta }{2} \) | \(p> \frac{2m(\gamma _{1}+1)}{n\beta }+1\) |
\(\gamma _{1}\in [-1,-1+\frac{n\beta }{2})\) | \(p> \max \biggl\lbrace \frac{2m(\gamma _{1}+1)}{n\beta }+1;\frac{2}{m} \biggr\rbrace \) |
In the same way, we can get the admissible range for q with respect to the parameters α and \(\gamma _{2}\).
Example 2.4
Let us choose the space dimension \(n=2\), the parameters \(\gamma _{1}=-1,\gamma _{2}=-\frac{1}{3}\), and the coefficients of the dissipation terms \(b_{1}(t)=(1+t)^{ -\frac{1}{2}}\) and \(b_{2}(t)=(1+t)^{ \frac{1}{2}}\), which implies \(\beta =\frac{1}{\alpha }=3\). Using (10) from the previous theorem for \(m=2\), we get \(\tilde{p}>1,\tilde{q}>\frac{7}{3}\). 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 exponent p̃ or q̃ is below the modified Fujita exponent, we distinguish four cases with respect to the values of α and β:
-
1.
\(\tilde{p}\leq 1+\frac{2m(\gamma _{1} +1)}{n},\tilde{q}>1+ \frac{2m(\gamma _{2} +1)}{n}\) with \(\min \{\alpha;\beta \}\geq 1\) or \(\min \{\alpha;\beta \}\leq 1\leq \max \{\alpha;\beta \}\).
-
2.
\(\tilde{p}>1+\frac{2m(\gamma _{1} +1)}{n},\tilde{q}\leq 1+ \frac{2m(\gamma _{2} +1)}{n}\) with \(\min \{\alpha;\beta \}\geq 1\) or \(\min \{\alpha;\beta \}\leq 1\leq \max \{\alpha;\beta \}\).
Theorem 2.5
Let \(m\in [1,2), \alpha \geq 1\), and \(\beta >0\). The data \((u_{0},u_{1}),(v_{0},v_{1})\) are assumed to belong to \(\mathcal{A}_{m,1} \times \mathcal{A}_{m,1}\). Moreover, let the modified exponents satisfy
Moreover, we assume that
and the exponents p and q of the power nonlinearities satisfy
Then there exists a constant \(\epsilon _{0}\) 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.
Remark 2.6
Choosing \(\tilde{p}=p_{Fuj,m}(n)\) in condition (12), we get an arbitrarily small loss of decay \(\kappa (\tilde{p})=\varepsilon \).
We summarize the remaining results for all cases with respect to \(\alpha,\beta,\tilde{p}\), and q̃ as follows:
-
If we assume in the statement of the previous theorem that \(\alpha <1\) and \(\beta \geq 1\), then, instead of (13), we get the condition
$$ \frac{n}{2}>m \biggl( \frac{\tilde{q}+1+\gamma _{1}\tilde{q}+\gamma _{2}+\frac{m}{2}(\alpha -1)(\gamma _{1}+1)}{ \tilde{p}\tilde{q}-1+\frac{m}{2}(\alpha -1)(\tilde{p}-1)} \biggr). $$ -
If \(\tilde{p}>\frac{2m(\gamma _{1}+1)}{n}+1,\tilde{q}\leq \frac{2m(\gamma _{2}+1)}{n}+1\), then, instead of (13), we have to assume that
$$\begin{aligned} &\frac{n}{2}>m \biggl( \frac{\tilde{p}+\beta +\gamma _{2} \tilde{p}+\gamma _{2}(\beta -1) +\gamma _{1}}{ \tilde{p}\tilde{q}-1+(\beta -1)(\tilde{q}-1)} \biggr) \quad \text{for } \alpha >0, \beta \geq 1, \\ &\frac{n}{2}>m \biggl( \frac{\tilde{p}+1+\gamma _{2} \tilde{p}+\gamma _{1}+\frac{m}{2}(\beta -1)(\gamma _{2}+1)}{ \tilde{p}\tilde{q}-1+\frac{m}{2}(\beta -1)(\tilde{q}-1)} \biggr) \quad \text{for } \alpha \geq 1, \beta < 1. \end{aligned}$$
2.2 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 \(L^{m}(\mathbb{R}^{n})\), \(m \in [1,2)\). 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.
Theorem 2.7
Let \(n\geq 4\), \(s_{1}\in (3+2\gamma _{1},\frac{n}{2}+1], s_{2}\in (3+2\gamma _{2}, \frac{n}{2}+1]\), \(0< s_{2}-s_{1}<1\), and \(\lceil s_{1}\rceil \neq \lceil s_{2}\rceil \). The data \((u_{0},u_{1}),(v_{0},v_{1}) \) are supposed to belong to \(\mathcal{A}_{m,s_{1}}\times \mathcal{A}_{m,s_{2}}\) with \(m\in [ 1,2)\). Furthermore, we assume that
and that the exponents p and q of the power nonlinearities satisfy the conditions
Then there exists a constant \(\epsilon _{0}\) such that if
then there exists a uniquely determined globally (in time) energy solution to (3) in
Furthermore, for \(l=0,1\), the solution satisfies the estimates
Particular cases:
-
If \(\beta \geq 1\) and \(s_{1}\geq 3+2\gamma _{1}\), then under the assumptions of Theorem 2.7, the condition \(p>\lceil s_{1}\rceil \) implies \(\tilde{p}>\frac{2m}{n} (\frac{s_{1}+1+2\gamma _{1}}{2} ) +1\).
