数理解析研究所講究録
1495 巻 2006 年 92-102

92

Large-time behavior of solutions forthe damped wave equation
早稲田大学政治経済学術院西原健二 (Kenji Nishih\‘arra)
School of Political Science and Economics,
Waseda University

Dedicated to Professor Tai-Ping Liu on his 60th birthday

1

Introduction

We consider the Cauchy problem for the semilinear damped wave equation$u_{tt}-\Delta u+u_{t}=f(u)$
$(D)$

,

$(t, x)\in \mathrm{R}_{+}\mathrm{x}\mathrm{R}^{N}$$\{$$(u, u_{t})(0, x)=(u_{0}, u_{1})(x)$,$x\in \mathrm{R}^{N}$,

corresponding to the semilincar heat equation$\phi_{t}-\Delta\phi=f(\phi)$

$(H)$

$\{$

$\phi(0, x)=\phi_{0}(x)$,,$(t, x)\in \mathrm{R}_{+}\mathrm{x}\mathrm{R}^{N}$$x\in \mathrm{R}^{N}$.

Many mathematicians have recognized that the damped wave cquation approaches to. Therefore, in the first part we treat
thc heat equation in some sense as(I) the linear damped wave equation and heat equation, that is, $f(u)\equiv 0$ ,and show precisely how the solutions to those equations behave as $tarrow\infty$ . Basedon these results we consider the auchy problem (D) for the semilincar damped wave
etc. withcquation rclated to (H). The semilinear term $f(u)$ is, it works as a sourcing term. On thethe exponent $\rho>1$ . Whcn
$f(u)=-|u|^{\rho-1}u$, itan absorbing term. So, in the second andother hand, whentreat
respectively
we
parts
third
,
(II) the semilincar problom (D) with a sourcing term, that is,term, that is, $f(\mathrm{u})=-|u|^{\rho-1}u$ ,(III) the semilinear problem (D) with an$tarrow\infty$$C$$\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{i}\mathrm{c},\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{y}\pm|u|^{\rho-1}u,$$\pm|u|^{\rho}$$f(u)=|u|^{\rho-1}u\mathrm{o}\mathrm{r}\pm|u|^{\rho}$$\mathrm{d}\mathrm{o}\mathrm{e}\mathrm{S}.r‘ 1\mathrm{b}^{\backslash }$$f(u)=|u|^{\rho-1}u\mathrm{o}\mathrm{r}\pm|u|^{\rho}$$\mathrm{a}\mathrm{b}\mathrm{s}\mathrm{o}\mathrm{r}\}_{)}\mathrm{i}\mathrm{n}\mathrm{g}$with relation to (H).
Original motivation to investigate (D) with (H) is coming from the results in the model
system of 1-dimensional compressiblc flow through porous media$v_{t}-u_{x}=0$

,

$\{$

$u_{t}+p(v)_{x}=-\alpha u$,$(0, x)=(x)arrow$ ,with$xarrow\pm\infty$,with $v_{\pm}>0$ , where $v(\geq 0),$ and arc, respectively, the specific volume, the velocity andthe pressure with $p’(v)<0(v>0)$ , and ae is a positive constant. The system wa.s firstconsidered by Nishida [24]. In Hsiao and Liu [14] they reformulated the problcm to the
$u$

$p$

93
second order wave cquation with dainping and asserted that the solution $(v, u)$ behavesaswhich is a solution to the parabolic systcm$(\tilde{v},\tilde{u})$$\tilde{v}_{t}-\tilde{u}_{x}=0$

,

$\{$

.

$p(\tilde{v})_{T}$$=-\alpha\tilde{u}$,due to the Darcy law. This, roughly speaking, implies that the dalnped wive equation isnear to the corresponding parabolic equation, in other words, the damped wave equation
has the diffusive vtructure as $tarrow\infty$ . So, we want to obtain the details how near ag$tarrow\infty$

2.

