Weakly Coupled Systems of Semilinear Effectively Damped Waves with Different Time-Dependent Coefficients in the Dissipation Terms and Different Power Nonlinearities

Abstract

We study the global existence of small data solutions to the Cauchy problem for the coupled system of semilinear damped wave equations with different effective dissipation terms and different exponents of power nonlinearities. The data are supposed to belong to different classes of regularity. We will show the interaction of the exponents p and q on the one hand and on the other hand the interaction of the dissipation terms b ₁(t)u _t and b ₂(t)v _t.

Appendix

In the Appendix we collect some background material which is helpful and important for our approach. Most of these tools are from the theory of harmonic analysis and function spaces. In particular, these tools allow us to estimate power nonlinearities in different scales of function spaces.

Proposition A.1

Let 1 < p, p ₀, p ₁ < ∞, σ > 0 and s ∈ [0, σ). Then it holds the following fractional Gagliardo-Nirenberg inequality for all $u\in L^{p_{0}}(\mathbb {R}^{n}) \cap \dot {H} _{p_{1}}^{\sigma }(\mathbb {R}^{n}):$

$$\displaystyle \begin{aligned} \|u\|{}_{\dot{H} _{p}^{s}(\mathbb{R}^{n})}\lesssim\|u\|{}_{L^{p_{0}}(\mathbb{R}^{n})}^{(1-\theta)}\|u\|{}_{\dot{H} _{p_{1}}^{\sigma}(\mathbb{R}^{n})}^{\theta}, \end{aligned} $$

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where $\theta =\theta _{s,\sigma }:=\frac {\frac {1}{p_{0}}-\frac {1}{p}+\frac {s}{n}}{\frac {1}{p_{0}}-\frac {1}{p_{1}}+\frac {\sigma }{n}}$ and $\frac {s}{\sigma }\leq \theta \leq 1.$

Proof

For the proof see [7] and [1, 4,5,6, 8, 9].

Proposition A.2

Let us assume s > 0 and 1 ≤ r ≤∞, 1 < p ₁, p ₂, q ₁, q ₂ ≤∞ satisfying the following relation

$$\displaystyle \begin{aligned} \frac{1}{r}= \frac{1}{p_{1}}+\frac{1}{p_{2}}= \frac{1}{q_{1}}+\frac{1}{q_{2}}. \end{aligned}$$

Then the following fractional Leibniz rule holds:

$$\displaystyle \begin{aligned} \||D|{}^{s}(fg)\|{}_{L^{r}(\mathbb{R}^{n})}\lesssim \||D|{}^{s} f \|{}_{L^{p_{1}}(\mathbb{R}^{n})} \|g\|{}_{L^{p_{2}}(\mathbb{R}^{n})}+\|f\|{}_{L^{q_{1}}(\mathbb{R}^{n})}\||D|{}^{s} g \|{}_{L^{q_{2}}(\mathbb{R}^{n})} \end{aligned} $$

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for all $f\in \dot {H} _{p_{1}}^{s}(\mathbb {R}^{n})\cap L^{q_{1}}(\mathbb {R}^{n})$ and $g\in \dot {H} _{q_{2}}^{s}(\mathbb {R}^{n})\cap L^{p_{2}}(\mathbb {R}^{n}).$

For more details concerning fractional Leibniz rules see [4].

Proposition A.3

Let us choose s > 0, p > ⌈s⌉ and 1 < r, r ₁, r ₂ < ∞ satisfying

$$\displaystyle \begin{aligned} \frac{1}{r}=\frac{p-1}{r_{1}}+\frac{1}{r_{2}}.\end{aligned}$$

Let us denote by F(u) one of the functions |u|^p, ±|u|^p−1 u. Then it holds the following fractional chain rule:

$$\displaystyle \begin{aligned} \||D|{}^{s}F(u)\|{}_{L^{r}(\mathbb{R}^{n})}\lesssim \|u\|{}_{L^{r_{1}}(\mathbb{R}^{n})}^{p-1} \||D|{}^{s}u\|{}_{L^{r_{2}}(\mathbb{R}^{n})}. \end{aligned} $$

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Proof

For the proof see [12].

Proposition A.4

Let p > 1 and $u\in H^{s}_{m}(\mathbb {R}^{n})$ , where $s\in \left (\frac {n}{m},p \right ) .$ Then the following estimates hold:

$$\displaystyle \begin{aligned}\ \||u|{}^{p}\|{}_{H^{s}_{m}(\mathbb{R}^{n})}\lesssim\|u\|{}_{ H^{s}_{m}(\mathbb{R}^{n})}\|u\|{}^{p-1}_{L^{\infty}(\mathbb{R}^{n})}, \end{aligned}$$

$$\displaystyle \begin{aligned} \|u|u|{}^{p-1}\|{}_{H^{s}_{m}(\mathbb{R}^{n})}\lesssim\|u\|{}_{ H^{s}_{m}(\mathbb{R}^{n})}\|u\|{}^{p-1}_{L^{\infty}(\mathbb{R}^{n})}. \end{aligned}$$

Proof

For the proof see [14].

We can derive from Proposition A.4 the following corollary.

Corollary A.5

Under the assumptions of Proposition A.4 it holds:

$$\displaystyle \begin{aligned} \||u|{}^{p}\|{}_{\dot{H}^{s}_{m}(\mathbb{R}^{n})}\lesssim\|u\|{}_{ \dot{H}^{s}_{m}(\mathbb{R}^{n})}\|u\|{}^{p-1}_{L^{\infty}(\mathbb{R}^{n})}, \end{aligned}$$

$$\displaystyle \begin{aligned} \|u|u|{}^{p-1}\|{}_{\dot{H}^{s}_{m}(\mathbb{R}^{n})}\lesssim\|u\|{}_{ \dot{H}^{s}_{m}(\mathbb{R}^{n})}\|u\|{}^{p-1}_{L^{\infty}(\mathbb{R}^{n})}. \end{aligned}$$

Proof

For the proof see [13].

Lemma A.6

Let 0 < 2s ^∗ < n < 2s. Then for any function $f\in \dot {H}^{s^{*}}(\mathbb {R}^{n})\cap \dot {H}^{s}(\mathbb {R}^{n})$ one has

$$\displaystyle \begin{aligned} \|f\|{}_{L^{\infty}(\mathbb{R}^{n})}\leq\|f\|{}_{\dot{H}^{s^{*}}(\mathbb{R}^{n})}+\|f\|{}_{\dot{H}^{s}(\mathbb{R}^{n})}. \end{aligned}$$

Proof

For the proof see [2].