A nonexistence result for a nonlinear wave equation with damping on a Riemannian manifold

Abstract

In this paper, we study the global nonexistence of solutions to a nonlinear wave equation with critical potential $V(x)$ on a Riemannian manifold, the form of which is more general than those in (Todorova and Yordanov in C. R. Acad. Sci., Sér. 1 Math. 300:557-562, 2000). The way we follow is motivated by the work of Qi S. Zhang (C. R. Acad. Sci., Sér. 1 Math. 333:109-114, 2001). We also prove the local existence and uniqueness result.

1 Introduction and main results

In this paper, we study the global nonexistence of solutions to the following nonlinear wave equation with a damping term:

$$ \textstyle\begin{cases}{ \Delta u(x,t)+W(x)\vert u\vert ^{p}(x,t)-u_{t}(x,t)-u_{tt}(x,t)=0 \quad \textrm{in } {\mathbb{M}}^{n}\times(0,\infty),} \\ { u(x,0)= u_{0} (x) \quad\textrm{in } \mathbb{M}^{n},} \\ { u_{t}(x,0)= u_{1} (x) \quad\textrm{in } \mathbb{M}^{n},} \end{cases} $$

(1.1)

where $\mathbb{M}^{n}$ ($n\geq3$) is a non-compact complete Riemannian manifold, Δ is the Laplace-Beltrami operator, and $\int u_{0}(x)\,dx,\int u_{1}(x)\,dx>0$, while the constant $p>1$.

In [1], Todorova and Yordanov proved the following result for (1.1) when $\mathbb{M}^{n}=\mathbb{R}^{n}$ and $W(x)\equiv1$:

• Let $1< p<1+\frac{2}{n}$. If we assume that $u_{0}(x)$, $u_{1}(x)$ is compactly supported and $\int u_{0}(x)\,dx, \int u_{1}(x)\,dx>0$, then the global solution of (1.1) does not exist.

However, whether or not the critical case $p=1+\frac{2}{n}$ belongs to the blow-up case was left open. In [2], Qi S. Zhang showed $p=1+\frac{2}{n}$ belongs to the blow-up case.

The investigation of nonexistence and existence of global solutions to evolution equations has a long history, We refer the reader to the surveys [3–7]. There are more recent contributions to the discussion of the test function method; we refer to [8–11] for a survey of the literature on this problem.

In this paper, we study the global nonexistence of solutions to a nonlinear wave equation with critical potential $V(x)$ on a Riemannian manifold, the form of which is more general than those in [1]. The way we follow is motivated by the work of Qi S. Zhang [2]. We also prove the local existence and uniqueness result.

Throughout the paper, for a fixed $x_{0}\in\mathbb{M}^{n}$, we make the following assumptions (see [2]):

• (i) $\frac{\partial\log g^{\frac{1}{2}}}{\partial r}\leq\frac{C}{r}$, when $r=d(x,x_{0})$ is smooth; here $g^{\frac{1}{2}}$ is the volume density of the manifold;

• (ii) there are positive constants $\alpha>2$ and $m>-2$, such that

  • $C^{-1}r^{\alpha}\leq \vert B_{r}(x_{0})\vert \leq Cr^{\alpha}$, when r is large and for all $x\in\mathbb{M}^{n}$;

  • $W(x)$ are non-negative $L_{loc}^{\infty}$ functions. For large $r=d(x,x_{0})$, $C^{-1}r^{m}\leq W(x)\leq Cr^{m}$.

Lemma 1

see [12]

Under assumptions (i) and (ii), there exist positive constants C and $R_{0}$, for $R\geq R_{0}$ and $\frac{1}{p}+\frac{1}{q}=1$, such that

$$ \int_{B_{R}(x_{0})}W^{-\frac{q}{p}}(x)\,dx\leq C\ln R+CR^{-\frac{qm}{p}+\alpha}. $$

Our result is as follows.

Theorem 1.1

Under assumptions (i) and (ii), let $p\in(1,1+\frac{2+m}{\alpha}]$. If we assume that $u_{0}(x)$, $u_{1}(x)$ is compactly supported and $\int u_{0}(x)\,dx,\int u_{1}(x)\,dx>0$, then the global solution of (1.1) does not exist.

Remark

Clearly $\mathbb{R}^{n}$ satisfies assumptions (i) and (ii), so if $\mathbb{M}^{n}=\mathbb{R}^{n}$ and $W(x)\equiv1(m=0)$, from the proof of Theorem 1.1, it is in accordance with (a).

