LARGE TIME BEHAVIOR OF SOLUTIONS TO THE GENERALIZED BURGERS EQUATIONS

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1 Kato, M. Osaka J. Math. 44 (27), LAGE TIME BEHAVIO OF SOLUTIONS TO THE GENEALIZED BUGES EQUATIONS MASAKAZU KATO (eceived June 6, 26, revised December 1, 26) Abstract We study large time behavior of the solutions to the initial value problem for the generalized Burgers equation. It is known that the solution tends to a self-similar solution to the Burgers equation at the rate t 1 log t in L as t. The aim of this paper is to show that the rate is optimal under suitable assumptions and to obtain the second asymptotic profile of large time behavior of the solutions. 1. Introduction This paper is concerned with large time behavior of the global solutions to the generalized Burgers equations: (1.1) (1.2) u t + ( f (u)) x = u xx, t, x ¾, u(x, ) = u (x), where u ¾ L 1 () and f (u) = (b2)u 2 +(c3)u 3 with b =, c ¾. The subscripts t and x stand for the partial derivatives with respect to t and x, respectively. It is well-known that the solution of (1.1) and (1.2) tends to a nonlinear diffusion wave defined by (1.3) (x, t) Ô 1 x Ô, t, x ¾, 1 + t 1 + t where (1.4) (1.5) (x) 1 b Æ (e bæ2 1)e (x 2 4) Ô + (e bæ2 1) Ê x2 e y2 dy, u (x) dx. 2 Mathematics Subject Classification. Primary 35B4; Secondary 35L65.

2 924 M. KATO By the Hopf-Cole transformation in Hopf [4] and Cole [1], we see that it is a solution of the Burgers equation b (1.6) t + 2 x 2 = xx, t, x ¾, satisfying (1.7) (x, ) dx = Æ. Concerning the convergence rate of the nonliner diffusion wave (x, t) to the original solution u(x, t), we can infer the following result from the argument given in Kawashima [7] and Nishida [1], which deal with a class of system, in the case where u(x, t) is a scalar unknow function without any essential difficulty: If u ¾ L 1 () H 1 () for some ¾ (, 1) and u H 1 + u L 1 is small, then we have (1.8) u(, t) (, t) L C(1 + t) 1+«(u H 1 + u L 1 ), t, where «= (1 )2. Here, for an integer k, H k () denotes the space of functions u = u(x) such that x l are L2 -functions on for l k, endowed, with Ê the norm H k, while L 1 () is a subset of L1 () whose elements satisfy u L 1 u(1 + x) dx. This observation lead to a natural question whether it is possible to take «= in (1.8) for the extreme case = 1 or not. An attempt to answer the question can be found Ê in Matsumura and Nishihara Ê [9]. To be more precise, we put x Û (x) = exp( (b2) u x (y) dy) exp( (b2) (y, ) dy) and we suppose that Û ¾ H 2 () L 1 () and Û H 2 + Û L 1 + u L 1 is small. Then the estimate (1.9) u(, t) (, t) L C(1 + t) 1 log(2 + t)(û H 2 + Û L 1 + Æ 32 ), t holds instead of (1.8). The aim of this paper is to show that the decay rate of estimate (1.9) is actually optimal, unless Æ = or c =. Indeed, the second asymptotic profile of large time behavior of the solutions is given by (1.1) V (x, t) cd 12 Ô V where x Ô (1 + t) 1 log(2 + t), t, x ¾, 1 + t (1.11) (1.12) V (x) (b (x) x)e x 2 4 (x), b x (x) exp (y) dy, 2

