The Rate of Asymptotic Convergence of Strong Detonations for a Model Problem
Lungan YINGt, Tong YANGtt and Changjiang ZHUttt tInstitute of Mathematics, Peking University,
P.R. China
ft Department of Mathematics, City University of Hong Kong, P.R. China
tttDepartment of Mathematics, Central China Normal University, Wuhan, P.R. China
Received December 9, 1996 Revised November 25, 1997
We prove that the strong detonation travelling waves for a viscous combustion model are nonlinearly stable, that is, for a given perturbation of a travelling wave, the solution converges to a shifted travelling wave asymptotically as t —+ oo. The rate of convergence is obtained. This work is a continuation of a previous work by done T.P. Liu and the first author in [7].
Key words: combustion, detonation wave, travelling wave, asymptotic stability
1. Introduction
We consider a viscous combustion model
(u + qz)t
+
f (u)„ _ ,ßu„., (1.1)zt = —KO(u)z, (1.2)
where u represents the lumped fluid variable, z E [0, 1] represents the concentration of the reactant, and the positive constants ß, q, and K represent the viscosity, heat release, and the reaction rate respectively. The flux function f is smooth and satisfies
f'(u) > 0, (1.3)
f"(u) > 0. (1.4)
The reaction rate function rß is smooth and satisfies 1, for u > d,
u 0, for u < 0, (1.5)
0'(u) > 0, for 0 <u < d. (1.6) The travelling wave solutions to (1.1) and (1.2) have been studied in [8]. The initial value problem of the system has been studied in [13]. Let
0
, +0 and468
K f +oc, then the limits of solutions depend on OK. In this respect the Riemann problem for the case of ßK = 0 has been studied in [14], where (1.3) is not necessary satisfied. A related problem is the nonlinear stability of travelling waves as t 3 +oo.
The nonlinear stability of strong detonation waves was proved in [7]. For the other results of this system, see [2, 3, 6, 12].
A weighted energy method has been proved to be an effective approach to study stability problems (see [1, 5, 9, 11]). The purpose of this paper is to apply the weighted energy method to study the nonlinear stability problem of (1.1) (1.2). The result of this paper is a continuation of [7]. We will show that if q is sufficiently small, then a perturbation of a travelling strong detonation wave leads to a solution which tends to a shifted travelling strong detonation wave as t k +oo, and the rate is determined by the rate of initial perturbation as 4 oo. T. Li has obtained a series of interesting results on another combustion model. In [4] the rate of asymptotic convergence of strong detonations was also proved. The previous works indicate that these two models share some common properties, which may hint some general results on more general models.
We make a brief statement of the results here. Let u* and u* be two constants with u* < u*, then it is known that there exist travelling shock waves so(e), C _ x  at, to a single conservation law,
st + f (s). = os^{y}.^, such that
f(u*)  f(u*) _ a(u^{*}  u*). (1.7)
lim s^{o}(C) = u^{*}, lim so(C) = u*.
^— +oo ^^—oo
Here (1.7) is the RankineHugoniot condition. For a fixed value of so(0) E (u*, u*), so(C) is unique. If u* > d, where d is the positive constant in (1.5), then under some restrictions on the parameters q, ß, K there exist travelling strong detonation waves s(C), ((C), C = x  Qt, such that
f (u*)  f (u+) _ a(u*  u^{+}  q), (1.8)
lim (s(C), (C)) _ (u+, 1), lim (s(C), ((C)) = (u*, 0).
And under some other restrictions on q, ,3, K there exist travelling weak detonation
waves s(C),
((C), C = x ^{} atsuch that
f (u*)  f (u+) = a(u^{*}  u_{+}  q), (1.9)
li n(s(C), ((C)) _ (u+, 1), lim (s(C), ((C)) _ (u,0).
Since we consider strong detonation waves only in this paper, for simplicity we assume that u,k <0 and u > d. It is easy to prove that strong detonation waves always exist for any positive numbers q, ß, and K. From (1.8) (1.7) we can get u+
<ui.
Let the perturbation of s(C) and ((^) be uo (C) and zo(C). We assume that uo(C) and zo (C) are sufficiently smooth and
lim (uo(C)  s(C), zo(C)  ((C)) = 0.
Without loss of generality we assume that
f
^{+}^(uo(^) + qzo(^)  s(C)  q((^)) dC = 0, (1.10) otherwise a translation of (s, () leads to (1.10). (1.10) implies that for all tf
(u+qzsqdx=0,^{+^} (1.11)where (u, z) is the solution to (1.1) (1.2) with initial condition
(u, z)1t=o = (u0, z0),(1.12)
and
s = s(x  Qt), ( = ((x  at).(1.11) enables us to consider the
antiderivativesX
v(x, t) = fU(Ct)dC ,
where
U = ul + q(z  (), ul =us.
(1.13)Our main result is the following:
THEOREM.