Theorem 1.2

Local existence and uniqueness

Let $\mathbb{M}^{n}$ be an n-dimensional smooth compact manifold, and $u_{0}$ be a smooth hypersurface immersion of $\mathbb {M}^{n}$ into $\mathbb{R}^{n+1}$. Then there exists a constant $T > 0$ such that the initial value problem

$$ \textstyle\begin{cases}{ \Delta u(x,t)+W(x)\vert u\vert ^{p}(x,t)-u_{t}(x,t)-u_{tt}(x,t)=0 \quad \textrm{in } {\mathbb{M}}^{n}\times(0,\infty),} \\ { u(x,0)= u_{0} (x) \quad\textrm{in } \mathbb{M}^{n},} \\ { u_{t}(x,0)= u_{1} (x) \quad\textrm{in } \mathbb{M}^{n},} \end{cases} $$

(1.2)

has a unique smooth solution $u(x,t)$ on $\mathbb{M}^{n}\times[0,T)$, where $u_{1} (x)$ is a smooth vector-valued function on $\mathbb{M}^{n}$.

Theorem 1.1 is proved in Section 2; Theorem 1.2 is proved in Section 3.

2 Global nonexistence of solutions

Proof of Theorem 1.1

From now on, C is always a constant that may change from line to line. Throughout the section, we let $\varphi,\eta\in C^{\infty}[0,\infty)$ be two functions satisfying

$$ \textstyle\begin{cases} { \varphi(r)\in[0,1],\quad\textrm{if } r\in[0,\infty),} \\ { \varphi(r)=1,\quad \textrm{if } r\in[0,\frac{1}{2}],} \\ { \varphi(r)=0,\quad\textrm{if } r\in[1,\infty];} \\ { \eta(t)\in[0,1],\quad \textrm{if } t\in[0,\infty),} \\ { \eta(t)=1,\quad \textrm{if } t\in[0,\frac{1}{4}],} \\ { \eta(t)=0,\quad \textrm{if } t\in[1,\infty];} \\ { \frac{\vert \nabla\varphi \vert ^{2}}{\varphi}\leq C, \quad \textrm{if } r\in[0,1];} \\ { \frac{\eta_{t}^{2}}{\eta}\leq C,\quad \textrm{if } t\in[0,1];} \\ { -C\leq\varphi(r)^{\prime}\leq0;\quad \quad \vert \varphi (r)^{\prime\prime} \vert \leq C;\quad \quad-C\leq\eta(t)^{\prime}\leq0; \quad \quad \vert \eta(t)^{\prime\prime} \vert \leq C.} \end{cases} $$

(2.1)

For $R>0$, we define $Q_{R}=B_{R}(x_{0})\times[0,R^{2}]$. We also need a cut-off function

$$ \psi_{R}=\varphi_{R}\bigl[d(x,x_{0})\bigr] \eta_{R} (t), $$

(2.2)

where $\varphi_{R}(r)=\varphi(\frac{r}{R})$ and $\eta_{R}(t)=\eta(\frac{t}{R^{2}})$. Clearly,

$$ \begin{aligned} &\frac{\partial\varphi_{R}}{\partial r}\in\biggl[-\frac{C}{R},0\biggr];\quad\quad \frac{\partial^{2}\varphi _{R}}{\partial r^{2}}\in\biggl[-\frac{C}{R^{2}},\frac{C}{R^{2}}\biggr];\quad\quad \frac{\partial \eta_{R}}{\partial t}\in\biggl[-\frac{C}{R^{2}},0\biggr]; \\ &\frac{\vert \nabla\varphi _{R}\vert ^{2}}{\varphi_{R}}\leq\frac{C}{R^{2}};\quad \quad \frac{(\partial_{t}\eta_{R})^{2}}{\eta_{R}}\leq \frac{C}{R^{4}}. \end{aligned} $$

(2.3)

We use the method of contradiction. Suppose that $u(x,t)$ is a global positive solution of (1.1). For $R>0$, we set

$$ I_{R}\stackrel{\triangle}{=} \int_{Q_{R}}W(x)\vert u\vert ^{p}(x,t)\psi _{R}^{q}(x,t)\,dx\,dt, $$

(2.4)

where $\frac{1}{p}+\frac{1}{q}=1$.

Since $u(x,t)$ is a solution of (1.1), we have

$$ I_{R}= \int_{Q_{R}}\bigl[u_{t}(x,t)-\Delta u(x,t)+u_{tt}(x,t)\bigr]\psi_{R}^{q}(x,t)\,dx \,dt=J_{1}+J_{2}, $$

(2.5)

where

$$ J_{1}\stackrel{\triangle}{=} \int_{Q_{R}}\bigl[u_{t}(x,t)-\Delta u(x,t)\bigr] \psi_{R}^{q}(x,t)\,dx\,dt, \quad\quad J_{2}\stackrel{ \triangle}{=} \int _{Q_{R}}u_{tt}(x,t)\psi_{R}^{q}(x,t) \,dx\,dt. $$

(2.6)

We will estimate $J_{1}$ and $J_{2}$ separately.