3 (1.13) ASYMPTOTIC BEHAVIO OF GENEALIZED BUGES EQUATIONS 925 d 1 (y) 3 (y) dy. Bisides, we can take the initial data from L 1 1 () H 1 (), analogously to the works of [7], [1]. And we set, for k, E k, u H k +u L 1. Then we have the following result. Theorem 1.1. Assume that u ¾ L 1 () H 1 () and E 1, is small. Then the initial value problem for (1.1) and (1.2) has a unique global solution u(x, t) satisfying u ¾ C ([, ); H 1 ) and x u ¾ L 2 (, ; H 1 ). Moreover, if u ¾ L 1 1 () H 1 () and E 1,1 is small, then the solution satisfies the estimate (1.14) u(, t) (, t) V (, t) L C E 1,1 (1 + t) 1, t 1. Here (x, t) is defined by (1.3), while V (x, t) is defined by (1.1). EMAK 1.2. In Liu [8], the initial value problem for the Burgers equations (1.1) and (1.2) is studied, provided c = implicity at p.42. After the proof of Theorem 2.2.1, it is mentioned, without proof, that if we assume (1 + x) 2 u (x) Æ and Æ is small, then the estimate u(, t) (, t) L C Æ(1 + t) 1, t 1 holds. However, from our result, the above estimate fails true for the case cæ =. We remark that the estimate similar to (1.14) was obtained for other types of Burgers equation such as KdV-Burgers in Hayashi and Naumkin [3] and Kaikina and uiz-paredes [5], and Benjamin-Bona-Mahony-Burgers in Hayashi, Kaikina and Naumkin [2]. 2. Preliminaries In order to prove the basic estimates given by Lemma 3.2, Lemma 3.3 and Lemma 3.5, we prepare the following two lemmas. The first one is concerned with the decay estimates for semigroup e t associated with the heat equation. For the proof, see Kawashima [6]. Lemma 2.1. estimate Let k be a positive integer. Suppose q ¾ H k () L 1 (). Then the (2.1) l x et q L 2 C(1 + t) (14+l2) q L 1 e C¼t l x q L 2, t holds for any l =, 1,, k.

4 926 M. KATO The second one is related to the diffusion wave (x, t) and the heat kernel G(x, t). The explicit formula of (x, t) and G(x, t) are given by (1.3) and (2.2) G(x, t) = 1 Ô 4t e (x 2 4t), t, x ¾, respectively. It is easy to see that (2.3) (x, t) CÆ(1 + t) (12) e x 2 (4(1+t)), t, x ¾. Moreover, we get the following (see e.g. [9]). Let «and be positive integers. Then, for p ¾ [1, ], the esti- Lemma 2.2. mates (2.4) (2.5) x «t (, t) L p CÆ(1 + t) (12)(1 1p) «2, t, x «(12)(1 1p) «2 t G(, t) L p Ct, t hold. For the latter sake, we introduce 1, 2 defined by x b x (2.6) 1 (x, t) Ô = exp 1 + t 2 (2.7) We easily have 2 (x, t) 1 1 (x, t). (y, t) dy, (2.8) (2.9) min1, e bæ2 1 (x, t) max1, e bæ2, min1, e bæ2 2 (x, t) max1, e bæ2. Moreover, we get Corollary 2.3. (2.1) (2.11) (2.12) Let l be a positive integer. For i = 1, 2, if Æ 1, then we have x l i(, t) L 1 CÆ(1 + t) (l 1)2, x l i(, t) L CÆ(1 + t) 12, x l i(, t) L 2 CÆ(1 + t) (l2 14). Proof. We shall prove only (2.1), (2.11) and (2.12) for i = 1, since we can prove (2.1), (2.11) and (2.12) for i = 2 in a similar way. We put q 1 (x) = e (b2)x,

5 ASYMPTOTIC BEHAVIO OF GENEALIZED BUGES EQUATIONS 927 Ê q 2 (x, t) = x (y, t) dy. Then we have, for l 1, (2.13) l x 1(x, t) = C l,m1,,m l d p q 1 dx p (q 2(x, t))( x q 2 (x, t)) m 1 (l x q 2(x, t)) m l = C l,m1,,m l b 2 p 1 (x, t)((x, t)) m 1 (l 1 x (x, t)) m l, where the sum is taken for all (m 1,, m l ) ¾ N k such that m 1 + 2m lm l = l, and p = m m l, 1 p l. First we derive (2.1). Since m 1 + 2m lm l = l, there exists j ¾ 1, 2,, l such that m j 1. Therefore, by Lemma 2.2 and (2.13), we have l x 1(, t) L 1 C (, t) m 1 L x (, t) m 2 L x j 1 (, t) m j L m j x l 1 (, t) m l L CÆ (1 + t) m 12 (1 + t) 2m 22 (1 + t) ( jm j 1)2 (1 + t) lm l 2 CÆ(1 + t) (l 1)2. Hence we get (2.1). Next we derive (2.11). By Lemma 2.2 and (2.13), we have l x 1(, t) L C (, t) m 1 L x (, t) m 2 L l x 1 CÆ (1 + t) m 12 (1 + t) 2m 22 (1 + t) lm l 2 CÆ(1 + t) l2. (, t) m l L Hence we get (2.11). Finally, by the interpolation inequality f L p f 1 1p L f 1p L 1 have from (2.1) and (2.11) l x 1(, t) L 2 l x 1(, t) 12 L l x 1(, t) 12 L 1 CÆ(1 + t) (l2 14). (1 p ), we This completes the proof. 3. Basic estimates We deal with the following linearized equations which coressponds to (4.1), (4.11) below: (3.1) (3.2) z t = z xx (bz) x, t, x ¾, z(x, ) = z (x). The explicit representation formula (3.4) below plays a crucial role in our analysis.

