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


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:

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

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.


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

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

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

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

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

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

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

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

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

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

where \(\frac{1}{p}+\frac{1}{q}=1\).

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

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


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

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

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

which implies, via integration by parts,

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

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

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

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

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

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

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

we can reduce (2.10) to

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

Since \(\varphi_{R}\) is radial, we have

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

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

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

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

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

By (2.3), we have

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

which yields

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

Therefore, as \(\varphi_{R}, \eta_{R}\leq1\),

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

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

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

By Lemma 1, we obtain

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


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

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

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

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

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

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

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

By (2.20), (2.24) yields

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

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

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

which yields

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

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

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

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

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

This shows

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


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


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

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

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

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 \(\{x^{i}\}\) (\(1\leq i\leq n\)).


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

the wave equation (1.1) can be equivalently rewritten as

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


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

it follows that

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

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

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

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


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

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

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

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

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

the initial value problem (3.7) can be rewritten as

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

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

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

We have

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

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. □


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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).

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Ru, Q. A nonexistence result for a nonlinear wave equation with damping on a Riemannian manifold. Bound Value Probl 2016, 198 (2016).

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