Linear damped wave equation and heat equation

By $S_{N}(t)g$ and$P_{N}(t)\phi_{0}$$v_{tt}-\Delta v+v_{t}=()$(2.1)

and the solution, we respectively denote the solution $v(t, x)$ to,$(t, x)\in \mathrm{R}_{+}\mathrm{x}\mathrm{R}^{N}$$(v, v_{t})(0, x)=(0, g)(x),$$\phi(t, x)$$\phi_{t}-\Delta\phi=0$

Then the solution$x\in \mathrm{R}^{N}$,$(t, x)\in \mathrm{R}_{+}\mathrm{x}\mathrm{R}^{N}$

$\{$

$u$

,

to

(2.2)

(2.3)

,

$\{$

,

.

$\phi(0, x)=\phi_{0}(x)$,$x\in \mathrm{R}^{N}$$u_{tt}-\Delta u+u_{t}=\mathrm{t}\mathrm{I}$,$(t, x)\in \mathrm{R}_{+}\mathrm{x}\mathrm{R}^{N}$

to
,

$\{$

$(u, u_{t})(0, x)=(u_{0}, u_{1})(x)$,$x\in \mathrm{R}^{N}$is given by(2.4)$u(t, \cdot)=S_{N}(t)(u_{0}+u_{1})+\partial_{t}(S_{N}(t)u_{0})$.

estimate on the difference of $S_{N}(t)g$
Our aim in this section is to show the precise
to (2.3) and (2.2). We
and $P_{N}(t)g$ , and so, the cliffercnce of solutions $u(t, x)$ andtreat the cases $N=1,2,3$ , mainly $N=3$ .
The solution $S_{N}(t)g$ has the explicit formula, which is found in Courant and Hilbert[3]. We decompose the formula to the following form:$L^{p_{-}}L^{q}$$\emptyset(t, x)$$S_{N}(t)g=e^{-t/2}W_{N}(t)g+J_{N}(t)g$ ,(2.5)

where $W_{N}(t)g$ is a solution to the linear wave equation without dissipation$w_{\iota\iota}-\Delta w=0$

(2.6)

,

$(t, x)\in \mathrm{R}_{+}\mathrm{x}\mathrm{R}^{N}$$\{$$(w, w_{t})(0, x)=(0, g)(x)$ ,$x\in \mathrm{R}^{N}$

,

.

94

Hence, when $N=1,2,3$ , we have the following form,

$S_{1}(t)g= \frac{e^{-\iota/2}}{2}.\int_{|z|\leq\downarrow}g(x+z)dz+\frac{e^{-t/2}}{2}\int_{|z|\leq\iota}(I_{0}(\frac{\sqrt{t^{2}-|z|^{2}}}{2})-1)g(x+z)dz$$=:e^{-t/2}W_{1}(t)g+J_{1}(t)g$$S_{2}(t)g= \frac{e^{-t/2}}{2\pi}\int_{|z|\leq\iota}\frac{g(x+z)}{\sqrt{t^{2}-|z|^{2}}}dz+\frac{e^{-t/2}}{2\pi}\int_{|z|\leq\iota}’\frac{(^{\tau}()\mathrm{s}\mathrm{h}(\frac{\sqrt{t^{2}-|z|^{2}}}{2})-1}{\sqrt{t^{2}-|z|^{2}}}g(x+z)dz$,$=:e^{-t/2}W_{2}(t)g+J_{2}(t)g$,$S_{3}(t)g=e^{-t/2} \frac{t}{4\pi}\int_{S^{2}}g(x+t\omega)d\omega+\frac{e^{-t/2}}{4\pi}\int_{|z|\leq\iota}\frac{I_{1}(\frac{\sqrt{t^{2}-|z|^{2}}}{2})}{2\sqrt{t^{2}-|z|^{2}}}g(x+z)dz$$=:e^{-t/2}W_{2}(t)g+J_{2}(t)g$,by the D’Alembert, Poisson and Kirchhoff forrnulas for the wave equation in each dimension. Then we have the following estimates.