By the Stokes formula and noting that $\psi_{R}=0$ on $\partial B_{R}(x_{0})$, we have

$$ \begin{aligned}[b] J_{1}&= \int_{Q_{R}}u_{t}(x,t)\psi _{R}^{q}(x,t) \,dx\,dt- \int_{0}^{R^{2}} \int_{\partial B_{R}(x_{0})}\frac{\partial u(x,t)}{\partial n}\psi_{R}^{q}(x,t) \,dS_{x}\,dt \\ &\quad{} + \int_{Q_{R}}\nabla u(x,t)\nabla\psi_{R}^{q}(x,t) \,dx\,dt \\ &= \int_{Q_{R}}u_{t}(x,t)\psi_{R}^{q}(x,t) \,dx\,dt+ \int_{Q_{R}}\nabla u(x,t)\nabla\psi_{R}^{q}(x,t) \,dx\,dt, \end{aligned} $$

(2.7)

which implies, via integration by parts,

$$ \begin{aligned}[b] J_{1}&= \int_{B_{R}(x_{0})}u\bigl(x,R^{2}\bigr)\psi _{R}^{q}\bigl(x,R^{2}\bigr)\,dx- \int_{B_{R}(x_{0})}u(x,0)\psi_{R}^{q}(x,0)\,dx \\ &\quad{} - q \int_{Q_{R}}u(x,t)\varphi_{R}^{q}(x) \eta_{R}^{q-1}(t)\eta^{\prime }_{R}(t)\,dx\,dt + \int_{0}^{R^{2}} \int_{\partial B_{R}(x_{0})}u(x,t)\frac{\partial \varphi_{R}^{q}}{\partial n}\eta^{q}_{R}(t) \,dS_{x}\,dt \\ &\quad{} - \int_{Q_{R}}u(x,t)\Delta\varphi_{R}^{q}(x) \eta ^{q}_{R}(t)\,dx\,dt. \end{aligned} $$

(2.8)

We observe that $\psi_{R}^{q}(x,R^{2})=0$; $\int u_{0}(x)\,dx> 0$, $\frac{\partial\varphi_{R}^{q}}{\partial n}=q\varphi_{R}^{q-1}\varphi^{\prime}_{R}(\frac{\partial r}{\partial n})= 0$ on $\partial B_{R}(x_{0})$, so we obtain

$$ J_{1} \leq{ -q \int_{Q_{R}}u(x,t)\varphi_{R}^{q}(x) \eta_{R}^{q-1}(t)\eta^{\prime }_{R}(t)\,dx\,dt - \int_{Q_{R}}u(x,t)\Delta\varphi_{R}^{q}(x) \eta ^{q}_{R}(t)\,dx\,dt.} $$

(2.9)

Since $\Delta\varphi_{R}^{q}(x)=q\varphi_{R}^{q-1}(x)\Delta\varphi _{R}(x)+q(q-1)\varphi_{R}^{q-2}(x)\vert \nabla\varphi_{R}(x)\vert ^{2}$, (2.9) yields

$$\begin{aligned} J_{1} \leq& -q \int_{Q_{R}}u(x,t)\varphi_{R}^{q}(x) \eta_{R}^{q-1}(t)\eta^{\prime }_{R}(t)\,dx\,dt -q \int_{Q_{R}}u(x,t)\varphi_{R}^{q-1}(x)\Delta \varphi_{R}(x)\eta ^{q}_{R}(t)\,dx\,dt \\ &{} - q(q-1) \int_{Q_{R}}u(x,t)\varphi_{R}^{q-2}(x)\bigl\vert \nabla \varphi_{R}(x)\bigr\vert ^{2} \eta^{q}_{R}(t)\,dx\,dt. \end{aligned}$$

(2.10)

Recalling the supports of $\varphi_{R}(x)$ and $\eta_{R}(t)$, that is,

$$ \textstyle\begin{cases} { \eta_{R}(t)=1,\quad\quad \eta_{R}^{\prime}(t)=0,\quad \textrm{if } t\in[0,\frac{R^{2}}{4}],} \\ { \varphi_{R}(x)=1,\quad\quad \Delta\varphi_{R}(x)=0,\quad \textrm{if } r\in[0,\frac{R}{2}],} \end{cases} $$

(2.11)

we can reduce (2.10) to

$$ \begin{aligned}[b] J_{1}&\leq -q \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}u(x,t)\varphi_{R}^{q}(x) \eta_{R}^{q-1}(t)\eta^{\prime }_{R}(t)\,dx\,dt \\ &\quad{} - q \int_{0}^{R^{2}} \int_{ B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})}u(x,t)\varphi _{R}^{q-1}(x)\Delta \varphi_{R}(x)\eta^{q}_{R}(t)\,dx\,dt \\ &\quad{} - q(q-1) \int_{0}^{R^{2}} \int_{ B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})} u(x,t)\varphi _{R}^{q-2}(x)\bigl\vert \nabla\varphi_{R}(x)\bigr\vert ^{2} \eta^{q}_{R}(t)\,dx\,dt. \end{aligned} $$

