Introduction

Let $\Omega \subset R^{N} (N \geq 3)$ be an exterior domain with compact smooth boundary $\partial\Omega$. Without loss of generality we may assume $0 \notin \bar{\Omega}$. In this paper we are concerned with the initial-boundary value problem for the linear dissipative wave equation:

$$v_{tt}(t,x) - \Delta v(t,x) + v_{t}(t,x) = 0,\quad (t,x) \in (0,\infty)\times\Omega,$$ (1.1)

$$v(0,x) = v_{0}(x),\quad v_{t}(0,x) = v_{1}(x), \quad x \in \Omega,$$ (1.2)

$$v\vert_{\partial\Omega} = 0,\quad t \in (0,\infty),$$ (1.3)

and the semilinear dissipative wave equation:

$$u_{tt}(t,x) - \Delta u(t,x) + u_{t}(t,x) = \vert u(t,x)\vert^{p},\quad (t,x) \in (0,\infty)\times\Omega,$$ (1.4)

$$u(0,x) = u_{0}(x),\quad u_{t}(0,x) = u_{1}(x), \quad x \in \Omega,$$ (1.5)

$$u\vert_{\partial\Omega} = 0,\quad t \in (0,\infty).$$ (1.6)

Throughout this paper, $\Vert\cdot\Vert_{q}$ and $\Vert\cdot\Vert_{H^{1}}$ mean the usual $L^{q}(\Omega)$-norm and $H_{0}^{1}(\Omega)$-norm, respectively, and in particular, we set $\Vert\cdot\Vert = \Vert\cdot\Vert_{2}$ for simplicity. Furthermore, we adopt

$$(f,g) = \int_{\Omega}f(x)g(x)dx$$

as the usual $L^{2}(\Omega)$-inner product. The total energy $E_{v}(t)$ to the equation (1.1) and (1.4) is defined by

$$E_{v}(t) = \frac{1}{2}\Vert v_{t}(t,\cdot)\Vert^{2} + \frac{1}{2}\Vert\nabla v(t,\cdot)\Vert^{2}.$$

The first purpose of this job is to derive certain decay estimates for the total energy $E_{v}(t)$ and the $L^{2}$-norm of a solution $v(t,x)$ to the linear problem (1.1)-(1.3) faster than the usual one through the (modified) time integral method developed in Ikehata-Matsuyama [3]. In that occasion, we do assume some further restrictions on the initial data as $\vert x\vert \to +\infty$. On the contrary, Ikehata-Matsuyama [3] and Saeki-Ikehata [11] adopted another weight condition on the initial data. For the exterior mixed problem, these restrictions on the initial data seem to be new (for conditions on the initial data with the compact support, see Dan-Shibata [1]). For these restrictions as $\vert x\vert \to +\infty$ on the initial data to the "Cauchy problem" of the equation (1.1), there are lots of related results and we refer the reader to Kawashima-Nakao-Ono [6], Matsumura [8] and the references thererein.

The second purpose of this paper is to determine the exponent $p$ of the semilinear exterior problem (1.4)-(1.6) for which the small data global existence property holds. Very recently, in Ikehata-Miyaoka-Nakatake [4] and Todorova-Yordanov [12] they have derived such a critical (Fujita type) exponent $p_c = 1+2/N$ to the Cauchy problem of (1.4) in the framework of $L^{1}\times L^{1}$ assumption on the initial data and of the initial data with compact support, respectively (for another type of critical exponents like $p_{c} = 1+2m/N$ for the Cauchy problem of (1.4) with $L^{m}\times L^{m}$ assumption on the initial data, see also Ikehata-Ohta [5]). These works are fully based on the decay estimates for the linear equations due to Matsumura [8] and Kawashima-Nakao-Ono [6]. Thus, it seems to be difficult to apply those decay estimates for the linear equations due to [6] and [8] to the present exterior mixed problem (1.4)-(1.6). On the other hand, in the framework of the compactly supported initial data Ikehata [2] has already constructed a small global solution to the exterior problem (1.4)-(1.6) with the power $1+ 6/(N+2) < p \leq N/(N-2) (N = 3)$ or $1+ 6/(N+2) < p < +\infty (N = 2)$. His result is based on the decay estimates for the linear equations which are developed in [3] and [11]. By using decay estimates for the linear equations developed in the former part instead of those developed in [3] and [11], we can exclude the compactness of the support on the initial data as in [2] to the problem (1.4)-(1.6) with further relaxed exponent, and we can also treat the higher dimensional case $N = 4,5$ (for another exponent $p_{c} = 1+4/N$, see Nakao-Ono [10]).