6 928 M. KATO (3.3) Lemma 3.1. If we set U[Û](x, t, ) = then the solutions for (3.1) and (3.2) is given by y x (G(x y, t ) 1 (x, t)) 2 (y, ) Û() d dy, (3.4) z(x, t) = U[z ](x, t, ), t, x ¾. t, x ¾, Proof. If we put (3.5) r(x, t) = x z(y, t) dy, then we see from (3.1), (3.2) that r(x, t) satisfies (3.6) (3.7) r t = r xx br x, t, x ¾, r(x, ) = x z (y) dy. Then a direct computation yields r(x, t) (3.8) 1 (x, t) where 1 is defined by (2.6). Therefore, we have (3.9) r(x, t) = 1 (x, t) Hence (3.9), (3.5), (3.7) and (2.7) yield (3.4). t = r(x, t), 1 (x, t) xx G(x y, t) 1 (y, ) 1 r(y, ) dy. Next we derive the decay estimates (3.1) and (3.22) below for the homogenous equation (3.1). Here, to prove the estimate, we follow the argument which is used to show the estimate for the semigroup e t in [7]. Lemma 3.2. Let Ê ¾ [, 1], k be a positive interger and p ¾ [1, ]. Assume that Æ 1, z ¾ L 1 () and z (x) dx =. Then, the estimate (3.1) l x U[z ](, t, ) L p Ct (1 1p+ +l)2 z L 1, t holds for any l =, 1,, k.

7 ASYMPTOTIC BEHAVIO OF GENEALIZED BUGES EQUATIONS 929 Ê Proof. Since z (x) dx =, by the integration by parts with respect to y, we have from (3.3) y (3.11) U[z ](x, t, ) = z (y) x (G(x, t) 1 (x, t)) 2 (, ) d dy. Using this expression, we shall show that (3.12) (3.13) l x U[z ](, t, ) L Ct x l U[z ](, t, ) L 1 Ct (1+ +l)2 z L 1, t, ( +l)2 z L 1, t. The desired estimate follows from the interpolation inequality. Actually, we have (3.14) l x U[z ](, t, ) L p x l U[z ](, t, ) 1 1p L l x U[z ](, t, ) 1p L 1 (1 Ct 1p+ +l)2 z L 1. First we shall prove (3.12). From (3.11), we have l x U[z ](x, t, ) z (y) y z (y)i 1 (x, y, t) dy. l+1 x (G(x, t) 1 (x, t)) 2 (, ) d dy By using (2.11), (2.9), (2.5) and the Hölder inequality, we have (3.15) I 1 C C Ct l+1 m= l+1 m= l+1 m x 1 (, t) L y (1 + t) (l+1 m)2 m x G(x, t) L (1+ +l)2 y. Hence (3.14) and (3.15) yield (3.12). Next we shall prove (3.13). From (3.11), we have x l U[z ](, t, ) L 1 where we put (3.16) I 2 (y, t) = z (y) y z (y)i 2 (y, t) dy, y m x G(x, t) 2(, ) d 1(1 )y x l+1 (G(x, t) 1 (x, t)) 2 (, ) d dy dx l+1 x (G(x, t) 1 (x, t)) 2 (, ) d dx.

8 93 M. KATO In order to get (3.13), it suffices to show (3.17) (3.18) I 2 Ct I 2 Cyt l2, (1+l)2. In fact, if we would have these estimates, then x l U[z ](, t, ) L 1 C z (y)i 1 2 I 2 dy z (y)t l(1 )2 t (1+l) 2 y dy Ct ( +l)2 z L 1. (3.19) First we shall prove (3.17). From (3.16) we have I 2 C = C l+1 m= m l+1 x 1 (x, t) 1(x, t) l m= I 2,1 + I 2,2. y m l+1 x 1 (x, t) y m x G(x, t) 2(, ) d dx l+1 x G(x, t) 2 (, ) d dx y By using the integration by parts with respect to, we have (3.2) I 2,1 C C Ct m x G(x, t) 2(, ) d dx l x G(x y, t) 2(y, ) l x G(x, t) 2(, ) dx y l2, x l G(x, t) 2 (, ) d dx l x G(x, t) dx 2 (, ) x l G(x, t) dx d where we have used (2.8), (2.9), (2.1) and (2.5). On the other hand, from (2.9), (2.1) and (2.5), we have (3.21) Hence we obtain (3.17). I 2,2 C C l m= l m= l+1 m x 1 (, t) L 1 m x G(, t) L 1 (1 + t) (l m)2 t m2 Ct l2.