Proposition 2.1 (cf. [21, 16, 25]) For$l$$\leq q\leq p\leq\infty$

and $1\leq N\leq 3$ , it holds that$||(J_{N}(t)-P_{N}(t))g||_{L^{\mathrm{p}}}\leq Ct^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{p})-1}||g||_{L^{q}}$,$||\partial_{t}(J_{N}(t)g)||_{L^{\mathrm{p}}}\leq C(1+t)^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{\mathrm{p}})-1}||g||_{L^{q}}$$t\geq t_{0}>()$,$t\geq 0$,.

Since it is well-known that
(2.7)$||’\partial_{t}(P_{N}(t)g)||_{L^{\mathrm{p}}}\leq Ct^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{p})}||g||_{L^{q}},$$t\geq t_{0}>0$,

Proposition 2.1 mcans that $J_{N}(t)g$ behave as $P_{N}(t)g$ as $tarrow\infty$ . By (2.5) and (2.4) thesolution $u(t, x)$ to (2.3) is given by(2.8)$\mathrm{u}(t, \cdot)=J_{N}(t)(u_{0}+u_{1})+\partial_{t}(J_{N}(t)u_{0})+e^{-t/2}\mathrm{W}(t;u_{0}, u_{1})$

,

where
$e^{-\iota/2}\mathrm{W}(t;u_{0}, u_{1})$$=e^{-t/2}W_{N}(t)(u_{0}+u_{1})+\partial_{t}(e^{-t/\mathrm{z}}W_{N}(t)u_{0})$(29)$=e^{-t/2} \{W_{N}(t)(\frac{1}{2}u_{0}+u_{1})+\partial_{t}(W_{N}(t)u_{0})\}$.

Hcnce we have the following IP-L estimate.

Theorem 2.1 (cf. [21, 16, 25]) Let $u(t, x)$ be a solution to (2.3) andto (2.2) with $\phi_{0}=u_{0}+u_{1}$ . Then, for $1\leq q\leq p\leq\infty$ , it holds that$\phi(t, x)$

be a solu-$t^{l}i,on$$||u(t, \cdot)-\phi(t, \cdot)-e^{-l/2}\mathrm{W}_{0}(t;u_{0}, u_{1})||_{L^{\mathrm{p}}}\leq Ct^{-\frac{N}{\lambda}(\frac{1}{q}-\frac{1}{\mathrm{p}})-1}||u_{0},$$u_{1}||_{L^{q}}$,$t\geq t_{0}>0$.

95

For this kind of cstimate, see also Hosono and Ogawa [13] in case of $N=2$ , Narazaki[23] in case of general dimension, Ikehata and Nishihara [15], Chill and Haraux [2] in the
abstract setting.
By Proposition 2.1 $J_{N}(t)g$ bchaves as $P_{N}(t)g$ a.s $tarrow\infty$ , and hcncc we call $J_{N}(t)g$ ,““the wave part”.
the parabolic part” of $S_{N}(t)g$ , while we call
Therefore, the solution $u(t, x)$ to the damped wave equation decomposed to the sum of
only, the wave
“the wave part” and “the parabolic part”. Since we assume
part may include the singularity in case of $N\geq 2$ . Thus, by Theorem 2.1 we can say that$e^{-t/2}\mathrm{W}_{0}(t;u_{0}, u_{1})$$\partial_{t}(J_{N}(t)g)$$u_{0},$$u_{1}\in L^{q}$

if we remove the singularity, though its strength decays exponentially, then the solution
to the damped rvave equation behaves as that to the $hmtequation_{f}$$\bullet$in other words,$\bullet$

if we resolve the regulanty problem from the solution to the damped wave equation,then we may show that the solution behaves as that to the heat equation.
Basicestimate was given by Matsumura [22] as$L^{\mathrm{P}_{-}}L^{q}$$||S_{N}(t)g||_{L^{\mathrm{p}}}\leq C(1+t)^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{\mathrm{p}})}||g||_{L^{q}}+Ce^{-\alpha t}||g||II^{|\#|}$

for some constant $\alpha>0$ , which is also interpreted on the same line.
Thus, when wc consider the semilinear problems (D) in the next two sections, it
becomes important to resolve the regularity problem. The parabolic problem (H) has asmoothing effect and a InaximtlIn principle and hence the nonnegativity of the solution
for the noImegativc data, roughly speaking. Our aim is to investigatc the damped wave
equation, which generally has not the nonnegativity of the solution nor the smoothing
effect. These raise up the difficulties in (D).