In the following, we set

$$I_{0,u} = \Vert u_{0}\Vert_{H^{1}} + \Vert u_{1}\Vert + \Vert u_{0} + u_{1}\Vert_{2N/(N+2)}.$$

Then our first result reads as follows.

Theorem 1.1   Let $N \geq 3$. For each $[v_{0},v_{1}] \in (H_{0}^{1}(\Omega)\cap L^{2N/(N+2)}(\Omega))\times (L^{2}(\Omega)\cap L^{2N/(N+2)}(\Omega))$, the weak solution $v \in C([0,+\infty);H_{0}^{1}(\Omega))\cap C^{1}([0,+\infty);L^{2}(\Omega))$ to the linear problem (1.1)-(1.3) satisfies the decay estimates:

$$\Vert v(t,\cdot)\Vert^{2} \leq CI_{0,v}^{2}(1+t)^{-1},$$

$$\Vert v_{t}(t,\cdot)\Vert^{2}+ \Vert\nabla v(t,\cdot)\Vert^{2} \leq CI_{0,v}^{2}(1+t)^{-2}$$ (1.7)

with some generous constant $C > 0$.

Next we shall make some assumptions before treating the semilinear problem (1.4)-(1.6).

$${\bf 1 + \displaystyle{\frac{4}{N+2}} < p \leq \displaystyle{\frac{N}{N-2}}},$$ (1.8)

Now based on these decay estimates for the linear equations as in Theorem 1.1 our second result to the semilinear problem reads as follows.

Theorem 1.2   Let $N = 3,4,5$. Under the assumption (1.8), there exists a real number $\delta > 0$ such that if the initial data $[u_{0},u_{1}] \in (H_{0}^{1}(\Omega)\cap L^{2N/(N+2)}(\Omega))\times (L^{2}(\Omega)\cap L^{2N/(N+2)}(\Omega))$further satisfies $I_{0,u} < \delta$, the problem (1.4)-(1.6) has a unique global solution $u \in C([0,+\infty);H_{0}^{1}(\Omega))\cap C^{1}([0,+\infty);L^{2}(\Omega))$ satisfying

$$\Vert u(t,\cdot)\Vert^{2} \leq CI_{0,u}^{2}(1+t)^{-1},$$

$$\Vert u_{t}(t,\cdot)\Vert^{2}+ \Vert\nabla u(t,\cdot)\Vert^{2} \leq CI_{0,u}^{2}(1+t)^{-2}$$

with some generous constant $C > 0$.

Remark 1.1   In the case when $N = 3$, Theorem 1.2 completely contains the result in Ikehata [2]. Furthermore, (1.8) is more relaxed condition than that in [2]. For $N = 2$, this is completely open. Note that in the case when $N = 2$, formally we get $1+ 4/(N+2) = 1+2/N$. On the other hand, the results in Theorems 1.1 and 1.2 seem sharp if we take $m = 2N/(N+2)$ in Ikehata-Ohta [5, Theorem 1.2]. Finally, for the nonexistence of global solution to the equation (1.4) in ${\bf R}^{N}$ with power $p$ less than the critical exponent we refer to Li-Zhou [7], Todorova-Yordanov [12] and Ikehata-Ohta [5].