9 ASYMPTOTIC BEHAVIO OF GENEALIZED BUGES EQUATIONS 931 Next we shall prove (3.18). From (3.16), (2.9), (2.11) and (2.5) we have I 2 C C l+1 m= l+1 m= x l+1 m 1 (, t) L x m G(, t) L 1y (1 + t) (l+1 m)2 t Hence we get (3.18). This completes the proof. m2 y Ct (1+l)2 y. Lemma 3.3. Let k be a positive interger. Assume that Æ 1, z ¾ H k () L 1 () and Ê z (x) dx =. Then the estimate (3.22) l x U[z ](, t, ) L 2 C(1 + t) (14+l2) z L 1 e t z H l, t holds for any l =, 1,, k. Proof. We have from (3.3) (3.23) l x U[z ](x, t, ) = where we put = l+1 m= y x l+1 (G(x y, t) 1 (x, t)) 2 (y, ) z () d dy l + 1 x l+1 m 1 (x, t) x m m G(x y, t) 2(y, ) = x l+1 1 (x, t) l+1 l m m=1 G(x y, t) 2 (y, ) l+1 m x 1 (x, t) (3.24) J(y) = 2 (y, ) y y z () d dy y x m 1 G(x y, t) y J(y) dy, z () d. Therefore we have from (2.9), (2.5) and Corollary 2.3 z () d dy (3.25) x l U[z ](, t, ) L 2 Cx l+1 1 (, t) L 2G(, t) L 1z L1 l+1 m=1 l+1 m x 1 (, t) L m 1 x e t [ x J] L 2 C(1 + t) (14+l2) z L1 l+1 m=1 (1 + t) (l+1 m)2 x m 1 e t [ x J] L 2.

10 932 M. KATO From (3.24), (2.9), (2.4) and Corollary 2.3, we have (3.26) x x J L 1 C (x, t) z () d z L 1 L 1 ( x ) Cz L 1, and for m 1 (3.27) m x m J L 2 C n= m x n 2 (x, )x n x C m x 2(, ) L 2z L 1 C(z L 1 + z H m 1). z () d L 2 ( x ) m n=1 m n x 2 (, ) L n 1 x z L 2 Hence, for 1 m l + 1, from (3.26), (3.27) and Lemma 2.1, we have (3.28) m 1 x e t [ x J] L 2 C(1 + t) (14+(m 1)2) x J L 1 e t m x J L 2 C(1 + t) (14+(m 1)2) z L 1 e t (z L 1 + z H m 1). Therefore by (3.25) and (3.28), we obtain (3.22). This completes the proof. From Lemma 3.2 and Lemma 3.3, we get the following uniform estimate. Corollary Ê 3.4. Let k be a positive integer. Assume that Æ 1, z ¾ L 1 1 () H k () and z (x) dx =. Then the estimate x l U[z ](, t, ) L 2 C E l,1 (1 + t) (34+l2), t holds for any l =, 1,, k. Here E l,1 = z H l + z L 1 1. Next we derive the decay estimate (3.29) below in the same way as Lemma 3.6 of [6]. The estimate will be used to get the decay rate of the solution Û(x, t) for the problem (4.1) and (4.11). Bisides, we denote the Fourier transform of f (x) Ô Ê ¾ L 2 () L 1 () by ˆf () F[ f ]() = (1 2) e i x f (x) dx, and set k H1 k () f : L 1 loc () f H 1 k x m f L µ. 1 m=

11 ASYMPTOTIC BEHAVIO OF GENEALIZED BUGES EQUATIONS 933 Lemma 3.5. Let k be a positive integer. Suppose Æ 1 and Û ¾ C (, ; H k ) C (, ; H1 k ). Then the estimate t (3.29) l x C U[ x Û()](x, t, ) d L 2 ( x ) t2 l m= (1 + t ) (34+l2) Û(, ) L 1 d t t2 l t m= (1 + t ) 34 (1 + ) (l m)2 m x Û(, ) L 1 d e (t ) (1 + ) (l m) m x Û(, )2 L 2 d 12 holds for any l =, 1,, k. Proof. From (3.3) and Corollary 2.3, we have (3.3) t l x where we put (3.31) I (x, t) = l+1 U[ x Û()](x, t, ) d C L 2 ( x ) t From Lemma 2.1, we have (3.32) I (, t) L 2 C C C t t t n= (1 + t) (l+1 n)2 n x I (, t) L 2, G(x y, t ) 2 (y, )Û(y, ) dy d. e (t ) [( 2 Û)()]( ) L 2 d (1 + t ) 14 Û(, ) L 1 + e (t ) Û(, ) L 2 d (1 + t ) 14 Û(, ) L 1 d t 12 e (t ) Û(, ) 2 L d, 2 because t t for t. In the following, let 1 n l + 1. It follows that (3.33) x n I (, t) L 2 (i)n Î (, t) L 2 (1) + (i) n Î (, t) L 2 (1) I 1 + I 2.

