Open Access

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

Boundary Value Problems20162016:198

https://doi.org/10.1186/s13661-016-0705-5

Received: 8 June 2016

Accepted: 31 October 2016

Published: 9 November 2016

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.

Keywords

nonexistence wave equation

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 [37]. There are more recent contributions to the discussion of the test function method; we refer to [811] 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]):
  1. (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;

     
  2. (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. □

Declarations

Acknowledgements

This research was supported by the Fundamental Research Funds for the Central Universities (2014QNA63) and the Natural Science Foundation of Jiangsu Province (BK20150172) and the TianYuan Special Funds of the National Natural Science Foundation of China (11526191).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, China University of Mining and Technology

References

  1. Todorova, G, Yordanov, B: Critical exponent for a nonlinear wave equation with damping. C. R. Acad. Sci., Sér. 1 Math. 300, 557-562 (2000) MathSciNetMATHGoogle Scholar
  2. Zhang, QS: A blow-up result for a nonlinear wave equation with damping: the critical case. C. R. Acad. Sci., Sér. 1 Math. 333, 109-114 (2001) MathSciNetMATHGoogle Scholar
  3. Levine, HA: The role of critical exponents in blow-up theorems. SIAM Rev. 32, 262-288 (1990) MathSciNetView ArticleMATHGoogle Scholar
  4. Zhang, QS: A new critical phenomenon for semilinear parabolic problems. J. Math. Anal. 219, 125-139 (1998) MathSciNetView ArticleMATHGoogle Scholar
  5. Kong, D-X, Liu, K: Wave character of metrics and hyperbolic mean curvature flow. J. Math. Phys. 48, 1-14 (2007) MathSciNetView ArticleGoogle Scholar
  6. He, C-L, Kong, D-X, Liu, K: Hyperbolic mean curvature flow. J. Differ. Equ. 246, 373-390 (2009) MathSciNetView ArticleMATHGoogle Scholar
  7. Zhu, X-P: Lectures on Mean Curvature Flows. Stud. Adv. Math., vol. 32. Am. Math. Soc./International Press, Providence (2002) MATHGoogle Scholar
  8. Zhang, QS: Blow-up results for nonlinear parabolic equations on manifolds. Duke Math. J. 97, 515-539 (1999) MathSciNetView ArticleMATHGoogle Scholar
  9. Wakasugi, Y: Critical exponent for the semilinear wave equation with scale invariant damping. In: Ruzhansky, M, Turunen, V (eds.) Trends in Mathematics. Trends in Mathematics, pp. 375-390. Birkhäuser, Basel (2014) Google Scholar
  10. Li, X: Critical exponent for the semilinear wave equation with critical potential. Nonlinear Differ. Equ. Appl. 20, 1379-1391 (2013) MathSciNetView ArticleMATHGoogle Scholar
  11. Lin, J, Nishihara, K, Zhai, J: Critical exponent for the semilinear wave equation with time-dependent damping. Discrete Contin. Dyn. Syst., Ser. A 32, 4307-4320 (2012) MathSciNetView ArticleMATHGoogle Scholar
  12. Zhang, QS: Blow-up results for nonlinear parabolic equations on manifolds. Duke Math. J. 97, 515-539 (1999) MathSciNetView ArticleMATHGoogle Scholar
  13. DeTurck, D: Some regularity theorems in Riemannian geometry. Ann. Sci. Éc. Norm. Supér. 14, 249-260 (1981) MathSciNetMATHGoogle Scholar
  14. Hörmander, L: Lectures on Nonlinear Hyperbolic Differential Equations. Math. Appl., vol. 26. Springer, Berlin (1997) MATHGoogle Scholar

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