3

Semilinear problem with a sourcing term

In this section we consider (D)(3.1)$u_{tt}-\Delta u+u_{t}=f(u)$,$(t, x)\in \mathrm{R}_{+}\cross \mathrm{R}^{N}$$\{$$(u, u_{l})(0, x)=(u_{0},u_{1})(x),$$x\in \mathrm{R}^{N}$,

related to(3.2)$\phi_{t}-\Delta\phi=f(\phi)$,$(t, x)\in \mathrm{R}_{+}\mathrm{x}\mathrm{R}^{N}$$\{$$\phi(0, x)=\phi_{0}(x),$$x\in \mathrm{R}^{N}$

,

where
(3.3)

$f(u)=|u|^{\rho-1}u$or$\pm|u|^{\rho}$.

therc are long literatures. Summing these up, we know that,
For (3.2) with
globally exists for small data, and that
if $\rho>\rho_{c}(N):=1+\frac{2}{N}$ , then the solution$f(\phi)=|\phi|^{\rho-1}\phi$$\emptyset(t, x)$

96. For these rcsults
blows up within a finite time if
the positive solution
refer Fujita [8], Hayakawa [11], Weisseler [29] etc. and thc scrvcy papcrs Lcvinc [19], Dengis called the Fujita exponent namcd after his
and Levine [4]. The critical exponent
work [8].
From the discussion in the preccding section we expect that the solution $u(t, x)$ to, Todorova and Yordanov [28] have shown that, if(3.1) behaves as to (3.2). In$\rho_{c}(N)<\rho<\frac{N+2}{N-2}$ , then the solution
to (3.1) with $f(u)=|u|^{\rho}$ globally exists for small
with compact support, and that, if $\rho<\rho_{c}(N)$ , then the local solution blows
data
Zhang [30] has
up within a finite time for suitable data. In the critical caseof solutions
globalexistence
the
small
data
up.
Note
that
to
blow
shown the solution$f(u)=\pm|\mathrm{u}|^{\rho-1}u,$. But, for the blow-up result they have treatedis available for$f(u)=|u|^{\rho}$ to get positivity.
For the related topics, when $N=1$ , Gallay and Raugel [10] has shown the agymptotic
profile in the supercritical casc, aPplying the scaling variables. When $N=1,2$ , Li andZhou [20] obtained the cstimatc of the blow-up time in the critical and subcritical case,using the explicit formula of solutions.
We present two theorems in case of $N=3$ , based on the preccding discussions. The
weak solution $u(t, x)$ to (3.1) is defined by the solution to$\phi(t, x)$$p\leq\rho_{\mathrm{c}}(N)$$\rho_{c},(N)$$\mathrm{f}\mathrm{a}\mathrm{c},\mathrm{t}$$\phi$$u$$u$$u_{0},$ $u_{1}$$\rho=\rho_{\mathrm{t}j}(N)$$\pm|u|^{\rho}$$=S_{N}(t)(u_{0}+u_{1})+ \partial_{t}(S_{N}(t)u_{0})+\int_{0}^{t}S_{N}(t-\tau)f(u)(\tau)d\tau$$u(t, \cdot)$(3.4)$=S_{N}(t)(u_{0}+u_{1})+\partial_{t}(S_{N}(t)u_{0})$.

$+ \int_{0}^{t}e^{-\frac{t-\tau}{2}}W_{N}(t-\tau)f(u)(\tau)d\tau+\int_{0}^{l}J_{N}(t-\tau)f(u)(\tau)d\tau$Also, denoting the Sobolcv space by
for $1\leq p\leq\infty$ , we have the followings.