12 934 M. KATO First, we evaluate I 1. From (3.31), we have I 1 t2 + t t2 I 1,1 + I 1,2. (i) n e (t )2 F[ 2 Û](, ) L 2 (1) d (i) n e (t )2 F[ 2 Û](, ) L 2 (1) d Since 1 j e 2(t )2 d C(1 + t ) ( j+1)2, for j, we have (3.34) I 1,1 C C t2 t2 While, we have from Corollary 2.3 (3.35) I 1,2 C C C t t2 t t2 t t2 12 sup F[ 2 Û](, ) 2n e d 2(t )2 d 1 1 (1 + t ) (2n+1)4 Û(, ) L 1 d. sup (i) n 1 F[ 2 Û](, ) 1 1 (1 + t ) 34 x n 1 ( 2 Û)(, ) L 1 d 2 e 2(t )2 d 12 d n 1 (1 + t ) 34 (1 + ) (n 1 m)2 x m Û(, ) L 1 d. m= Therefore, we get from (3.34) and (3.35) (3.36) I 1 C t2 t (1 + t ) (2n+1)4 Û(, ) L 1 d n 1 (1 + t ) 34 (1 + ) (n 1 m)2 x m Û(, ) L 1 d. t2 m= Next we evaluate I 2. For 1, by using the Schwarz inequality, we have (i) n Î (, t) C C t t e (t )2 (i) n 1 F[ 2 Û](, ) d e (t )2 (i) n 1 F[ 2 Û](, ) 2 d 12.

13 ASYMPTOTIC BEHAVIO OF GENEALIZED BUGES EQUATIONS 935 Therefore we find from (3.33) and Corollary 2.3 t I 2 C e (t ) (i) n 1 F[ 2 Û](, ) 2 d d (3.37) C C t t 1 e (t ) x n 1 ( 2 Û)(, ) 2 L 2 d e (t ) n (1 + ) (n 1 m) x m Û(, )2 L d. 2 m= This com- Summarizing (3.3), (3.32), (3.33), (3.36) and (3.37), we obtain (3.29). pletes the proof. 4. Proof of Theorem 1.1 In order to prove our result, we introduce the following auxiliary problem: (4.1) (4.2) Ú t = Ú xx (bú) x Ú(x, ) =. c 3 x 3, t, x ¾, We derive the decay estimate for the solution Ú(x, t) to the above problem. Lemma 4.1. Let l be an integer. Then we have (4.3) l x Ú(, t) L 2 CÆ3 (1 + t) (34+l2) log(2 + t), t. Proof. By the Duhamel principle, we see from Lemma 3.1 that (4.4) Ú(x, t) = From Lemma 3.5, we get t c U x 3 3 () (x, t, ) d. (4.5) t2 x l Ú(, t) L 2 C (1 + t ) (34+l2) 3 (, ) L 1 d l m= t t2 l t m= I 1 + I 2 + I 3. (1 + t ) 34 (1 + ) (l m)2 m x ( 3 (, )) L 1 d e (t ) (1 + ) (l m) m x ( 3 (, )) 2 L 2 d 12

14 936 M. KATO It follows from Lemma 2.2 that for p ¾ [1, ] (4.6) x m ( 3 (, )) L p C Using (4.6), we obtain m n= m n s= CÆ 3 m n= x n (, ) L s x (, ) L m x n s (, ) L p m n s= CÆ 3 (1 + ) (3 (1p)+m)2. (1 + ) (1+n)2 (1 + ) (1+s)2 (1 + ) (1 (1p)+m n s)2 (4.7) (4.8) t2 I 1 CÆ 3 (1 + t ) (34+l2) (1 + ) 1 d CÆ 3 (1 + t) (34+l2) log(2 + t), t I 2 CÆ 3 (1 + t ) 34 (1 + ) (1+l2) d t2 CÆ 3 (1 + t) (34+l2), and (4.9) t I 3 CÆ 3 e (t ) (1 + ) (52+l) d CÆ 3 (1 + t) (54+l2). 12 This completes the proof. Our first step to prove Theorem 1.1 is the following. Proposition 4.2. Let k 1 be an integer. Assume that u ¾ L 1 () H k () and E k, is small. Then the initial value problem for (1.1) and (1.2) has a unique global solution u(x, t) satisfying u ¾ C ([, ); H k ) and x u ¾ L 2 (, ; H k ). Moreover, if u ¾ L 1 1 () H k () and E k,1 is small, then the estimate l x (u(, t) (, t) Ú(, t)) L 2 C E k,1(1 + t) (34+l2), t holds for l k. In particular, u(, t) (, t) Ú(, t) L C E 1,1 (1 + t) 1, t. Here (x, t) is defined by (1.3), while Ú(x, t) is the solution for the problem (4.1) and (4.2).

