Asymptotics of dissipative nonlinear evolution equations with ellipticity: different end states

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1 J. Math. Anal. Appl. 33 5) Asymptotics of dissipative nonlinear evoltion eqations with ellipticity: different end states enjn Dan, Changjiang Zh Laboratory of Nonlinear Analysis, Department of Mathematics, Central China Normal University, Whan 4379, P China eceived 5 Janary 4 Available online 3 November 4 Sbmitted by P. Smith Abstract In this paper, we consider the global existence and asymptotic behaviors of soltions to the Cachy problem for the following nonlinear evoltion eqations with ellipticity and dissipative effects: ψt = α)ψ θ x + αψ xx, E) θ t = α)θ + νψ x + ψθ x + αθ xx, with initial data ψ, θ)x, ) = ψ x), θ x) ) ψ ±,θ ± ) as x ±, I) where α and ν are positive constants sch that α<, ν<4α α). Under the assmption that ψ + ψ + θ + θ is sfficiently small, we show that if the initial data is a small pertrbation of the diffsion waves defined by.5) which are obtained by the diffsion eqations.), soltions to Cachy problem E) and I) tend asymptotically to those diffsion waves with exponential rates. The analysis is based on the energy method. The same problem was stdied by Tang and Zhao [J. Math. Anal. Appl ) ] for the case of ψ ±,θ ± ) =, ). 4 Elsevier Inc. All rights reserved. Keywords: Evoltion eqations; Diffsion waves; Decay rate; Energy method; A priori estimates * Corresponding athor. address: cjzh@ccn.ed.cn C. Zh). -47X/$ see front matter 4 Elsevier Inc. All rights reserved. doi:.6/j.jmaa.4.6.7

2 6. Dan, C. Zh / J. Math. Anal. Appl. 33 5) Introdction Many systems, describing the essential mechanism of nonlinear interaction between ellipticity and dissipation, arise broadly from physical and mechanical fields, cf. [3,6,7,9]. These systems are far from being completely investigated in mathematics so far de to their complexity. However, the well nderstanding of these mathematical models, reversely, will give s mch help to yield insight into the physical phenomenon and mechanical law. A set of simplified eqations was ths proposed by Hsieh [5], ψt = σ α)ψ σθ x + αψ xx,.) θ t = β)θ + νψ x + ψθ x + βθ xx, where α, β, σ and ν are positive constants sch that α<σ and β<. Ignoring the damping and diffsion terms temporarily, system.) is simplified as ) ) ) ψ σ ψ =..) θ ν ψ θ t x It is easy to see that system.) is elliptic for ψ < σν, and hyperbolic for ψ > σν. Arond the zero eqilibrim, system.) sbject to initial small distrbance is nstable owing to the ellipticity and ψ will grow becase of the inherent instability of system.). When the growth of ψ leads to ψ > σν, system.) becomes hyperbolic at once and ψ ceases to grow. Ths, a switching back and forth phenomenon is expected de to the interplaying among ellipticity, hyperbolicity and dissipation for sitable coefficients, which makes system.) qite complicated. These have been nmerically verified in [4,5]. Bt we may predict that the damping and diffsion terms joining to system.) will prevent ψ from growing and make the soltions stable. As to the stdy of system.), there are only a few rigoros reslts available so far de to the complexity of system.) as we have mentioned above. To make the analysis easier for reading, Tang and Zhao in [] discssed the Cachy problem of the system.) with α = β and σ =. That is, ψt = α)ψ θ x + αψ xx,.3) θ t = α)θ + νψ x + ψθ x + αθ xx, with initial data ) ψx,), θx, ) = ψ x), θ x) )..4) Under the assmptions that ν<4α α) and the initial data ψ x), θ x) ) L W,, ),.5) they proved the global existence of the soltions to Cachy problem.3),.4) and obtained the decay rates of the soltions by the Forier analysis and the energy method. However, the assmption.5) implies ψ x), θ x) ), ) as x ±,.6) which is a rigoros restriction on the initial data ψ x), θ x)).

3 . Dan, C. Zh / J. Math. Anal. Appl. 33 5) To consider more general problems in application, one always assmes that the initial data ψx, ), θx,)) = ψ x), θ x)) satisfies ψx,), θx, ) ) = ψ x), θ x) ) ψ ±,θ ± ) as x ±,.7) where ψ ±, θ ± are constant states and ψ + ψ,θ + θ ), ). This case was considered in []. In this paper, we will frther stdy the global existence and decay rates of soltions to the Cachy problem.3) and.7) by applying the energy method. For this, we discss in Section the linear diffsion eqations obtained by approximating the system.3) and give the decay estimates of the corresponding diffsion waves. In Sections 3, the global existence reslts are obtained from the local existence and a priori estimates. Finally, we get in Section 4 decay rates to the diffsion waves for soltions to.3) and.7). Frthermore, we also obtain the optimal decay rates. Notations. Throghot this paper, we denote positive constants by C. Moreover, the character C may differ in different places. L p = L p ) p ) denotes the sal Lebesge space on =, ) with its norm f L p = fx) p dx) /p, p<, f L = sp fx),andwhenp =, we write L ) =. H l ) denotes the sal lth order Sobolev space with its norm f H l ) = f l = l i= x i f ) /. For simplicity, f,t) L p and f,t) l are denoted by ft) L p and ft) l, respectively.. Analysis of the linear diffsion waves As in [,], we expect that the soltions of.3) time-asymptotically behave as those of the following linear system: ψ t = α) ψ + α ψ xx,.) θ t = α) θ + α θ xx. Since both eqations in.) are independent, it sffices to solve.). By setting the transformation ψx,t) = φx,t)e α)t, we can derive a heat eqation from.),that is φ t = α φ xx..) We hope to find the soltion φx,t) of the following form: x pξ) = p ), <ξ<,.3) + t satisfying bondary conditions p± ) = ψ ±,whereξ = x/ + t. It follows from.) and.3), ξp ξ) = αp ξ), p± ) = ψ ±..4)