Let p be a positive real number. We assume thatf+oe
J
(1+lxl),(v2(x,0)+u1(x,0)+ul1(x,0))dx is bounded, andZU(X)  ((x)j, j(zo(x)  ((x))xl = O(Ix^^{}p2
), as ixj — oe, besides we assume that
f
^{+} ^{(v}^{2}^{(x,}0) + ui(x, 0)) dx470
and q are sufficiently small, then
ui(x,t)I+Iz(e+ort,t)—S(C)I o(1)(t^{}Pl2), as t
decay rate for scalar viscous conservation laws was obtained in [9] [10].
Applying the weighted energy estimate, we will prove the theorem in the next section. In the proof we will always denote any generic constant by C.
2. Stability Analysis
The local solvability of the problem (1.1) (1.2) and (1.12) can be demostrated by iteration. Applying the Poisson formula to the heat equation we have
n=
J
^{1}+^ 2 ^FO^{e}^{}t (x ^)21(4at)uo() dt +00
_ 1ef)2/(4a(t T))(gzT + f (u)^) d dr
0 x 2 ßß(t — r)
— p+oc
J
^{2}e—(x—^)2l(4/jt)u0(S) d^00 t +oo
+ 12 /(40(t—r))
tiKO(u(^ 7))zo(e)
0 00 2 7fß(í — T)
. eK f. 0(u(^,ii )) din dd r
+ t+0 X 
2/(4a(tT))f (u(^, r)) ddr.
0
j_„
4V"(N(t—T))3/2U The existence of u follows from Picard's method of contraction mappings, then z is obtained by solving the ordinary differential equation (1.2) with initial condition (1.12). The global existance of (u, z) is a consequence of the following energy estimate.
By (1.7) there is a constant c E (u*, u*) such that f'(c) = a. A shift of the origin, if necessary, leads to s(0) = c. So one gets f'(s(0)) = a.
The functions s(x  at) and ((x  at) satisfy the equations (1.1) and (1.2), that is
(s + q^)t + f (s)„ ^{= i3s,} ^{(2.1)}
St = K(s)(, (2.2)
By integrating the difference of (1.1) and (2.1) we have
vt + f (u)  f (s) = Qul .. (2.3)
We define a weight with a > 0, 6> 0, y > 0,
(t + 1)°W(^) = (t + 1)a(1 + b^2)y12, ^ = x — at. (2.4) Multiplying the equation (2.3) by (2.4) and v and taking integration we get
 (t_{ + 1)aWvv} t
t dxdt + J J +^
/.t
J
(t + 1)"Wv(f (u) — f (s)) dxdt0 ^{x} ^{0} ^{oc}
=ß JJ ^{(t+1)'Wvu}
^{lx}^{dxdt.} ^{(2.5)}
The first term of (2.5) is equal to
1
1+co(t + 1)aWv^{2} dx —
1
_{2} _{2}_{J }
^{+oe}^{ W(x)v}^{2}^{(x, 0) dx}+ 2 ^{J J} ^{r}
^{t }^{r}
^{+00}(t + 1)aQW'v2 dxdt —J J
^{f}^{t }^{r}
2 (t + 1)a ^{+00} 1Wv2 dxdt.0 00 0 00
The second term of (2.5) is equal to
• f^{+oe}
J
^{(t + 1)}^{°}
Wv(f'(s)ul + O(1)u1) dxdt 1J 0x^{!!}t+00

(t + 1)aWv{ f'(s)(v
^{x}— q(z
_{—}c)) + O(1)ui } dxdt
0 oc
t +x v2 v2
= Jo (t + 1)^{a}{ — W f"(s)s^{x} 2 — W' f'(s) 2 — qWv(z — C).f'(s)
+ O(1)Wvu1 } dxdt, where we have used integration by parts. Analogously the right hand side of (2.5) is equal to
ƒ01.
—ß1 t +00
] (t + 1)"(WUu
i+ W'vu
^{l}) dxdt
o oc
(t +1)^{a}(Wui + gWul (z — () + W'vul) dxdt.
We define
h(^) _ —W( )f"(s( ))s'( )+W'(^)(ff —
(2.5) is reduced to
1
J ^{(t + 1)aWv}
^{2}^{ dx + 1} J ^{I}
^{t}J ^{(t + 1)ahv}
^{2}^{ dxdt}
2 2
472
(t
J
^{,.}^{+ce}+ß
J
(t+1)Wuidxdt o_< f W (x)v2(x, 0) dx + a ff(t+ 1) 1Wv2 dxdt
ft f+°°
— ,ß
JJ
(t + 1)cW'vui dxdto
— q J
^{t }^{J}
^{f}^{+00}^{(t+ }
^{1)"}^{W(Qu}
^{l}— f'(s)v)(z — ^) dxdt
J
o^{o}
^{t} ^{f}^{+00}+ C J
(t + 1)"Wvu^{2} dxdt. (2.6)00
To get an energy estimate from (2.6), we need to study the property of the function h, which is summerized in the following lemma:
LEMMA 1. If q is small enough and y 0, then h> 0 and h(^) ^^ry 1 as
—oo.