15 ASYMPTOTIC BEHAVIO OF GENEALIZED BUGES EQUATIONS 937 Proof. We can show the local existence and uniqueness of the solution to (1.1) and (1.2) by standard argument. Using the argument of Theorem 8.2 in [7], we can continue the local solution globally in time. We shall prove only the decay estimate. We note that the smallness of E k,1 implies that of Æ by (1.5). We put Then Û(x, t) satisfies Û(x, t) = u(x, t) (x, t) Ú(x, t). (4.1) (4.11) Û t = Û xx (bû) x + (g(û,, Ú)) x, t, x ¾, Û(x, ) = Û (x), where we have set Û (x) = u (x) (x, ) and (4.12) g(û,, Ú) = b (Û + Ú)2 2 c 3 [Û3 + Ú 3 + 3(Û + Ú)(Û + )( + Ú)]. Since u (x), (x, ) ¾ L 1 1 () H k (), we have Û (x) ¾ L 1 1 H k. Bisides by (1.5) and (1.7), (4.13) Now, we define N(T ) by (4.14) N(T ) = sup tt k m= Û (x) dx =. (1 + t) 34+m2 m x Û(, t) L 2. For l k 1, we have from (4.14) and the Sobolev inequality (4.15) l x Û(, t) L N(T )(1 + t) (1+l2). We shall show that for l k (4.16) (4.17) l x g(, t) L 1 C(1 + t) (32+l2) ((Æ log(2 + t)) 2 + N(T ) 2 ), l x g(, t) L 2 C(1 + t) (74+l2) ((Æ log(2 + t)) 2 + N(T ) 2 ). We shall prove only (4.16), since we can prove (4.17) in a similar way. Here and later, Æ and N(T ) are assumed to be small. We put h 1 (x, t) = Û(x, t) + Ú(x, t), h 2 (x, t) = Û(x, t) + (x, t) and h 3 (x, t) = (x, t) + Ú(x, t). Then, for m k, we

16 938 M. KATO have from (4.14), (4.3) and (2.4) (4.18) (4.19) (4.2) m x h 1(, t) L 2 C(1 + t) (34+m2) (Æ log(2 + t) + N(T )), m x h 2(, t) L 2 C(1 + t) (14+m2) (Æ + N(T )), m x h 3(, t) L C(1 + t) (12+m2) Æ. Hence, we have from (4.18), (4.19), (4.2), (4.14), (4.15), (4.3) and (2.4) (4.21) (4.22) (4.23) and (4.24) x l ((Û + Ú)2 (, t)) L 1 C x l (Û3 (, t)) L 1 C x l (Ú3 (, t)) L 1 C x l (h 1h 2 h 3 )(, t) L 1 C l 1 l m m= n= l m= m x h 1(, t) L 2 l m x h 1 (, t) L 2 C(1 + t) (32+l2) ((Æ log(2 + t)) 2 + N(T ) 2 ), m x Û(, t) L n x Û(, t) L 2l m n x Û(, t) L 2 l x Û(, t) L 2Û(, t) L 2Û(, t) L C(1 + t) (52+l2) N(T ) 3, l l m m= n= m x Ú(, t) L n x Ú(, t) L 2l m n x Ú(, t) L 2 C(1 + t) (52+l2) (Æ 3 log(2 + t)) 3 C(1 + t) (32+l2) (Æ log(2 + t)) 2 l l m m= n= x m h 1(, t) L 2x n h 2(, t) L 2x l m n h 3 (, t) L C(1 + t) (32+l2) (Æ log(2 + t) + N(T ))(Æ + N(T )) C(1 + t) (32+l2) ((Æ log(2 + t)) 2 + N(T ) 2 ). Summing up these estimates, we obtain (4.16) from (4.12). Applying the Duhamel principle for the problem (4.1) and (4.11), we have (4.25) Û(x, t) = U[Û ](x, t, ) + t U[ x g(û,, Ú)()](x, t, ) d, t, x ¾.

