Global existence for small amplitude semilinear wave equations with time-dependent scale-invariant damping

Consider the Cauchy problem of the semilinear wave equation with time-dependent damping

where $\mu\geq 0$, $\beta\geq 0$, $p>1$, $n\geq 2$ and $t_{0}\geq 0$. When $\mu=0$ and $t_{0}=0$ is , (1.1) becomes

For (1.2), Strauss in [43] proposed the following well-known conjecture (Strauss’ conjecture):

Let $p_{s}(n)$ denote the positive root of the quadratic algebraic equation

If $p>p_{s}(n)$, (1.2) admits a unique global solution with small data; whereas if $1<p\leq p_{s}(n)$, the solution of (1.2) can blow up in finite time.

So far the Strauss’ conjecture has been systematically studied and solved well, see [11, 12, 13, 21, 31, 40, 41, 55]. Especially, in [50], one can find the detailed history on the studies of (1.2).

When $\beta=0$ holds and $\mu=1$ is assumed without loss of generality, (1.1) with $t_{0}=0$ becomes

Nowadays the study of (1.4) is almost classic, it is shown that the solution $u$ can blow up in finite time when $1<p\leq p_{f}(n)$ with the Fujita exponent $p_{f}(n)=1+\frac{2}{n}$ as defined in [8] for the semilinear parabolic equations, while the small data solution $u$ exists globally for $p>p_{f}(n)$ with $n=1,2$ and $p_{f}(n)<p<\frac{n}{n-2}$ for $n\geq 3$, see [7], [29], [44], [47] and [56]. Recently, Chen and Reissig in [1] considered (1.4) with data from Sobolev spaces of negative order, they obtained a new critical exponent $p_{c}=1+\frac{4}{n+2\gamma}$ and established small data global existence result for $1<p<p_{c}$ and blow up result for $1<p<p_{c}$ provided that $1\leq n\leq 6$, while the global existence for $p=p_{c}$ is proved by D’Abbicco [3] in Euclidean setting and in the Heisenberg space.

For $\beta>1$ or $0<\beta<1$, consider the corresponding linear homogeneous equation

then the long time decay rate of $v$ just corresponds to that of linear wave equation or linear parabolic equation, respectively, see Wirth [51, 52]. When $0<\beta<1$ and $\mu>0$, if $p>p_{f}(n)$ with $n=1,2$ or $p_{f}(n)<p<\frac{n+2}{n-2}$ with $n\geq 3$, then $\eqref{equ:eff}$ has a global small data solution $u$, see D’Abbicco, Luencte and Reissig [4], Lin, Nishihara and Zhai [30] and Nishihara [33]; if $1<p\leq p_{f}(n)$, the solution $u$ generally blows up in finite time, one can be referred to Fujikawa, Ikeda and Wakasugi [9]. When $\beta>1$ and $\mu>0$, the global existence or blowup results on (1.1) are analogous to the ones on (1.2). It is shown that for $p>p_{s}(n)$, the global small data solution exists (see Liu and Wang [32]), while the solution may blow up in finite time for $1<p\leq p_{s}(n)$ (see Lai and Takamura [24] and Wakasa, Yordanov [49]).

When $\beta=1$, the equation in (1.5) is invariant under the scaling

and hence we say the damping term is scale-invariant. In this case, the behavior of (1.2) depends on the scale of $\mu$. If $n=1$ with $\mu\geq\frac{5}{3}$ or $n=2$ with $\mu\geq 3$ or $n\geq 3$ with $\mu\geq n+2$ and $p_{f}(n)<p<\frac{n}{n-2}$, then the global existence of small data solution $u$ has been established in the important work [2] by D’Abbicco. For relatively small $\mu$, so far there are few results on the global solution of (1.1) with $\beta=1$, recently, for $n=3$ and the radial symmetrical case, the global small solution is shown in Lai and Zhou [26] for $\mu\in[\frac{3}{2},2)$ and $p_{s}(3+\mu)<p\leq 2$. On the other hand, there have been systematic results for blow up: for the case of $n=1$, if $0<\mu<\frac{4}{3}$ and $1<p\leq p_{s}(1+\mu)$, or $\mu\geq\frac{4}{3}$ and $1<p\leq p_{f}(1)$, the solution will blowup in finite time; for the case of $n\geq 2$, the blowup results are also established when $\mu>0$ and $1<p<p_{s}(n+\mu)$, or $0<\mu<\frac{n^{2}+n+2}{n+2}$ and $1<p\leq p_{s}(n+\mu)$ (see the works in [20, 25, 45, 46, 47, 48] by Ikeda, Sobajima, Lai, Takamura, Wakasa, Tu, Lin and Wakasugi). These results enlighten us the critical exponent for $0<\mu<\frac{n^{2}+n+2}{n+2}$ should be $p_{\text{crit}}(n,\mu)=p_{s}(n+\mu)$ where $p_{s}(n+\mu)$ is the positive root of the following quadratic equation:

From our last paper [19], we start to study the global existence of small data solutions to problem (1.1) with $\beta=1$, $\mu\in(0,1)\cup(1,2)$ and $p>p_{s}(n+\mu)$. That is, we focus on the following regular Cauchy problem

where $(u_{0},u_{1})\in C^{\infty}_{c}(\mathbb{R}^{n})$ and $n\geq 2$. Comparing (1.3) and (1.6), we see that we have a shift from $n$ to $n+\mu$ in the coefficients of the quadratic equations of $p$. Thus the conformal exponent for (1.2) should be

In [19], we have obtained global existence result for small data solution for $n\geq 2$, $\mu\in(0,1)$ and $p\geq\max\{p_{\text{conf}}(m,\mu),3\}$, or $\mu\in(1,2)$ and $p\geq\max\{p_{\text{conf}}(n,\mu),\frac{4}{\mu}-1\}$. The main results in this article can be stated as follows.

Note that when $0<\mu<1$, as in D’Abbicco, Lucente and Reissig [6], if setting $\mu=\frac{k}{k+1}$ with $k\in(0,\infty)$ and $t=T^{k+1}/(k+1)$, then the equation in (1.7) is essentially equivalent to

when $1<\mu<2$, if setting

then the equation in (1.7) can be written as

where $\tilde{\mu}=2-\mu\in(0,1)$, and the unimportant constant coefficients $C_{\mu}>0$ before the nonlinearities in (1.10) and (1.11) are neglected. Let $\tilde{\mu}=\frac{\tilde{k}}{\tilde{k}+1}$ with $\tilde{k}\in(0,\infty)$ and $t=T^{\tilde{k}+1}/(\tilde{k}+1)$. Then for $t\geq 1$, (1.11) is actually equivalent to

where $\alpha=2\tilde{k}+1-p$, and the unimportant constant coefficient $C_{\tilde{k}}>0$ before the nonlinearity of (1.12) is also neglected. Based on (1.10) and (1.12), in order to prove Theorems 1.1, it is necessary to study the following semilinear generalized Tricomi equation

where $m>0$ and $\alpha\in\mathbb{R}$.

The local existence result for (1.13) was first considered by the pioneering work [53] of Yagdjian. Under the conditions of $n\geq 2$, $\alpha>-1$ and

it is proved in [53, Theorem 1.3] that (1.13) addmits no global solution $u\in C([0,\infty),L^{p+1}(\mathbb{R}^{n}))$. On the other hand, if $n\geq 2$, $\alpha>-1$ and

then [53, Theorem 1.2] that (1.13) has a global small data solution $u\in C([0,\infty),L^{p+1}(\mathbb{R}^{n}))\cap C^{1}([0,\infty),{\mathcal{D}}^{\prime}(\mathbb{R}^{n}))$. By checking (1.14) and (1.15) one see that the results in [53] were not completed since there is a gap between the intervals for $p$ where blowup happens or small data solution exists globally, also an upper bound of $p$ exists for the global existence part. Later Galstian in [10] investigated Cauchy problem for (1.13) for the case $n=1$, $\alpha>-1$ and $m\geq 0$. She proved the blowup result for $1<p<1+\frac{4}{m}$ and established the global existence for small data solution under the condition (1.15).

Witt, Yin and the first author of this article considered the case $\alpha=0$ and $m>0$, and for the initial data problem of (1.13) starting from some positive time $t_{0}$, it has been shown that there exists a critical index $p_{\text{crit}}(n,m)>1$ such that when $p>p_{\text{crit}}(n,m)$, the small data solution $u$ of (1.13) exists globally; when $1<p\leq p_{\text{crit}}(n,m)$, the solution $u$ may blow up in finite time, where $p_{\text{crit}}(n,m)$ for $n\geq 2$ and $m>0$ is the positive root of

while for $n=1$, $p_{\text{crit}}(1,m)=1+\frac{4}{m}$ (see [14, 15, 16, 17, 18]).

It is pointed out that about the local existence and regularity of solution $u$ to (1.13) with $\alpha=0$ and $m\in\mathbb{N}$ under the weak regularity assumptions of initial data $(u,\partial_{t}u)(0,x)=(u_{0},u_{1})$, the reader may consult Ruan, Witt and Yin [36, 37, 38, 39].