Proof. Let y = qC, then the integration of the equation (2.1) yields
s' = Q ^{1}(—o (s + y) + f (s) + c), (2.7)
where
c=—f(u*)
+cru
*.The critical point A divides the curve {(s(^), y(^))( —oo < < +oo} into two parts:
s'(^) <0 for >
e
o and s'(e) > 0 for <eo,
cf. Fig. 1.6
Fig. 1.
We have known that u* <s(0) <u, and it is shown in Fig. 1 that s(o) > u*, so f'(s(^o)) > a, o < 0.
If > 0 then W' > 0, f'(s) < a, and if o < < 0, then W' < 0, f(s) > v, hence h > 0 for the case of > o. We turn now to consider the case of <
Since s'(eo) = 0, we get from the equation (2.7) that (f^{1}  o,)(s  u*) = cry
at the point , where the argument of f' is a mean value., therefore
* cry
s(o) = u + fl or,
but y E [0, q], so
s(o) E
lu
^{.},
^{ u* + }^{f}aq _{ j}
. ^{(2.8)}
Q
From s > u*
> dwe have 0 __ 1. The integration of the equation (2.2) yields
y = y(^o)e^{K(E}^^{o)la}. (2.9)
We rewrite (2.7) as
s' = Q1(f^{1}  Q)(s  u*)  /3^{1}Qy. (2.10)
Let
p = Q^{}'(Q  f'),the solution of (2.10) is
s = u* + e f£ _{( (s(^0) } u*)  f /ß'ay( o)eK(^ o)/a
e
f o pd' d1
^{1}{o
then noting (2.8) we get
s  u* < Cge^{b}^°), _{(2.11)}
where b > 0. Applying (2.9) and (2.11) to (2.10) we have
s"I < Cgemin(b,Kla)(f—Co)The second term of h(^) is positive, so if q is small enough, h(^) is positive for Finally the definition (2.4) gives the degree of h(^) as * oc. q Let us turn now to the estimate of (2.6). For the third term on the right hand side we have
ft f+oc
4<
JJ (t + 1)"W'vul dxdtl
0
c +oo t +oo W12
(t + 1)'Wui dxdt + ß j J (t + 1)' W v2 dxdt. (2.12)
0 00 0 00
If 6 is small enough, then Lemma 1 implies
'2 < h.
The right hand side of (2.12) is majorized by the left hand side of (2.6). To estimate the fourth and fifth terms of (2.6), we need the following a priori assumptions:
CM < 0, (2.13)
ul < min (
I
2I u2 d) , (2.14)where d is given in (1.5). Using (2.13) the fifth term can be easily estimated. It remains to estimate the fourth term. Let x^{1i}x2 E ll such that s(xi) = 2 and s(x2) = d• Notice that x1 — x2 has an upper bound which is independent of q.
We divide the domain S? = ll x [0, t) into four parts:
S^{1} =Qn{x>x^{1} +ot}, Q2=Qn{x2+ot<x<xl+ot}, Q3 = Q n {x2 < x < x2 + O't}, Q4 = n n {x < x2 }.
From (2.14) we see that O(s) = c(u)  0 in f2^{1} and (s) = 0(u)  1 in S?3 U S?4.