17 ASYMPTOTIC BEHAVIO OF GENEALIZED BUGES EQUATIONS 939 For l k, we have from (4.25), Corollary 3.4 and Lemma 3.5 (4.26) x l Û(, t) L 2 C(1 + t) (34+l2) E k,1 l m= t t2 l t m= I 1 + I 2 + I 3 + I 4. First we evaluate I 2. From (4.16), we have t2 (1 + t ) (34+l2) g(, ) L 1 d (1 + t ) 34 (1 + ) (l m)2 m x g(, ) L 1 d e (t ) (1 + ) (l m) m x g(, )2 L 2 d 12 (4.27) I 2 C t2 (1 + t ) (34+l2) (1 + ) 32 ((Æ log(2 + )) 2 + N(T ) 2 ) d C(1 + t) (34+l2) (Æ 2 + N(T ) 2 ). Next we evaluate I 3. From (4.16), we have I 3 C (4.28) l m= t t2 (1 + t ) 34 (1 + ) (3+l)2 ((Æ log(2 + )) 2 + N(T ) 2 ) d C(1 + t) (54+l2) ((Æ log(2 + t)) 2 + N(T ) 2 ). Finally we evaluate I 4. From (4.17), it follows that I 4 C (4.29) l t m= e (t ) (1 + ) (72+l) ((Æ log(2 + )) 4 + N(T ) 4 ) d C(1 + t) (74+l2) ((Æ log(2 + t)) 2 + N(T ) 2 ). Since Æ E k,1, if E k,1 is small, then we obtain the inequality (4.3) (1 + t) 34+l2 x l Û(, t) L 2 C(E k,1 + N(T ) 2 ). Therefore, (4.3) gives the desired estimate N(T ) C E k,1. This completes the proof. 12 To prove Theorem 1.1, it is sufficent to show Proposition 4.3 below by virtue of Proposition 4.2. Although the similer estimate was shown by Lemma 3 in [5], but we need to modify the proof of it in order to avoid the logarithmic term in the right-hand side.

18 94 M. KATO Proposition 4.3. Assume that Æ 1. Then the estimate (4.31) Ú(, t) V (, t) L CÆ 3 (1 + t) 1, t 1 holds. Here, Ú(x, t) is the solution for the problem (4.1) and (4.2), while V (x, t) is defined by (1.1). (4.32) Proof. It follows from (4.4) and (3.3) that Ú(x, t) = c 3 = c 3 c 3 t t t2 t2 I 1 + I 2. U[ x 3 ()](x, t, ) d x (G(x y, t ) 1 (x, t)) 2 (y, ) 3 (y, ) dy d x (G(x y, t ) 1 (x, t)) 2 (y, ) 3 (y, ) dy d First we evaluate I 1. By the integration by parts with respects to y, we have I 1 = c t 3 1(x, t) x G(x y, t ) + b2 (x, t)g(x y, t ) t2 2 (y, ) 3 (y, ) dy d = c t 3 1(x, t) G(x y, t ) t2 y ( 2 (y, ) 3 (y, )) + b 2 2(y, )(x, t) 3 (y, ) dy d. Therefore, we get from Lemma 2.2, (2.8) and (2.9) (4.33) I 1 (, t) L C t t2 G(, t ) L 1(, ) 4 L + (, )2 L x (, ) L t CÆ 3 (1 + ) 2 d t2 CÆ 3 (1 + t) 1. + (, t) L (, ) 3 L d

19 ASYMPTOTIC BEHAVIO OF GENEALIZED BUGES EQUATIONS 941 Next we evaluate I 2. If we put (4.34) (x, t, y, ) c 3 x(g(x y, t ) 1 (x, t)), then we have (4.35) I 2 = t2 (x, t, y, ) 2 (y, ) 3 (y, ) dy d. Spliting the y-integral at y = and making the integration by parts, we have (4.36) I 2 = t2 + t2 t2 y (x, t, y, ) y (x, t, y, ) (x, t,, ) I 3 + I 4 + I 5. y y 2 (, ) 3 (, ) d dy d 2 (, ) 3 (, ) d dy d 2 (, ) 3 (, ) d d First we consider I 3. From (4.34) and Lemma 2.2, we have (4.37) sup t2 C C sup x¾ sup y (x, t, y, ) y¾ sup (x 2 G(, t ) L + (, t) L x G(, t ) L ) t2 sup (t ) 32 Ct t2 32. From (4.36) and (4.37), we have I 3 (, t) L Ct t2 32 y (, ) 3 d dy d. Then, by the integration by parts with respect to y, it follows from Lemma 2.2 and (2.3) that (4.38) I 3 (, t) L Ct CÆ 3 t t2 32 t2 32 CÆ 3 (1 + t) 1 y(y, ) 3 dy d (1 + ) 1 y Ô 1 + e y2 (4(1+)) dy d for t 1. Similarly, we have (4.39) I 4 (, t) L CÆ 3 (1 + t) 1.