(2.12)

Since $\varphi_{R}$ is radial, we have

$$ \Delta\varphi_{R}=\varphi_{R}^{\prime\prime}+ \biggl[ \frac {n-1}{r}+\frac{\partial \log g^{\frac{1}{2}}}{\partial r} \biggr]\varphi_{R}^{\prime}. $$

(2.13)

Taking R sufficiently large, by assumption (i), that is, $\frac{\partial\log g^{\frac{1}{2}}}{\partial r}\leq\frac{C}{r}$, we obtain

$$ \Delta\varphi_{R}\geq-\frac{C}{R^{2}}, $$

(2.14)

when $x\in B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})$. Merging (2.12), (2.14), and (2.3), we know

$$ \begin{aligned}[b] J_{1} &\leq \frac{Cq}{R^{2}} \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}u(x,t)\varphi_{R}^{q}(x) \eta_{R}^{q-1}(t)\,dx\,dt \\ &\quad{} + \frac{Cq}{R^{2}} \int_{0}^{R^{2}} \int_{ B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})}u(x,t)\varphi _{R}^{q-1}(x) \eta^{q}_{R}(t)\,dx\,dt \\ &\quad{} - q(q-1) \int_{0}^{R^{2}} \int_{ B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})} u(x,t)\varphi _{R}^{q-2}(x)\bigl\vert \nabla\varphi_{R}(x)\bigr\vert ^{2} \eta^{q}_{R}(t)\,dx\,dt. \end{aligned} $$

(2.15)

By (2.3), we have

$$ \varphi_{R}^{q-2}(x)\bigl\vert \nabla\varphi_{R}(x) \bigr\vert ^{2}=\varphi _{R}^{q-1} \frac{\vert \nabla\varphi_{R}(x)\vert ^{2}}{\varphi_{R}}\geq-\frac {C}{R^{2}}\varphi_{R}^{q-1}, $$

(2.16)

which yields

$$ \begin{aligned}[b] J_{1} &\leq \frac{Cq}{R^{2}} \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}u(x,t)\varphi_{R}^{q}(x) \eta_{R}^{q-1}(t)\,dx\,dt \\ &\quad{} + \frac{Cq}{R^{2}} \int_{0}^{R^{2}} \int_{ B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})}u(x,t)\varphi _{R}^{q-1}(x) \eta^{q}_{R}(t)\,dx\,dt \\ &\quad{} + \frac{Cq(q+1)}{R^{2}} \int_{0}^{R^{2}} \int_{ B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})}u(x,t)\varphi _{R}^{q-1}(x) \eta^{q}_{R}(t)\,dx\,dt \\ &\leq \frac{Cq}{R^{2}} \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}u(x,t)\varphi_{R}^{q}(x) \eta_{R}^{q-1}(t)\,dx\,dt \\ &\quad{} +\frac{Cq}{R^{2}} \int_{0}^{R^{2}} \int_{ B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})}u(x,t)\varphi _{R}^{q-1}(x) \eta^{q}_{R}(t)\,dx\,dt. \end{aligned} $$

(2.17)

Therefore, as $\varphi_{R}, \eta_{R}\leq1$,

$$\begin{aligned} J_{1} \leq &\frac{Cq}{R^{2}} \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}u(x,t)\psi_{R}^{q-1}(x,t)\,dx\,dt \\ &{} + \frac{Cq}{R^{2}} \int_{0}^{R^{2}} \int_{ B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})}u(x,t)\psi _{R}^{q-1}(x,t)\,dx\,dt \\ \leq& \frac{Cq}{R^{2}} \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}W^{\frac{1}{p}}(x)\bigl\vert u(x,t)\bigr\vert \psi_{R}^{q-1}(x,t)W^{-\frac {1}{p}}(x)\,dx\,dt \\ &{} + \frac{Cq}{R^{2}} \int_{0}^{R^{2}} \int_{ B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})}W^{\frac {1}{p}}(x)\bigl\vert u(x,t)\bigr\vert \psi_{R}^{q-1}(x,t)W^{-\frac{1}{p}}(x)\,dx\,dt. \end{aligned}$$

(2.18)