Let all assumptions of the theorem be satisfied. We set F(x, t) = q(t + 1)aW (,3u1 — f'(s)v)(z — () and estimate f f1 F(x, t) dxdt. In the domain S^{1}
z — ( = zo(x) — ^(x). (2.15)
The CauchySchwarz inequality gives
fJ _{n}
_{l} ^{F(xt)dxdt}< cqJ J (t + ^{1})aW (Izo(x) — (x)I1/2(ui + v2) + zo(x) — S(x)I3/2) dxdt n,
For small q it can be verified that
CqW l zo (x) — (x)11/2 <
1
^{ h(x — }^{ut).}It follows that
ff
_{n}_{l} ^{F(xt)dxdt}< g (t + 1)ah(x — Qt)v
f/
2 + Cq(t + 1YWu)))
1 } dxdt + Cq. (2.16)
nl
In the domain .0
^{2}we define
0^{,}
t
i(x) =
x — xlQ
x — x2
t2(x) = Q ,
t,
for x < x
^{l}, for x > xl,
for x < Qt + x2, for x > Qt + x2, then 22 = {(x,t);tl(x) < t < t2(x)} and
z — ^ = zo(x)e—K fá <P(u) dT — S(x)e—K fó O(s) dT
= z0(x)e—K ^{fe}^{t}^{a(=) O(u)}^{ dr} — ((x)e_K f,'() c(s) d/r
= zo (x) (e—K f,'^{e} (=) <P(u) ^{dr }— e—K ftl (x)s O(s) dr
/ _K ft
+ (zOlx) — ^(x))e ^{tl ( }q(s) dr'
therefore
fc
jz — ^1 < C
J
IulId^r + jzo(x) — ((x) t^(x)(2.17)
It follows that
ff
F(xt)dxdt < CqJ J
^{(t+1)tW}22 n2
(f
^{t}^{(}^{x)} ^{\l}x
{u+v
^{2}+ ull dr
/
I+(zo(x) —(x))
^{2}JJJ ^{}dxdt.}
,
(2.18)
Due to the CauchySchwarz inequality we have
f ^{(t+} ^{l)awI}
^{'}_{J}
^{t}^{ Iu}
^{l}^{IdTI}
^{Z}^{dxdt}
\\\ t^{1}(x) /
n2
476
f f
^{(t+}(
I
^{t^}^{,}^{\}
<C
J (x)
^{1} ^{W} ^{u} ^{d r I dxdt}líí
2z
r
It ^
^{,}^{z(x)} ^{f}^{t}=C
J
dx(t+1)"Wdt^{(x)}J
^{ti(x)}^{uidyrB.}But (t + 1)"/(T + 1)" is bounded in 112, so
f
I
^{t}^{t}^{2}^{(x)} ^{t}B < C
J
^{dx} ^{dt } (T +1)"Wu1 dT1(x) ti(x)
/'
i
^{t}t3(x)
< C J ^{dx} (rr+1)"Wul dr
i ()
= c f l
(t + 1)"Wui dxdt. (2.19)02
We substitute (2.19) into (2.18) and notice that
Wand
hare in fact bounded functions in
_{Q2,}then we get
Jf
F(x, t) dxdt < CqJ J
(t + 1)"(Wui + hv2) dxdt + Cq. (2.20)f2 d22
We have the following estimate in 113:
1z  /' 1
C1 <
^{(}t2(x)
C J
^{Jul }^{dr }+ I zo(x) — ((x)I)
eK(tt2(X)) (2.21)
t1(x) therefore
J f
F(xt)dxdt l < CqJ J
(t + 1)"WeK(tt2(X))2^{3} n3
{U2 /'tz(X)
x
+ v
^{2}+ J ui dr + (zo(x) —ß(x))2} dxdt.
t^{i}(x)
For small q it can be verified that
CgWeK(tt2(2)) < 1h(x — vt).
And we have
t
(t + 1)"We^{K(tt2(}')) dt < C(t2(x) + 1)" <_ C(T + 1)", t2(x)
and
J f
(t + 1)"We((tt2(X)) (zo
(x) — ((x))2 dxdt < C,n3
thus
ff
F(x, t) dxdt < Cq (J J
(t + 1)"Wui dxdt +ff
^{(t + 1)}^{"u}
^{ dxdt + 1)}923 ^3 n2
11
+
1J J (t + 1)"hv
2dxdt.
(2.22)f23
Since 0  1 in .fl4i we have
z — _ (zo(x) — ((x))e xt, therefore
ff
^{F(xt)dxdt}.Q4
< Cq
J J
(t + 1)" (Wui+ Wlzo(x) — C(x)je
^{Kt}(v
2+ 1)) dxdt
n4
< Cq f/ (t + 1)"Wui dxdt + g ff t + 1)"hv
2dxdt + Cq.
(2.23)f24 ^4
Let q be sufficiently small, then we use (2.12) (2.13) (2.16) (2.20) (2.22) (2.23) to estimate the right hand side of (2.6) and obtain
/
+^ t f^{+Qc} 1 % (t + 1)"
W v2 dx + 1 ^{)}
J J
^{(t + 1} "hv2 dxdt24 0
/^t /^+oc
+
J _{J J }
(t + 1)"Wu1 dxdt 4 0p
^{+ao}j
^{t +oo}^{j} ^{(t}
<_
1J ^{W(x)v}
^{2}(x, 0) dx + +l)1Wv2dxdt+Cq.
(2.24)
We now turn to estimate the derivatives. We derive from (1.1) and (2.1) that
Ut + (ƒ(u) — f(s)). = Qui... (2.25)
We multiply both sides of (2.25) by the weight (t + 1)"W and ul and then take integration. For the first term we have
t +co
(t + 1)"Wu
l Utdxdt
o oc
=
1J
^{ +oe}(t + 1)"Wui dx —
1J
+oGW(x)u2(x, 0) dx
2 2
478 L. YING, T. YANG and C. ZHU
+ 2 ff
^{ o}(t + 1)QW'udxdt
—JJ
^{f}^{t }^{f}^{+00}^{a(t + 1)^ }
^{1 W }^{2}^{dxdt}
+ qj
^{t}(t + 1)aWu^ (—KO(u)z + KO(s)() dxdt. (2.26)3o
J
We notice that
—K4(u)z + KO(s)( = —KO(z — () — KO'u^{1}C.