-
If \(\alpha \geq 1\) and \(s_{2}\geq 3+2\gamma _{2}\), then under the assumptions of Theorem 2.7, the condition \(p>\lceil s_{2}\rceil \) implies \(\tilde{q}>\frac{2m}{n} (\frac{s_{2}+1+2\gamma _{2}}{2} ) +1\).
2.3 Large regular data
This case has been classified to benefit from the embedding in \(L^{\infty }(\mathbb{R}^{n})\), where the data are supposed to have a high regularity, which means that
Theorem 2.8
Let \(n\geq 4\), \((u_{0},u_{1}),(v_{0},v_{1})\in \mathcal{A}_{m,s_{1}}\times \mathcal{A}_{m,s_{2}}\), \(m\in [ 1,2)\), \(\min \{s_{2}; s_{1}\}>\frac{n}{2}+1\), and \(s_{1}-s_{2}\in (-1,1)\). Moreover, let
and
Then there exists a constant \(\epsilon _{0}\) 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
3 Philosophy of our approach and proofs
3.1 Some tools
First, we recall the following result from [5].
Lemma 3.1
The primitive \(B=B(t,\tau )\) of \(\frac{1}{b} \) 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 \(t>\tau, \tau \geq 0\):
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].
3.2 Proofs
We define the norm of the solution space \(X(t)\) by
where we will choose \(M_{1}(\tau,u)\) and \(M_{2}(\tau,v)\) with respect to the goals of each theorem.
Let N be the mapping on \(X(t)\) defined by
where
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 \(\tau \in (0,t]\):
To prove (23), we need to estimate all terms appearing in \(\|(u^{nl},v^{nl})\|_{X(t)}\). Let us begin to estimate \(\| u^{nl}_{t}(t,\cdot ) \| _{L^{2}}\). Using (19) with \(m=2\) for \(\tau \in [\frac{t}{2},t]\), 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 \(B_{1}(t,\tau ) \); indeed,
where ν sufficiently small.
We distinguish two cases with respect to the value of β. If \(\beta \geq 1\), then we get
We can conclude from \(-\frac{n}{2m}(\tilde{p}-1)+\gamma _{1}<-1 \), which is equivalent to \(\tilde{p}>\frac{2m(\gamma _{1} +1)}{n}+1\), that \({\int _{0}^{\frac{t}{2}}}b_{1}(\tau )^{-1} (1+B_{1}( \tau,0))^{-\frac{n}{2m}(\tilde{p}-1)+\gamma _{1}} \,d\tau \) is bounded. Hence
If \(0<\beta <1\), then we get
for \(\tilde{p}>\frac{2m(\gamma _{1} +1)}{n}+1\). Finally, we obtain
Analogously, we can prove that
For the second component \(v^{nl}\), using the Gagliardo–Nirenberg inequality, from Proposition 4.1 we get for \(\tau \in (0,t]\) the following estimates:
Taking into account the last estimates, we can prove, similarly to (27)–(29), the estimates
for \(\tilde{q}>\frac{2m(\gamma _{2} +1)}{n}+1\). Finally, (27)–(32) imply (23).
The proof of (22) is completely analogous to that of (21). In this way, we complete the proof of Theorem 2.1. □
Proof of Theorem 2.5
We choose the same space of energy solutions \(X(t)\) with the norm \(M_{2}(\tau,v)\) used in the proof of Theorem 2.5. We modify the norm \(M_{1}(\tau,v)\) as follows:
where \(\kappa (\tilde{p}) =\gamma _{1}-\frac{n}{2m}(\tilde{p}-1)+1\). We begin the proof of (23) by estimating the norm \(\| u^{nl}_{t}(t,\cdot ) \| _{L^{2}}\). Using (19) with \(m=2\) for \(\tau \in [\frac{t}{2},t]\) together with the Gagliardo–Nirenberg inequality and following the same steps of the proof of (27), we get
for \(\beta >0\). Then we have
In the same way, we can prove
Now for \(v^{nl}\), using the Gagliardo–Nirenberg inequality and the definition of the solution space \(X(t)\), we can prove the following estimates:
Taking into account the last estimates together with (19), we obtain
where we use the condition
which is equivalent to condition (13). Then
Analogously, we can prove
Consequently, (33)–(38) imply (23).