$W^{\mathrm{m},p}=W^{m,p}(\mathrm{R}^{N})=\{f;\partial_{qj}^{\prime i}f\in L^{\mathrm{p}}(i=0,1, \cdots, m)\}$Theorem 3.1 ([25]) Let $N=3$ andThen, when $\rho>\rho_{c}(3)=5/3$ , there exists a uniqueto (3.1)

be small.
solution $u\in C([(), \infty))L^{1}\cap L^{\infty})$$(u_{0}, u_{1})\in(W^{1,\infty}\cap W^{1,1})\mathrm{x}(L^{\infty}\cap L^{1})=:Z_{0}$$\uparrow veak$$satisf\uparrow/ing$$||u(t, \cdot)||_{L^{p}}\leq C(1+t)^{-\frac{8}{2}(1-\frac{1}{\mathrm{p}})}||u_{0},$$(1\leq p\leq\infty)$$u_{1}||_{Z_{0}},$Remark 3.1. Since we assume suitable regularity, i.e.
data yields thc global existence theorem evcn for the exponent.
critical exponent$u_{0},$$u_{1}\in Z_{0}$.

, only smallness of thethan the Sobolev$\}_{)}\mathrm{i}\mathrm{g}\mathrm{g}\mathrm{e}\mathrm{r}$$1+ \frac{4}{N-2}$

Theorem 3.2 ([26]) Let $f(u)=|u|^{\rho}$ (resp. $f(u)=|u|^{\rho-1}u$ ) and$(\epsilon u_{0}, \epsilon u_{1})\in Z_{0},0<\epsilon<<1with/\mathrm{R}^{3(u_{0}}+u_{1})(x)dx>0(r\cdot es_{l^{J}}$. Then, whenwith(3.1) blows up within a finite tirne, and the blow-up time$u_{1}(x)\geq 0,$(3.5)$\int_{\mathrm{R}^{3}}u_{1}(x)dx>0)$$\mathrm{e},\mathrm{x}\mathrm{p}(C\epsilon^{-\alpha})$$p\leq p_{c}(3)$$\rho=:1+\alpha=\rho_{\mathrm{c}}(N)$$C\epsilon^{2\alpha/(2-N\alpha)}$$p=1+\alpha<\rho_{\mathrm{r}_{d}}(N)$,,be replaced by$(0, \epsilon u_{1})\in Z_{0},0<\epsilon<<1$, the time local solutionof is estimated as$T_{\epsilon}$$T_{\epsilon}\leq\{$.

$(u_{0}, u_{1})$$u$$(N=3)$ .

$u$

to

97
Remark 3.2. When the data iswith $u_{1}(x)\geq 0,$$f(u)=|u|^{\rho-1}u$solution to (3.1) withbecomcs positive by$((), \epsilon u_{1})$$u$$\int_{\mathrm{R}^{3}}u_{1}(x)dx>0$$\mathrm{t}_{}\mathrm{h}\mathrm{e}$, the local

solution formula$u(t, \cdot)=S_{N}(t)u_{1}+\int_{0}^{t}S_{N}(t-\tau)|u|^{\rho-1}u(\tau, \cdot)d\tau$.

For $f(u)=-|u|^{\rho}$ we don’t know the blow-up result.
is available even in $N=1,2$ , which was givcn by Li and
The estimate (3.5) ofZhou [20]. In fact, to prove Theorem 3.2 we use the explicit formula (3.4) and apply thefollowing lemma given in [20].
$T_{\epsilon}$

Lemma 3.1 If $I(t)$ satisfies,$I”(t)+I’(t)\geq c_{0^{\frac{I^{1+\alpha}(t)}{(1+l)^{\beta}}}}$

$t>0$

$\{$

$I(\mathrm{O})\geq\epsilon>0$,$I’(0)\geq 0$.

with $\alpha>0,0\leq\beta\leq 1$ , then $I(t)$ blows up in a finite time. More Pretiisely, the life spanis estimated from above as$T_{\epsilon}$

cxp$\beta=1$$(C\epsilon^{-\alpha})$$T_{\epsilon}\leq\{$$0\leq\beta<1$$C\epsilon^{-\frac{\alpha}{1-\beta}}$

4.