The same procedure in estimating F(x, t) is applied here, then the last term of (2.26) is majorized as the following:
+00
q
l
J (t + 1)"Wu^{t} i (—Kq(u)z + K^(s)O dxdt00
pt f^{+OO}
< Cq
J / (t + 1)aWui dxdt + Cq.
0 I oo
For the second term of (2.25) we have
%
^{t}J (t + 1)' Wulf (u) — f (s))„ dxdt
o ^
J
ƒ
^{+oe}^{ oo}/'t r+00
= J
^{J} ^{(t + }^{1)"(Wu}^{l x}^{ + W'u}^{l})(f (u) — f (s)) dxdt1o 00
= (.t (+00
J
J(t + 1)^(Wu^{l}u1 + W'u1)f' dxdt1o 00
4
J J ^{t }
^{+}^{^} ^{(t+1}^WU}
^{2}^{x}^{dxdt+C J }
J
t r+^
<
(t+l)°(W+IW'J)uidxdt.
o 00 000
For the right hand side of (2.25) the same procedure in deriving (2.12) is applied.
We get the following estimation from (2.25) that
f+00 ft
r
^{+00}1
J
(
t+1)
aWuidx+— J J (t+1)°Wui dxdt
2 002 0 00
/
'
^{t }(
^{+00}+ J J
^{o} ^{00} ^{(t + 1)}^{"hui }^{dxdt}<_ 2 +00W(x)ui(x,0)dx+C^ t +^
!!00
(t+1)aWuidxdt blo
If
+00
+ /(t +1)'
1Wu1 dxdt + Cq. (2.27)
2 .
We set
U2 = U,
and u2 = ul,, then from (2.25) we have the equation U2t + (f(u) — f(s))xx _ OU2xxMutiply this equation by (t + 1)"Wu2 and integrate over [0, t] x R. Since U2t = U2t + q(z — ()mot,
we have
+00
(t + 1)"Wu2(u2t + q(z — ()„t) dxdt
10 1 ^{.}
t +^
+ J
^{J} (t + 1)aWu2(f (u) — f (s))xx dxdto 00
ft f+OO
=
ß JJ
(t + 1)C'Wu2u2XX dxdt. (2.28)0 oc
We now estimate (2.28) term by term. For the first term, we have
t
+ce
(t + 1)” Wu2(u2t + q(z — ^)„t)dxdt
0 Ioo
f^{+oo} f+0O
= 1 (t + 1)"Wu2 dx — 1
J
W(x)u2(x, 0) dx2 2 L
t
+oe
oc (t + 1)a1Wu2 dxdt a
2 0_{ f}
I
^{t}^{ J } f
^{+^}+ (t + 1)W'u2 dxdt
I
^{t +00}— qK (t + 1)"W'u2(0(z — ^) — O'ui() dxdt
0 oc
t +00
— q J ^{I }
^{f}J (t + 1)" Wu2
x(O(z — () — O
'u^{i} )dxdt.
JOoo
The second and the third term can be written as follows t
+ce
(t + 1)"Wu2(f (u) — f (s))
o oc xx dxdt
t +°°
(t + 1)"W'u2(f (u) — f (s))x dxdt
0 +oo
(t + 1)" Wu2x(f'(u)ux + f'(s)s^{x}) dxdt
ƒo
^{ 00}^{t}t +00
(t + 1)"W'u2(f'u2 — f'ulsx) dxdt
o 00
L
^{c}(t + 1)"Wu2x(f'u2 — f'ulsx) dxdt,ƒo
^{t}and
t +00
ß (t + 1)aWu2u2xx dxdt
o oc
480 L. YING, T. YANG and C. ZHU ft
f
^{.}^{+oe}(t + 1)"Wu2„ dxdt
o/'
tf
^{.}^{ +00}+ ß
^{2}^{ o }J ^{(t + 1)}
^{a}^{W}
^{tt}^{u2dxdt.}
Combining all the above estimation for (2.28) and using Cauchy inequality and q being sufficiently small, we have
1 ^{ (t + 1)}
^{+^}^{00} ^{0}Wu2 dx + /3
^{]}^{f}^{t }^{p}^{+c}J (t + 1)
aWu2
xdxdt
ft f +00 o
+ JJ ^{(t + } ^{1)'hu2 } ^{dxdt}
o 00
J
^{+<>Q}W(x)u2(x, 0) dx + ct J t J ^{(t + 1)^} ^{+^}
^{1}^{Wu2 dxdt}
00 0 00
t
L
^{e}+C f
(t + 1)^W(u1 +u2)
dxdt +Cq.(2.29)
oCombining (2.4), (2.27) and (2.29), we have for y 54 0
+^ ^{(t + 1)}
^{0}^{W (v}
^{2}+ ui + u2} dx + J t J +^
J (t + ^{1)}
^{a}^{h(v}
^{2}+ ui + u2) dxdt
o 00 t
f
^{+00}^{.}+ ß (t +1)"W (ui +u2+ u2
x) dxdt
o
t
<c W (x) (v
2(x, 0) + ui (x, 0) + u2 (x, 0)) dx
0
+C ^{a} ^{J}
^{f}^{t}f
^{}^{+00}^{.} ^{(t+1)}^{ii1}^{W}^{(v}
^{2}^{+ui} +u2)dxdt+Cq.