By the Hölder inequality and noticing $\frac{1}{p}+\frac{1}{q}=1$, we have

$$\begin{aligned} J_{1} \leq& \frac{Cq}{R^{2}} \biggl[ \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}W(x)\vert u\vert ^{p}(x,t) \psi_{R}^{q}(x,t)\,dx\,dt \biggr]^{\frac {1}{p}}\times \biggl[ \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}W^{-\frac{q}{p}}(x)\,dx\,dt \biggr]^{\frac{1}{q}} \\ &{} +\frac{Cq}{R^{2}} \biggl[ \int_{0}^{R^{2}} \int_{ B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})}W(x)\vert u\vert ^{p}(x,t) \psi_{R}^{q}(x,t)\,dx\,dt \biggr]^{\frac{1}{p}} \\ &{} \times \biggl[ \int_{0}^{R^{2}} \int_{ B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})}W^{-\frac {q}{p}}(x)\,dx\,dt \biggr]^{\frac{1}{q}} \\ \leq& \frac{Cq}{R^{2}} [I_{R} ]^{\frac{1}{p}}\times \biggl[ \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}W^{-\frac{q}{p}}(x)\,dx\,dt \biggr]^{\frac{1}{q}} \\ &{} + \frac{Cq}{R^{2}} [I_{R} ]^{\frac{1}{p}} \times \biggl[ \int_{0}^{R^{2}} \int_{ B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})}W^{-\frac {q}{p}}(x)\,dx\,dt \biggr]^{\frac{1}{q}}. \end{aligned}$$

(2.19)

By Lemma 1, we obtain

$$ \begin{aligned}[b] \biggl[ \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}W^{-\frac{q}{p}}(x)\,dx\,dt \biggr]^{\frac{1}{q}} & \leq \biggl\{ \int_{\frac{R^{2}}{4}}^{R^{2}} \bigl[C\ln R+CR^{-\frac{qm}{p}+\alpha} \bigr] \,dt \biggr\} ^{\frac{1}{q}} \\ &\leq CR^{\frac{2}{q}} \ln R+CR^{-\frac{m}{p}+\frac {2+\alpha}{q}}. \end{aligned} $$

(2.20)

Hence,

$$ \begin{aligned}[b] J_{1} &\leq \frac{Cq}{R^{2}} [I_{R} ]^{\frac{1}{p}}\times \bigl[CR^{\frac{2}{q}}\ln R+CR^{-\frac{m}{p}+\frac{2+\alpha }{q}} \bigr] +\frac{Cq}{R^{2}} [I_{R} ]^{\frac{1}{p}} \times \bigl[CR^{-\frac{m}{p}+\frac{2+\alpha}{q}} \bigr] \\ &\leq \frac{Cq}{R^{2}} [I_{R} ]^{\frac{1}{p}}\times \bigl[CR^{\frac{2}{q}}\ln R+CR^{-\frac{m}{p}+\frac{2+\alpha }{q}} \bigr] \\ &= C [I_{R} ]^{\frac{1}{p}}\times \bigl[CR^{\frac{2}{q}-2}\ln R+CR^{-\frac{m}{p}+\frac{2+\alpha }{q}-2} \bigr]. \end{aligned} $$

(2.21)

Now let us estimate $J_{2}$. Using integration by parts, we obtain

$$ \begin{aligned}[b] J_{2}&= \int_{Q_{R}}u_{tt}(x,t)\psi _{R}^{q}(x,t) \,dx\,dt \\ &= \int_{B_{R}(x_{0})}u_{t}(x,t)\psi _{R}^{q}(x,t) \bigg\vert ^{R^{2}}_{0}\,dx-q \int_{B_{R}(x_{0})}u\varphi _{R}^{q}(x) \eta_{R}^{q-1}(t)\partial_{t}\eta _{R} \bigg\vert ^{R^{2}}_{0}\,dx \\ &\quad{} + q \int_{Q_{R}}u(x,t)\varphi_{R}^{q}(x)\eta _{R}^{q-1}(t)\partial^{2}_{t} \eta_{R}(t)\,dx\,dt \\ &\quad {}+q(q-1) \int _{Q_{R}}u(x,t)\varphi_{R}^{q}(x) \eta_{R}^{q-2}(t) \bigl(\partial_{t}\eta _{R}(t)\bigr)^{2}\,dx\,dt. \end{aligned} $$

(2.22)

We observe that $\psi_{R}^{q}(x,R^{2})=\eta_{R}(R^{2})=0$; $\int u_{0}(x)\,dx,\int u_{1}(x)\,dx> 0$ and (2.3), The above implies

$$ \begin{aligned}[b] J_{2} &\leq q \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}\vert u\vert \varphi_{R}^{q} \eta_{R}^{q-1}\bigl\vert \partial^{2}_{t} \eta _{R}\bigr\vert \,dx\,dt \\ &\quad{} +q(q-1) \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}\vert u\vert \varphi_{R}^{q} \eta_{R}^{q-1}\frac{(\partial _{t}\eta_{R})^{2}}{\eta_{R}}\,dx\,dt. \end{aligned} $$

(2.23)

Again by (2.3) and the Hölder inequality, we have

$$ \begin{aligned}[b] J_{2} &\leq \frac{C}{R^{4}} \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}\vert u\vert \varphi_{R}^{q} \eta_{R}^{q-1}\,dx\,dt \\ &\leq \frac{C}{R^{4}} \biggl[ \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}W(x)\vert u\vert ^{p}(x,t) \psi_{R}^{q}(x,t)\,dx\,dt \biggr]^{\frac {1}{p}} \\ &\quad{} \times \biggl[ \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{ B_{R}(x_{0})}W^{-\frac{q}{p}}(x)\,dx\,dt \biggr]^{\frac{1}{q}}. \end{aligned} $$