Semilinear problem with an absorbing term

In this section we consider(4.1)$u_{tt}-\triangle u+u_{t}+|u|^{\rho-1}u=0$$\{$,$(t, x)\in \mathrm{R}_{+}\mathrm{x}\mathrm{R}^{N}$,$(u, u_{t})((), x)=(u_{0}, u_{1})(x)$,$x\in \mathrm{R}^{N}$$\phi_{t}-\Delta\phi+|\phi|^{\rho-1}\phi=0$,$(t, x)\in \mathrm{R}_{+}\mathrm{x}\mathrm{R}^{N}$

related to
(4.2)

$\{$

$\phi(0, x)=\phi_{0}(x)$,$x\in \mathrm{R}^{N}$.

Again we remind the results on (4.2). For big data , in the supercritical case the solution, where $G(t, x)=(4\pi t)^{-N/2}$ cxp $(-|x|^{2}/4t)$ is theasapproaches toGauss kernel and the constant is given by$\phi_{0}$$\phi(t, x)$$tarrow\infty$$\theta_{0}G(t, x)$$\theta_{0}$$\theta_{0}=\int_{\mathrm{R}^{N}}\phi_{0}(x)dx-\int_{0}^{\infty}\int_{\mathrm{R}^{N}}|\phi|^{\rho-\perp}\phi(t, x)dxdt$.

approaches to $\theta_{0}G(t, x)(\log t)^{-N/2}$ for some constant . In the
In the critical casesubcritical casc there is a unique positive similarity solution $w_{0}(t, x)=t^{-1/(\rho-1)}f(|x|/\sqrt{t})$to (4.2), where $f(r),$ $r=|x|$ , is a positive solution to$\phi(t, x)$$\theta_{0}$$-f”-( \frac{r}{2}+\frac{N-1}{r})f’+|f|^{\rho-1}f=\frac{1}{\rho-1}f$

(4.3)

$\{$

$f’(\mathrm{O})=0$,$\lim_{rarrow\infty}r^{\frac{2}{\rho-1}}f(r)=0$.

98

to (4.2) with
Then the positive solution
tcnds to the similarity solution $w_{0}(t, x)$ in the scnse that$\phi_{0}\in L^{1},1\mathrm{i}\mathrm{I}\mathrm{n}_{|x|arrow\infty}|x|^{2/(\rho-1)}\phi \mathrm{o}(x)=0$$\emptyset(t, x)$,$\phi_{0}(x)\geq 0$(4.4)$\lim_{tarrow\infty}t^{\frac{1}{\rho-1}}||\phi(t, x)-w_{0}(t, x)||_{L^{\infty}}=0$.

in (4.3), theninstead ofIf we impose$w_{b}(t,x)=t^{-1/(\rho-1)}f(|x|/\sqrt{t})$. Thenwe still have a solution and a similarity solution, thesatisfying
positive,necessarily, notfor the data
tends to $w_{b}(t, x)rT8tarrow\infty$ , in the sarnc scnse in (4.4), However, if wesolution, at least 4 non-trivial solutions when
do not impose the positivity in (4.3), then weto (4.2),$1<p<1+2/(N+1)$ , so that we don’t know until now how the solution. For these results, refer Brezis, Pelctier andaswhich may change its sign, })$[5,
6]$
Escobedo,
Kavian and Matano [7], Galaktionov,,
andKavianTerman [1], Escobedo
Kurdyumov and Samarskii [9] and the references therein.
We now go back to (4.1) and remcmber some known results. First, Kawashima,
Nakao and Ono [18] showed the existencc of a unique and global solution $u(t, x)$ to (4.1)
if $1<\rho<1+4/(N-2)$
in $C([0, \infty);H^{1})\cap C([0, \infty);L^{2})$ for any big data
( $1<\rho<\infty$ when $N=1,2$ ). Moreovcr, the dccay property$\lim_{rarrow\infty}r^{\frac{2}{\rho-1}}f(r)=b>0$$1\mathrm{i}\mathrm{I}\mathrm{n}_{rarrow\infty}r^{\frac{2}{\rho-1}}f(r)=$