(2.30) oWe need one more estimate of
J
^{ ce}+00^{ (t + 1)}^{0} W {(z — ()2 + ((z — ()^)2} dxfxl+<7t
^{+}lx,+,t
^{t+}2+fit
f'.
2 +
1
1X2^{X2} o0I1+12 +I3+14.
We notice (2.15) and t ' ó then get
I'll
^{ < C.}We notice (2.17) and (t + 1)a /(r + 1) < C, then get 11^{2}
<C + c f f ^{(t +}
^{ 1)a(u}^{ + u)dxdt.}
22
We notice (2.21) then get
(/^t2(x)
I31 < C 1
^{X2+Crt}(t + 1)aW { J (ul +u2) dT + (zo(x)
_{}((x))
22 l t,(x)
+ ((zo(x) — ^(x))x)
^{2}1e
2K(tt2('))dx.
To estimate the first term of it we use the inequality
(t + 1)a e K(tt2(ß)) _(t + 1)^{a eKt} (t2(x) + 1 )a < C, (T + 1)^{a}(t2(x) + 1)aeKt2(^)(T + 1)a and to estimate the second term we use
(t + 1)n e^{}K(t^{}t2(x)) < (t + 1)a eKtI^{2} < C, for t2 (x) < 2,
and
(t + 1)^{a}(Izo(x)  ((x)I + I (zo(x)  ^(x))xl )
< (2(x  x2) + 1) a (Izo(x)  ^(x)I + I (zo(x)  c(x))xI) C C,
_
^{o}J
for t2(x) > 2,t
then we get
1
^{131 }^{<C+C} J 1(t+1)a(ui+u2)dxdt.
nz
Finally it is easy to see that 1141 < C. It follows that
+ ^(t + 1)oW {(z f )2 + ((z  ^)^)2} dx
ao
<c + c f f
(t+l)a(u1+u2)dxdt. (2.31) n2Based on the above estimates, we are going to prove the theorem as follows.
Proof of Theorem.
Under thea priori
estimates (2.13) and (2.14), all the estimates given above hold.When
p = N
> 0 is an integer, we inductively let a = 0 and 'y = N in (2.30), the left hand side of (2.30) can be bounded only by the initial data and q. Then we let a = 1 and 'y = N  1 in (2.30), using Lemma 1, we get a similar estimate. This procedure can be continued up toa = N  1
and y = 1.For y = 0, we need the estimate for _{f_00}
(t +
1)av2dx
because the functionh
is no longer positive everywhere.By the equation (4.4) in [7], we have
p +Do t^{T }
1
^{00}^{+00}^{/'}^{T }^{/}^{'+o0}J v^{2}(x, T) dx + ( v^ dxdt + (J]f"(s)s^{x}lv2 dxdt
/o Jo oo
< J
^{ r+^ v}^{2} (x, 0) dx + C +ooIn
^{ f (q}2
(z ^{}()2+ giv(z  ()I) dxdt.
00J o0
By the definition of v, v^{x} in the second term can replaced by u^{I}. From the deduction it is easy to see (0, T) can be replaced by any interval (t^{1}, t^{2}).
Let at = M , t^{i} = iat. Then taking summation yields
m ( f+O0 ft; f
+oe
^(ti + l)" { i=1 l
J
oo ^{v}^{2}^{(x, t}^{Z}^{) dx + }J J
t1 00 ^{u1 dxdt}ft f+0o
+1 ^{ J}
_{t}_{ß}_{_}_{1}_{00}v ^{"(s)s}
^{x}^{Jv}
^{2}^{ dxdt^}
< (ti + 1)" 1 f v2 (x, ti1) dx
i—I l o0
+ C
J
^{tz}^{^}J
+(q2(z  c)^{2} + giv(z  ^)I) dxdt ).e 1 00 )))
Let M * +oo, we have
f+ce ^{t }f+00
J
^{(t+} ^{1)"v}^{2}^{dx+}JJ
(t+1)"uidxdtl
^{t}^{ f}^{+00} o+
^{ 00}J ^{i } ^{f"(s)s}
^{x}^{ iv}
^{2}^{ dxdt}
+00 p^{t }r^{+00}
< J 0) dx + ca J J
(t + l)"^{I}V^{2} dxdto 00 t+0C
+ C f (t + 1)"(q2 (z  ^)^{2} + qIv(z  ^) j) dxdt.