(2.24)

By (2.20), (2.24) yields

$$ J_{2} \leq \frac{C}{R^{4}} [I_{R} ]^{\frac{1}{p}}\times \bigl[CR^{\frac{2}{q}}\ln R+CR^{-\frac{m}{p}+\frac{2+\alpha }{q}} \bigr]. $$

(2.25)

Combining (2.5), (2.21), and (2.25), we obtain, for large R,

$$ \begin{aligned}[b] I_{R} &= J_{1}+J_{2} \\ &\leq \frac{C}{R^{2}} [I_{R} ]^{\frac{1}{p}}\times \bigl[CR^{\frac{2}{q}}\ln R+CR^{-\frac{m}{p}+\frac{2+\alpha }{q}} \bigr]+\frac{C}{R^{4}} [I_{R} ]^{\frac{1}{p}}\times \bigl[CR^{\frac{2}{q}}\ln R+CR^{-\frac{m}{p}+\frac{2+\alpha }{q}} \bigr] \\ &\leq C [I_{R} ]^{\frac{1}{p}}\times \bigl[CR^{\frac{2}{q}-2}\ln R+CR^{-\frac{m}{p}+\frac{2+\alpha }{q}-2} \bigr], \end{aligned} $$

(2.26)

which yields

$$ I_{R}^{\frac{1}{q}} \leq CR^{\frac{2}{q}-2}\ln R+CR^{-\frac{m}{p}+\frac{2+\alpha}{q}-2}. $$

(2.27)

If $p\in(1,1+\frac{2+m}{\alpha})$, then $-\frac{m}{p}+\frac{2+\alpha}{q}-2<0$. Let $R\rightarrow\infty$, we have

$$ \int_{0}^{\infty} \int_{\mathbb{M}^{n}}W(x)\vert u\vert ^{p}(x,t)\,dx\,dt=0. $$

(2.28)

Hence, (2.28) is a contradiction when R is large. This is because the left-hand side of (2.28) is positive and non-decreasing while $R\longrightarrow\infty$.

If $p=1+\frac{2+m}{\alpha}$, then $-\frac{m}{p}+\frac{2+\alpha}{q}-2=0$. Therefore, when R is large, (2.27) becomes

$$ I_{R} \leq C \bigl[CR^{\frac{2}{q}-2}\ln R+C \bigr]^{q}\leq C. $$

(2.29)

This shows

$$ \int_{0}^{\infty} \int_{\mathbb {M}^{n}}W(x)u^{p}(x,t)\,dx\,dt={ \lim _{R\rightarrow\infty }I_{R}}< \infty. $$

(2.30)

Hence

$$ { \lim_{R\rightarrow\infty }} \int_{\frac{R^{2}}{4}}^{R^{2}} \int_{B_{R}(x_{0})}W(x)u^{p}(x,t)\,dx\,dt=0 $$

(2.31)

and

$$ { \lim_{R\rightarrow\infty}} \int_{0}^{R^{2}} \int_{ B_{R}(x_{0})\backslash B_{\frac{R}{2}}(x_{0})}W(x)u^{p}(x,t)\,dx\,dt=0. $$

(2.32)

Using the last two equalities, (2.19) and (2.24) again, we obtain

$$ \int_{0}^{\infty} \int_{\mathbb {M}^{n}}W(x)\vert u\vert ^{p}(x,t)\,dx\,dt={ \lim_{R\rightarrow\infty }I_{R}}=0. $$

(2.33)

This is a contradiction.

Thus, the proof of Theorem 1.1 is completed. □

3 Local existence and uniqueness

Proof of Theorem 1.2

Let $u(\cdot,t):{\mathbb{M}}^{n}\longrightarrow{\mathbb{R}}^{n+1}$ be a one-parameter family of smooth hypersurface immersions in ${\mathbb {R}}^{n+1}$ and $g=\{g_{ij}\}$ be the induced metric on $\mathbb{M}$ in a local coordinate system $\{x^{i}\}$ ($1\leq i\leq n$).