Now let us turn back to the problem (1.13). Here and below denote

note that $T_{0}\in(1,e^{1/e}]$ for all $m>0$. We next focus on the global existence of the solution to the following problem

where $m>0$, $\alpha\in\mathbb{R}$, $p>1$, $f(x),g(x)\in C_{c}^{\infty}(\mathbb{R}^{n})$ with $n\geq 2$ and supp $f$, supp $g\in B(0,M)$ with $M>0$. It has been shown in Theorem 1 of Palmieri and Reissig [35] that there exists a critical exponent $p_{crit}(n,m,\alpha)>1$ for $\alpha>-2$ such that when $1<p\leq p_{crit}(n,m,\alpha)$, the solution of (1.17) can blow up in finite time with some suitable choices of $(u_{0},u_{1})$, where

with $p_{1}(n,m,\alpha)=1+\frac{2(2+\alpha)}{(m+2)n-2}$ and $p_{2}(n,m,\alpha)$ being the positive root of the quadratic equation:

It is not difficult to verify that when $n\geq 3$ and $\alpha>-2$ or $n=2$ and $\alpha>-1$, $p_{crit}(n,m,\alpha)=p_{2}(n,m,\alpha)$ holds; when $n=2$ and $-2<\alpha\leq-1$, $p_{crit}(n,m,\alpha)=p_{1}(n,m,\alpha)$ holds. However, as we will see in next subsection, in order to prove Theorem 1.1, it suffices to consider $-1<\alpha\leq m$ for $n\geq 2$ in (1.17) , thus we set $p_{crit}(n,m,\alpha)=p_{2}(n,m,\alpha)$ in this article.

In [19, (1.24)], we have determined a conformal exponent

Then we can state the global existence result for (1.17)

Take the results in Theorem 1.2 as granted, we can prove the main theorem.

We now sketch some of the ideas in the proof of Theorem 1.2. To prove Theorem 1.2, motivated by [11, 31], where some weighted Strichartz estimates with the characteristical weight $1+|t^{2}-|x|^{2}|$ were obtained for the linear wave operator $\partial_{t}^{2}-\triangle$, we need to establish some Strichartz estimates with the weight $\big{(}(\phi_{m}(t)+M)^{2}-|x|^{2}\big{)}^{\gamma}t^{\beta}$ for the generalized Tricomi operator $\partial_{t}^{2}-t^{m}\Delta$ with suitable index $\beta$ and $\gamma$. As we will see in Lemma 2.1, it is not difficult to get the linear homogeneous Strichatz estimate once we have the pointwise decay estiamtes.

In next step, we turn to establish weighted Strichartz estimates for the linear inhomogeneous problem:

which is the most difficult part in this article. The main result is the following:

Based on Theorem 1.3, repeating the proof of Theorem 1.4 in [17] we can prove the following modified version of (1.28).

With Lemma 2.1 and Theorem 1.4, we can prove Theorem 1.2 by a standard Picard iteration, the detail is given in Section 5.

It remains to give the proof of Theorem 1.3. By Stein’s analytic interpolation theorem (see [42]), in order to prove (1.28), it suffices to establish (1.28) for the two extreme cases of $q=q_{0}=\frac{2((m+2)n+2+2\alpha)}{(m+2)n-2}$ and $q=2$:

where $\gamma_{1}<\frac{1}{q_{0}}<\gamma_{2}$; and

where $\gamma_{1}<-\frac{1}{2}+\frac{m-\alpha}{m+2}$ and $\gamma_{2}>\frac{1}{2}$.

In order to derive (1.30), we use the idea from [11] and split the integral domain $\{(t,x):\phi^{2}_{m}(t)-|x|^{2}\leq 1\}$ into some pieces, which correspond to the “relatively small times” part and the “relatively large times” part respectively.

For the case of “relatively small times”, the key point is to establish the inhomogeneous Strichartz estimates without characteristic weight at the endpoint $\mathbf{q=q_{0}}$ for all $\mathbf{n\geq 2}$, $\mathbf{m>0}$ and $\mathbf{0<\alpha\leq\frac{n}{n-1}\cdot m}$ :

(1.32) is an essential improvement of Lemma 2.2 in [19] (see Remark 2.1 for explanation in details), it also make the proof in this article more self-contained. A local in time version of (1.32) can also be established for all $n\geq 2$, $m>0$ and $-1<\alpha\leq 0$ :