o ^{00}
We now estimate the last term as follows.
t +00
(t + 1)"jv(z  ^)I dxdt
o
< 2
ff ^{(t+}
^{1)_}^{1}^{v}^{2}^{d}
^{x}dt+ 2 ff ^{(t+} ^{1)+}
^{1}^{(}
^{z}^{_}
^{2}^{d}
^{x}^{dt}
Q1 U24Q1 U24
+ ff(t + 1)"v2 dxdt + 1
f
^{/(t + }^{1)(z )}^{2}^{ dxdt}+ 1 fJ ^{(t + }
^{1)"e}^{x(tt21}^{Xi}^{)}^{v}^{2}^{ dxdt}
23
+
J f
^{(t} + 1)ae^{K(tt2}(z — ()2 dxdt.g7_{3}
By (2.15), we have
ff
_{n,}^{(t + } ^{1y}
^{1}^{ (z } ^{— ^)}
^{2}^{ dxdt}
<
CJ J
^{(t+ }1)a+l(IxI + l)2`v4dxdt < C.nl
and
f»t +
^{1)a +1(z }—
^{)}2 dxdt
=
J J
^{(t + }^{1)a+i(}^{zo}^{(}^{x}^{) }^{_ ((x))}^{2}^{e}^{2Kt} dxdt <C.f14
Using (2.17) and (2.19) yields
J J ^{(t+}
^{1)°(z—()}^{2}^{ dxdt} ^{ <C} ƒƒ ^{(t+}
j)aU2 dxdt + C.22 22
By (2.21), we have
f
(t + 1)aex(tt2(X))(z — )_{2} dxdt < C f f(t + 1)aui dxdt + C.23 f12
Since I f"(s)s.,l > C > 0 in Q2, then
ff
^{(t+ 1)av}^{2} dxdt < cf f
^{fF(}^{s}^{)}^{sxv}^{2}^{d}^{x}^{dt.}f22f22
Let E (x2 + at, x1 + ut), we have
f
1(t + 1)"ex(tt2(X))v^{2}(x, t) dxdtf13
x
= J f(t + 1)neK(tt2(x))_{ (V(^1} t) + J v„(r7, t) dr7) dxdt
f13
f
(t + ^{1)Ck} 3.1 +Ut
(v(
\2
e x(tt2(x)) [^ t) + f v, t) dry I
f xl  X2 //
523
484 L. YING, T. YANG and C. ZHU
f p l+at
< C J J ^{ (t + 1)ae}
^{K(tt2(}^{x}
^{))}ƒ
^{. ,}v2t) d^dxdt
^{2}t?3 +ot
f r
l+atf
^{x}^{^}+ C J J ^{ (t + }
^{1}^{)}^{a}^{e}^{K(tt2(}^{x))} I ^{( —x)}
^{X}v^(r), t) drjd dxdt
2+at 3
=Il +I2,
where I
Iand 12 are estimated as follows.
I
^{l}< c J J (t + 1)"v
2(ß, t) d^dt,
33 2z
and
r
^{t}1 ^{X}
^{2^+0't}^{+0t} ^{f}^{x2+at}I2 < C J(t + 1Y dt d^
v2(77,t) dr1J
ex(tt2(x))(^ — x) dxo x2 x2
< c ff(t+1)v(mt)ddt
t?2 U 23
< _{ C U} ^{(t + 1)} ^{"ui } ^{dxdt + Cq}
^{2}E (t + 1)"(z — f)
2dxdt.
Therefore
+00
(t+1)'$v(z—O^dxdt
ƒo
^{ 00}^{t}< Q1 U24
ff
^{(t+ 1)a}^{1}^{v}^{2}^{ dxdt + C }^{J J }
172^{i }
^{f"(s)s}^{x}^{Iv}^{2}^{ dxdt}+C
J
_{172Uí}^{f (t+1)au}_{3} ^{2 dxdt+C.}Finally
+00
J ^{(t+1)av}
^{2}^{dx+(1—Cq)} J
^{o fl}^{ r} t +^ (t+1)'uldxdt
ft p+00
+(i _C)
]
]
1f'(s)s.Iv^{2}dxdt
o 00
/'^{t }r^{+00}
(a +
^{C}JJ JJJ
q) (t + 1)ct
lv
2dxdt 0
^{00}f^{+00}
+ J ^{v}
^{2}(x, 0) dx + Cq.