To prove (22), we suppose the existence of \((u,v)\) and \((\tilde{u},\tilde{v})\) belonging to the space of solution \(X(t)\). Then we have
Similarly to (25) and (26), using (5) and (6), we can prove the following estimates:
Analogously to (33)–(38), using (39)–(42), we can get
where \(j+l\leq 1\). The proof is completed. □
Proof of Theorem 2.7
Let us choose the space of energy solutions with suitable regularity
with the norm
where
and
To prove (23), we show how to estimate the norms \(\||D|^{s_{1}-1}u^{nl}_{t}(t,\cdot )\|_{L^{2}(\mathbb{R}^{n})}\) and \(\||D|^{s_{2}-1}\times v^{nl}_{t}(t,\cdot )\|_{L^{2}(\mathbb{R}^{n})}\). From estimate (20) it follows that
Under the assumptions of Theorem 2.7 and the choice of the above introduced norm, for \(0\leq \tau \leq t\), the inequalities (25) and (26) remain true. We calculate the norm
Using (56) and (57), for \(p > \lceil s_{1}-1\rceil \) and \(0\leq \tau \leq t\), we get the following estimate:
where
To satisfy the last conditions for the parameters \(\theta _{1}\) and \(\theta _{2}\), we choose \(q_{2}=\frac{2n}{n-2}\) and \(q_{1}=n(p-1)\). This choice implies the condition
Consequently, for \(\tau \in (0,t]\), we obtain the estimate
Summarizing all estimates implies
where \(\tilde{p}>\frac{2m}{n} (\frac{s_{1}+1+2\gamma _{1}}{2} ) +1\).
Then
Under the first condition of (14), in the same way, we can prove the following estimates:
Using the second condition of (14), we get
From (46)–(53) we get (23), which completes the proof of (21).
To prove (22), we use the same steps used in the previous proof. Indeed, from the fractional Leibniz rule (see Proposition 4.2) and the fractional chain rule (see Proposition 4.3) we may conclude for \(0\leq \tau \leq t\) the following estimates:
and
where we use condition (15). From (39)–(42) without loss of decay and (54)–(55) we can complete the proof. □
Remark 3.4
Theorem 2.8 can be proved by using a similar approach as in the proof of Theorem 2.7, with modifications in the estimates of some terms. Then using Proposition 4.4, Corollary 4.5, and Lemma 4.6, we can obtain the estimates
Using these estimates, provided that condition (16) is satisfied, we can follow steps in the proof of Theorem 2.7 to complete our proof.
Availability of data and materials
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Acknowledgements
The author expresses a sincere thankfulness to the Deanship of preparatory year in King Faisal university for their hospitality. The author thanks the reviewer for his or her comments and suggestions. Moreover, the author thanks the organizer of ISAAC congress in Aveiro (2019) for the opportunity to communicate these results.
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This work was fully supported by the Deanship of Scientific Research, King Faisal University through the Nasher Track under Grant No. 206150.
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Appendix
Appendix
Here we state some inequalities, which come into play in our proofs.
Proposition 4.1
Let \(1< p,p_{0},p_{1}<\infty \), \(\sigma >0\), and \(s\in [0,\sigma )\). Then the following fractional Gagliardo–Nirenberg inequality holds for all \(u\in L^{p_{0}} \cap \dot{H} _{p_{1}}^{\sigma }\):
where
For the proof, see [11] and [2, 8–10, 14, 15].
Proposition 4.2
Let \(s>0\), \(1\leq r\leq \infty \), and \(1< p_{1},p_{2},q_{1},q_{2}\leq \infty \) satisfy the relation
Then we have the following fractional Leibniz rule:
for all \(f\in \dot{H} _{p_{1}}^{s}\cap L^{q_{1}}\) and \(g\in \dot{H} _{q_{2}}^{s}\cap L^{p_{2}}\).
For more details concerning fractional Leibniz rule, see [8].
Proposition 4.3
Let us choose \(s>0,p>\lceil s \rceil \), and \(1< r,r_{1},r_{2}<\infty \) satisfying
Let \(F(u)\) be one of the functions \(|u|^{p}, \pm |u|^{p-1}u\). Then we have the following fractional chain rule:
For the proof, see [24].
Proposition 4.4
Let \(p>1\) and \(u\in H^{s}_{m}\), where \(s\in (\frac{n}{m},p)\). Then we have the following estimates:
For the proof, see [26].
From Proposition 4.4 we can derive the following corollary.
Corollary 4.5
Under the assumptions of Proposition 4.4, we have
For the proof, see [6] and [25].
Lemma 4.6
Let \(0<2s^{*}<n<2s\). Then for any function \(f\in \dot{H}^{s^{*}}\cap \dot{H}^{s}\), we have the estimate
For the proof, see [4].
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Mohammed Djaouti, A. Modified different nonlinearities for weakly coupled systems of semilinear effectively damped waves with different time-dependent coefficients in the dissipation terms. Adv Differ Equ 2021, 66 (2021). https://doi.org/10.1186/s13662-021-03215-0
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DOI: https://doi.org/10.1186/s13662-021-03215-0