$()$

$f$

$\lim_{|x|arrow\infty}|x|^{2/(\rho-1)}\phi_{0}(x)=b$$\phi_{0}\in L^{1}$$\phi(t, x)$$\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{t}^{\backslash }$$\phi(t, x)$$\mathrm{e}\mathrm{h}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{s}$$tarrow\infty$$(u_{0}, u_{1})\in H^{1}\cross L^{2}$(4.5)$||u(t)||_{L^{2}}\leq C(1+i)^{-\frac{N}{2}(\frac{1}{q}-\frac{1}{2})}$was provcd providcd that $(u_{0}, u_{1})\in(H^{2}\cap L^{q})\mathrm{x}(H^{1}\cap L^{q})(1\leq q\leq 2)$ and$1+ \frac{4}{N-2}(N\leq 4)$ . Thc decay rate (4.5) seems to be bcst possible, because we expect theGauss kernel to be an asymptotic profile. Based on [18], Karch [17] showcd that, when
$1+ \frac{4}{N}<\rho<$$N=1,2,3$ ,(4.6)$||u(t, \cdot)-\theta_{0}G(t, \cdot)||_{L^{\mathrm{p}}}=o(t^{-\frac{N}{2}(1-\frac{1}{\mathrm{p}})})$$\leq\infty$for$2\leq p\{$$<\infty$$\leq\frac{2N}{N-2}$

(4.7)

$N=1$
$N=2$
$N=3$

as

$tarrow\infty$

and$\theta_{0}=\int_{\mathrm{R}^{N}}(u_{0}+u_{1})(x)dx-\int_{0}^{\infty}\int_{\mathrm{R}^{N}}|u|^{\rho-1}u(t, x)dxdt$.

Recently, Hayashi, Kaikina and Naumkin [12] have shown that, when $N=1$ , the solution$u(t, x)$ behaves as$tarrow\infty$$u(t, x)\sim\{$$w_{0}(t, x)$$\rho_{c}(1)-\epsilon<\rho<\rho_{c}(1)$$\theta_{0}G(t, x)(\log t)^{-1/2}$$\rho=\rho_{c}(1)$$\theta_{0}G(t, x)$$\rho>\rho_{c}(1)$,.
for suitably small
In these situations we, roughly speaking, show the decay properties in the subcritical. Ill the critical oase we cannot
case and the asymptotic profile in the supercriticalhave any sharp estimatc. First theorem is the following.
$\epsilon>0$$\mathrm{C},r\gamma \mathrm{s}\mathrm{e}$99
Theorem 4.1 ([16]) Let $1< \rho\leq 1+\frac{4}{N}(<1+\frac{4}{N-2})$ . Suppose that
(4.8)$(1+|x|)^{m}(u_{0}, \nabla u_{0}, u_{1}, |u_{0}|^{(\rho+\mathrm{i})/2})\in L^{2}$

where for any small fixed constant$\delta>0$$m= \frac{2}{\rho-1}-\frac{N-\delta}{2}$

(4.9)

$\{$

$m>N/2$

Then the solution $u(t, x)$ to$(\mathit{4}\cdot \mathit{1})$

,

when
when

$\rho<\rho_{c,}(N)$$\rho\geq\rho_{\mathrm{c}}(N)$.

satisfies the decay properties
$N=1$
$N=2$

$\leq\infty$(4.10)$||u(t, \cdot)||_{L^{\mathrm{p}}}\leq C(1+t)^{-\frac{1}{\rho-1}+\frac{N}{2\mathrm{p}}}$for$1\leq p\{$$<\infty$$N\geq 3$$\leq\frac{2N}{N-2}$

We give some remarks. In the subcritical.