(2.32)Combining (2.32) and the above estimates yields
J
r+^(t+l)°W(v2+u1+u2)dx/^{'}t /^{'}+oe
+ßJ / (t+l)^W(u1+u2+u2^)dxdt
0 JJ 00
r
^{+00}< C
J
^{W(N)}^{ (v}^{2} (x, 0) + u1(x, 0) + u2 (x, 0)) dx + Cq, (2.33)for 0 < a
< Nand
y + a = N.Here
w°is the weight defined in (2.4) with ry
= 0.(2.31) gives
J
^{+}^{^(t + }^{1}^{)}^{a}^{W ((z — ()}^{2} + ((z — ()^{X})2) dx+oe
< C W(N)(x)(v2(x 0) + ui(x,0) + u2(x,0)) dx + C. (2.34) Now it remains to verify the
a priori
estimates (2.13) and (2.14). By (4.4) in [7] and the discussion above, it is easy to show that whenq
andf ±^ (v
2(x, 0) +ui (x, 0)) dx
are sufficiently small,+00 f^{+00} f^{+00}
J
v
2dx, J ^{ui } ^{dx,} ^{U2 dx}
are sufficiently small. Since
ft u2 dx
is bounded, using the Sobolev inequalityg2(x)<2( ^{(}
l
^{x}_{ g}2 dx I if 2
ƒ
^{,}^{ce}^{ (dg)2}dx ^{1/2}
^ /dx
and a simple continuity argument, we get (2.13) and (2.14). Therefore, by choosing
a = Nin (2.33) and (2.34) and using the Sobolev inequality again, we get the decay rate in the Theorem.
When p> 0 is not an integer. As in [9] [10], using the above inductive procedure
and the property of h, we haveff
^{1}+00
".
(t + 1)IPI
W(p[ ] (v
^{2}+ ui + u2) dx
+ (t +
^{1)W)(v2}+ u1 + u) dxdt
cc
/^
t +00
/^
+ß
J J
^{o} ^{oo} {t+1)^p^Wlpip>>(ui+u2+uz^)dxdtf+00
< C J ^{Wlpl (v}
^{2}(x, 0) + ui(x, 0) + u2(x, 0)) dx + Cq.
(2.35)486
For p < a < [p] + 1 and y = 0, (2.32) (2.27) and (2.29) give
+xt
1
^{".}^{+^}(t+1)CJ (v2
+ui
+uz)dx+ß J (t+l)° (ui
+u2+u2^)dxdtx 0
/^+^
<C
J
^{(v}^{2}(x,0)+u1(x,0)+u2(x,0))dx+Cq rt1
^{,.}^{+o}+ C
J
^{(t + }^{1)`^}^{1}^{(v}^{2}^{+ul }^{+ u2)dxdt.} ^{(2.36)}o
For simplicity, we denote I0(r)I2e f_+^{cX} _{+u1 }+u2) dx. Now we only need to estimate the last term of (2.36). As in [9] [10], this can be done by using the Hölder inequality:
f
^{0}^{t}^{(r+}_{ f}
^{1}^{)^ }^{'( (T)I0dr}+^
=
I
t(T+ ^{)}a1 W(P[P])([P]+1P)(P[PJ)([P]+1P)
°^ (q2)([Pl+1 P)+(P [P]) dedr
f t + [P]+1p
J
(r + 1)a1 ^ f WP1P102 de) O+oe p [p]
l W([P]+1p)02 dC)
dr
l
^{ + 1)}^{t}^{(,} ^{r}
^{}^{([PJ+}^{1}a) ((T + 1)[P] I0I2.J
^{[P]+1P}P [P]
r(r+ l)[P]I0IPIPI1)
dr
`PIP]
<C(IIp([P]+1p) + ql_{1}
1'(
^{T}^{ + 1)}^{}^{([P]+}^{1}^{ ^) ((}^{r + 1)1thi}^{2} IPI1)dr
pt [PJ+1P
< C (10(0)12([P]+1P) + q)^{ (I }^{(r} + 1)_ }+_)/([p1+1 )
d r )
fo
P [P]
(
^ (T+ t 1)[P1
IWI [P]1 dr 111
t [P]+1p
< C (10(0){p + q)
(1
^{0}^{ (r + }1)([P1+1c')/([P1+1P)dr
We have
f
+"0(t+l) ^{^} J (v2+u1+u2)dx
<CJ
( ( )u
2 x,0+ 0 u
^{2}1\ ) 2( ( )^{/}x, +u
2x,0))dx+Cq+Ct+ 1 a P
/_{ll^l )}^{/ }0
Ip2+q).
Combining this estimate and the same argument for the case when p is an integer, we get the decay rate in the Theorem. This completes the proof of the Theorem.
Acknowledgement. This work was finished when the first author visited the Department of Mathematics of the City University of Hong Kong. He is grateful to the department and to Professor R. Wong for their hospitality. Part of the research of the first author was supported by Doctorial Program Foundation of State Commission of Education. Part of the research of the second author was supported by the RGC Competitive Earmarked Research Grant #9040190. Part of the research of the third author was supported by the grant of the National Youth Science Foundation of China #19301038. Finally the authors express their thanks to the referees for bringing their attention to the works of T. Li [4] and M.
Nishikawa [10], and for the valuable comments to improve the paper.
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