Noting

$$ \Delta u=\Delta_{g} u=g^{ij}\nabla_{i} \nabla_{j}u=g^{ij}\biggl(\frac {\partial^{2}u}{\partial x^{i}\,\partial x^{j}}- \Gamma^{k}_{ij}\frac {\partial u}{\partial x^{k}}\biggr), $$

(3.1)

the wave equation (1.1) can be equivalently rewritten as

$$ u_{tt}(x,t)=g^{ij}\biggl(\frac{\partial^{2}u}{\partial x^{i}\,\partial x^{j}}- \Gamma^{k}_{ij}\frac{\partial u}{\partial x^{k}}\biggr)+W(x)\vert u\vert ^{p}(x,t)-u_{t}(x,t). $$

(3.2)

Since

$$ \Gamma^{k}_{ij}=g^{kl}\biggl(\frac{\partial^{2}u}{\partial x^{i}\,\partial x^{j}}, \frac{\partial u}{\partial x^{l}}\biggr), $$

(3.3)

it follows that

$$ u_{tt}(x,t)=g^{ij}\frac{\partial^{2}u}{\partial x^{i}\,\partial x^{j}}-g^{ij}g^{kl} \biggl(\frac{\partial^{2}u}{\partial x^{i}\,\partial x^{j}}, \frac{\partial u}{\partial x^{l}}\biggr)\frac{\partial u}{\partial x^{k}}+W(x)\vert u\vert ^{p}(x,t)-u_{t}(x,t). $$

(3.4)

We note that equation (3.4) is not strictly hyperbolic. Therefore, in order to consider equation (3.4), we need to follow a trick of DeTurck [13] by modifying the flow through a diffeomorphism of ${\mathbb{M}}^{n}$, under which (3.4) turns out to be strictly hyperbolic, so that we can apply the standard theory of hyperbolic equations.

Suppose $\hat{u}(x,t)$ is a solution of equation (3.2) and $\phi _{t}:{\mathbb{M}}^{n}\longrightarrow{\mathbb{M}}^{n}$ is a family of diffeomorphisms of $\mathbb{M}^{n}$. Let

$$ u(x,t)=\phi_{t}^{\ast}\hat{u}(x,t), $$

(3.5)

where $\phi_{t}^{\ast}$ is the pull-back operator of $\phi_{t}$. We now want to find the evolution equation for the metric $u(x, t)$.

Denote

$$ y(x,t)=\phi_{t}(x)=\bigl\{ y^{1}(x,t)y^{2}(x,t)y^{3}(x,t) \cdots y^{n}(x,t)\bigr\} , $$

(3.6)

in local coordinates, and define $y(x,t)=\phi_{t}(x)$ by the following initial value problem:

$$ \textstyle\begin{cases}{ \frac{\partial^{2}y^{\alpha }}{\partial t ^{2}}=\frac{\partial y^{\alpha}}{\partial x^{k}}g^{jl}(\Gamma_{jl}^{k}-\hat{\Gamma}_{jl}^{k}),} \\ { y^{\alpha}(x,0)= x^{\alpha},\quad\quad y^{\alpha }_{t}(x,0)= 0,} \end{cases} $$

(3.7)

where $\hat{\Gamma}_{jl}^{k}$ is the connection corresponding to the initial metric $\hat{g}_{ij}(x)$. Since

$$ \Gamma_{jl}^{k}=\frac{\partial y^{\alpha}}{\partial x^{j}}\frac {\partial y^{\beta}}{\partial x^{l}} \frac{\partial x^{k}}{\partial y^{\gamma}}\hat{\Gamma}_{\alpha\beta}^{\gamma}+\frac{\partial x^{k}}{\partial y^{\alpha}} \frac{\partial^{2} y^{\alpha}}{\partial x^{j}\,\partial x^{l}}, $$

(3.8)

the initial value problem (3.7) can be rewritten as

$$ \textstyle\begin{cases}{ \frac{\partial^{2}y^{\alpha }}{\partial t ^{2}}=g^{jl}(\frac{\partial^{2} y^{\alpha}}{\partial x^{j}\,\partial x^{l}}+\frac{\partial y^{\beta}}{\partial x^{j}}\frac {\partial y^{\gamma}}{\partial x^{l}}\hat{\Gamma}_{\beta\gamma }^{\alpha}-\frac{\partial y^{\alpha}}{\partial x^{k}}\hat{\Gamma }_{jl}^{k})}, \\ { y^{\alpha}(x,0)= x^{\alpha},\quad\quad y^{\alpha }_{t}(x,0)= 0.} \end{cases} $$

(3.9)

Obviously, (3.9) is an initial value problem for a strictly hyperbolic system. On the other hand, we note that