the similarity solution $w_{0}(t, x)$ satisfies$\mathrm{c},\mathrm{a}_{\iota}\mathrm{s}\mathrm{e}$for $1\leq p\leq\infty$$||w_{0}(t, \cdot)||_{L^{\mathrm{p}}}=t^{-\frac{1}{\rho-1}}(\int_{\mathrm{R}^{N}}t^{N/2}|f(\frac{|x|}{\sqrt{t}})|^{p}\frac{dx}{t^{N/2}})^{1/\mathrm{p}}=Ct^{-\frac{1}{\rho-1}+\frac{N}{2\mathrm{p}}}$.

Hence the decay rate (4.10) is best possible, though there are restrictions on . Our final
goal is to obtain the asymptotic profile. As we stated above, even in the parabolic problem
which may change its(4.2) the asymptotic proPle is not known for the solution
sign. In our problem (4.1) the solution $u(t, x)$ generally changes its sign. Therefore, toget our goal seems to be difficult in the subcritical case.
While in the supercritical casc we cannot have -boundedness from (4.10) since. We expect that the asymptotic profile is the Gauss kernel and hcnce
the rate (4.10) is less sharp. However, standing on this less sharp rate we will improve to
get the asymptotic profile in Theorem 4.2 below.
The rates of the weight in $(4.8)-(4.9)$ seem to bc reasonable. In fact, in the subcriticalin (4.3), which corresponds to (4.9) in thecase we imposedto get the asymptotic
should be insolutionsupercritic\‘althecaseframework. In the, $etc$ .
.
profle $G(t, x)$ . Clearly $(4.8)-(4.9)$ mean$p$$\phi(t, x)$$L^{1}$$- \frac{1}{\rho-1}+\frac{N}{2}>0$$L^{2_{-}}$$\lim_{|x|arrow\infty}|x|^{\frac{2}{\rho-1}f(|X|)}=0$$L^{1}$$\in L^{1}$$u_{0},$ $u_{1}$Theorem 4.2 ([16, 27]) Let(4.11)$\rho_{c}(N)<\rho\{\leq<1+1+=\# N4$$(N=1, 2, 3)(N=4).$

Suppose that $(u_{0}, u_{1})\in H^{2}\cross H^{1}(N\leq 3)$ [resp. $H^{3} \cross H^{2}(N=4)\int$ and(4.12)$(1+|x|)^{m}(u_{0}, \nabla u_{0}, \Delta u_{0}, |u_{0}|^{\frac{\rho+1}{2}}, u_{1}, \nabla u_{1})\in L^{2}(\mathrm{R}^{N})$with $m>N/2$ . Then it holds that
(4.13)

with$\theta_{0}$$||u(t, \cdot)-\theta_{0}G(t, \cdot)||_{L^{\mathrm{p}}}=o(t^{-\frac{N}{2}(1-\frac{1}{p})})$

in$(\mathit{4}\cdot 7)$.

as$tarrow\infty$,$1\leq p\leq\infty$

100

Theorem 4.2 covers the gap remained open in thc supcrcritical case. For the proof of
Theorem 4.1 we empl the weighted energy mcthod, originally developed by Todorova
and Yordanov [28]. By applying the Gagliardo-Nirenberg inequality to the -decay rcsults obtained, -decay rate (4.10) is derive, . For Theoreln 4.2 we combine the weighted
energy method with the explicit formula$o\mathrm{y}$$L^{2}$$L^{p}$$\mathrm{d}$$u(t, \cdot)=S_{N}(t)(u_{0}+u_{1})+\partial_{t}(S_{N}(t)u_{0})$(414)$- \int_{0}^{t}e^{-\frac{t-\tau}{2}}W_{N}(t-\tau)|u|^{\rho-1}u(\tau, \cdot)d\tau-\int_{0}^{t}J_{N}(t-\tau)|u|^{\rho-1}u(\tau, \cdot)d\tau$(cf. (3.4)) developed in Section 2. It is a key point how to treat the wave part.

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