20 942 M. KATO Next we consider I 5. From (2.6), (2.7), (1.3) and (1.13), we have hence it follows from (4.34) and (4.36) that 2 (, ) 3 (, ) d = d(1 + ) 1, (4.4) I 5 = d t2 = cd 3 1(x, t) (x, t,, )(1 + ) 1 d t2 (1 + ) 1 ( x G(x, t ) x G(x, t + 1)) + b 2 (x, t)(g(x, t ) G(x, t + 1)) d cd 3 1(x, t) x G(x, t + 1) + b2 2 + t (x, t)g(x, t + 1) log 2 I 5,1 + I 5,2. In order to evaluate I 5,1, we shall use (4.41) l x G(x, t ) l x G(x, t + 1) C(t ) (3+l)2 (1 + ) for l =, 1 and t2. This estimate can be shown by observing that 1 x l G(x, t ) l x G(x, t + 1) = (1 + ) ( t x l G)(x, 1 + t (1 + )) d and by recalling (2.5). Since d CÆ 3 by (1.13), we have from (4.41) (4.42) t2 I 5,1 (, t) L CÆ 3 (1 + ) 1 (t ) 2 (1 + ) + (1 + t) 12 (t ) 32 (1 + ) d t2 CÆ 3 (t ) 2 d CÆ 3 (1 + t) 1. Finally, we deal with I 5,2. From (4.4), (2.2), (1.3) and (2.6), it follows that I 5,2 = cd x x 12 Ô x Ô b Ô Ô e x 2 (4(1+t)) (1 + t) t 1 + t 1 + t (log(t + 2) log 2). We have from (1.1) and (1.11), (4.43) I 5,2 (, t) V (, t) L CÆ 3 (1 + t) 1.

21 ASYMPTOTIC BEHAVIO OF GENEALIZED BUGES EQUATIONS 943 Summarizing (4.32), (4.33), (4.36), (4.38), (4.39), (4.4), (4.42) and (4.43), we obtain (4.31). This completes the proof. ACKNOWLEDGMENT. The author would like to express his sincere gratitude to Professor Hideo Kubo for his unfailing guidance and valuable advices. The author is also grateful to Professor Nakao Hayashi and Professor Soichiro Katayama for thier stimulating discussions. eferences [1] J.D. Cole: On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. IX (1951), [2] N. Hayashi, E.I. Kaikina and P.I. Naumkin: Large time asymptotics for the BBM-Burgers equation, to appear in Ann. Inst. H. Poincaré. [3] N. Hayashi and P.I. Naumkin: Asymptotics for the Korteweg-de Vries-Burgers equation, to appear in Acta Math. Sin. (Engl. Ser.). [4] E. Hopf: The partial differential equation u t + uu x = u xx, Comm. Pure Appl. Math. 3 (195), [5] E.I. Kaikina and H.F. uiz-paredes: Second term of asymptotics for KdVB equation with large initial data, Osaka J. Math. 42 (25), [6] S. Kawashima: The asymptotic equivalence of the Broadwell model equation and its Navier- Stokes model equation, Japan. J. Math. (N.S.) 7 (1981), [7] S. Kawashima: Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. oy. Soc. Edinburgh Sect. A 16 (1987), [8] T.-P. Liu: Hyperbolic and Viscous Conservation Laws, CBMS-NSF egional Conference Series in Applied Mathematics 72, SIAM, Philadelphia, PA, 2. [9] A. Matsumura and K. Nishihara: Global Solutions of Nonlinear Differential Equations Mathematical Analysis for Compressible Viscous Fluids, Nippon-Hyoron-Sha, Tokyo, 24, (in Japanese). [1] T. Nishida: Equations of motion of compressible viscous fluids; in Patterns and Waves (ed. T. Nishida, M. Mimura, H. Fujii), Kinokuniya, Tokyo, North-Holland, Amsterdam, 1986, Department of Mathematics Osaka University Toyonaka, Osaka Japan smv648km@ecs.cmc.osaka-u.ac.jp