$$ \begin{aligned}[b] \Delta_{\hat{g}} \hat{u}&= \hat{g}^{\alpha\beta}\nabla_{\alpha }\nabla_{\beta}u= \hat{g}^{\alpha\beta}\biggl(\frac{\partial^{2}\hat {u}}{\partial y^{\alpha}\,\partial y^{\beta}}-\hat{\Gamma}^{\gamma }_{\alpha\beta} \frac{\partial\hat{u}}{\partial y^{\gamma }}\biggr) \\ &= g^{kl}\frac{\partial y^{\alpha}}{\partial x^{k}}\frac{\partial y^{\beta}}{\partial x^{l}}\biggl\{ \frac{\partial }{\partial y^{\alpha}}\biggl(\frac{\partial u}{\partial x^{i}}\frac {\partial x^{i}}{\partial y^{\beta}}\biggr)- \frac{\partial u}{\partial x^{i}}\frac{\partial x^{i}}{\partial y^{\gamma}}\hat{\Gamma }^{\gamma}_{\alpha\beta} \biggr\} \\ &= g^{kl}\frac{\partial^{2}u}{\partial x^{k}\,\partial x^{l}}+g^{kl}\frac{\partial y^{\alpha}}{\partial x^{k}} \frac {\partial y^{\beta}}{\partial x^{l}}\frac{\partial u}{\partial x^{i}}\frac{\partial^{2}x^{i}}{\partial y^{\alpha}\,\partial y^{\beta }}-g^{kl} \frac{\partial u}{\partial x^{i}}\biggl(\Gamma^{i}_{kl}-\frac {\partial x^{i}}{\partial y^{\gamma}} \frac{\partial^{2}uy^{\gamma }}{\partial x^{k}\,\partial x^{l}}\biggr) \\ &=g^{ij}\nabla_{i}\nabla_{j}u= \Delta_{g} u. \end{aligned} $$

(3.10)

We have

$$\begin{aligned}& \frac{\partial u}{\partial t}=\frac{\partial\hat{u}}{\partial t}+\frac{\partial\hat{u}}{\partial y^{k}}\frac{\partial y^{k}}{\partial t}, \end{aligned}$$

(3.11)

$$\begin{aligned}& u_{tt} = \frac{\partial^{2}\hat{u}}{\partial y^{\alpha}\,\partial y^{\beta}} \frac{\partial y^{\alpha}}{\partial t}\frac{\partial y^{\beta}}{\partial t}+2\frac{\partial^{2}\hat {u}}{\partial t\,\partial y^{\beta}}\frac{\partial y^{\beta}}{\partial t}+ \frac{\partial^{2}\hat{u}}{\partial t^{2}}+\frac{\partial\hat {u}}{\partial y^{\alpha}}\frac{\partial^{2}y^{\alpha}}{\partial t^{2}} \\& \hphantom{u_{tt}} = \frac{\partial^{2}\hat{u}}{\partial y^{\alpha }\partial y^{\beta}}\frac{\partial y^{\alpha}}{\partial t}\frac {\partial y^{\beta}}{\partial t}+2\frac{\partial^{2}\hat {u}}{\partial t\,\partial y^{\beta}} \frac{\partial y^{\beta}}{\partial t}+\Delta_{\hat{g}} \hat{u}+\frac{\partial u}{\partial x^{k}} \frac {\partial x^{k}}{\partial y^{\alpha}}\frac{\partial^{2}y^{\alpha }}{\partial t^{2}} \\& \hphantom{u_{tt}}= \Delta_{g} u +\frac{\partial u}{\partial x^{k}}g^{ij}\bigl( \Gamma^{k}_{ij}-\hat{\Gamma}^{i}_{kl} \bigr)+\frac{\partial ^{2}\hat{u}}{\partial y^{\alpha}\,\partial y^{\beta}}\frac{\partial y^{\alpha}}{\partial t}\frac{\partial y^{\beta}}{\partial t}+2\frac {\partial^{2}\hat{u}}{\partial t\,\partial y^{\beta}} \frac{\partial y^{\beta}}{\partial t} \\& \hphantom{u_{tt}} = g^{ij}\biggl(\frac{\partial^{2}u}{\partial x^{i}\,\partial x^{j}}-\Gamma^{k}_{ij} \frac{\partial u}{\partial x^{k}}\biggr)+\frac {\partial u}{\partial x^{k}}g^{ij}\bigl( \Gamma^{k}_{ij}-\hat{\Gamma }^{i}_{kl} \bigr)+\frac{\partial^{2}\hat{u}}{\partial y^{\alpha}\partial y^{\beta}}\frac{\partial y^{\alpha}}{\partial t}\frac{\partial y^{\beta}}{\partial t}+2\frac{\partial^{2}\hat{u}}{\partial t\,\partial y^{\beta}} \frac{\partial y^{\beta}}{\partial t} \\& \hphantom{u_{tt}} = g^{ij}\frac{\partial^{2}u}{\partial x^{i}\,\partial x^{j}}-\frac{\partial u}{\partial x^{k}}g^{ij}\hat{ \Gamma }^{i}_{kl}+\frac{\partial^{2}\hat{u}}{\partial y^{\alpha}\,\partial y^{\beta}}\frac{\partial y^{\alpha}}{\partial t} \frac{\partial y^{\beta}}{\partial t}+2\frac{\partial^{2}\hat{u}}{\partial t\,\partial y^{\beta}}\frac{\partial y^{\beta}}{\partial t}. \end{aligned}$$

(3.12)

By the standard theory of hyperbolic equations (see [14]), we obtain a local existence and uniqueness result. Thus, the proof of Theorem 1.2 is completed. □