4 8. Dan, C. Zh / J. Math. Anal. Appl. 33 5) 5 35 We get by the direct calclation φx,t) = pξ) = ψ + ψ 4πα + t) x exp y 4α + t) ) dy + ψ, which gives the soltions of.) ψx,t)= e α)t ψ + ψ ) x Gy, t + )dy+ ψ ), θx,t)= e α)t θ + θ ) x Gy, t + )dy+ θ ), where Gx, t) = ) exp x 4παt 4αt.5) is the heat kernel fnction. It is easy to show ψx,t) ψ ± e α)t, x ±, θx,t) θ ± e α)t.6), x ±. Now we will consider the asymptotic behavior of ψx,t), θx,t) and their derivatives in L p ). First for the heat kernel fnction, it has the following properties. Lemma.. When p +, l, k<+, we have l t k x Gt) L p Ct /) /p) l k/. From the above lemma, we can get the following reslts by simple calclations. Lemma.. The soltions ψx,t) and θx,t) to.) satisfy the following properties: i) l t ψt) L Ce α)t, l t θt) L Ce α)t, l =,,,...; ii) for any p with p +, it holds that l t x k ψt) L p C ψ + ψ e α)t + t) /p) k/, k =,,..., l=,,,..., l t x k θt) L p C θ + θ e α)t + t) /p) k/, k =,,..., l=,,, Global existence of soltions 3.. eformlation of the problem Let x, t) = ψx,t) ψx,t), vx,t) = θx,t) θx,t). 3.)

5 . Dan, C. Zh / J. Math. Anal. Appl. 33 5) Then from.), we can rewrite problem.3) and.7) as follows: t = α) v x + α xx θ x, v t = α)v + ν x + v x + αv xx + ψv x + θ x + Fx,t), with initial data x, ) = x) = ψ x) ψx,), x ±, vx,) = v x) = θ x) θx,), x ±, where 3.) 3.3) Fx,t)= ν ψ x + ψ θ x. 3.4) We seek the soltions of 3.), 3.3) in the set of fnctions X,T)defined by X,T)=, v), v L,T; H ) L,T; H 3 ) }. Now we state or first main reslts as follows. Theorem 3.. Let x), v x)) H, ). Frthermore, Sppose that both δ = ψ + ψ + θ + θ and δ = + v are sfficiently small. Then for any <α<, ν<4α α), the Cachy problem 3.), 3.3) admits a niqe global soltion x, t), vx, t)) X,T)satisfying t) + vt) t + τ) 3 + vτ) ) 3 dτ Cδ + δ ) 3.5) and sp, v)x, t) + x,v x )x, t) ) as t. 3.6) x 3.. Local existence of soltions In this sbsection, we will stdy the local existence of Cachy problem 3.) and 3.3). To this end, we rewrite the Cachy problem 3.) and 3.3) in the following integral forms: x, t) = Gx, t) x) α) t Gx, t s) x, s) ds + t G xx, t s) vx,s)ds t Gx, t s) θ x x, s) ds, vx,t) = Gx, t) v x) α) t Gx, t s) vx,s)ds ν t G xx, t s) x, s) ds + t Gx, t s) v x)x, s) ds + t Gx, t s) ψv x )x, s) ds + t Gx, t s) θ x )x, s) ds + t Gx, t s) Fx,s)ds, where the convoltions are taken with respect to the space variable x. We can constrct the approximate soltion seqences and obtain the local existence by implementing the standard argments with Brower fixed point principle [].

6 . Dan, C. Zh / J. Math. Anal. Appl. 33 5) 5 35 Lemma 3. Local existence). If x), v x)) H, ), then there exists t depending only on x), v x)) H, ), sch that the Cachy problem 3.), 3.3) admits a niqe smooth soltion x, t), vx, t)) X,t ) satisfying x,t),vx,t) ) H, ) x), v x) ) H, ). 3.7) 3.3. Global existence of soltions By the local existence reslt, in order to get the global existence of Cachy problem 3.) and 3.3), it is sfficient to get a priori estimates. Precisely, we need to prove that there exists a constant C depending only on x), v x)) H, ) sch that any soltion, v)x, t) in X,T) satisfy + v C for any t in [,T]. Next, we devote orselves to the estimate of the soltion x, t), vx, t)) of 3.), 3.3) nder the a priori assmption NT)= sp k x t) + k x vt) } δ, 3.8) <t<t k= k= where <δ. By Sobolev ineqality f L f / f x /,wehave, x,v,v x ) L δ, 3.9) which will be sed later. Moreover, if ν<4α α), we can find ε, ), c > sch that c α α)ε >, α) c ν αε >. 3.) In fact, from ν<4α α), we know that there exists a constant k, ), sch that ν = 4kα α). Choosing ε = k + and c = α α) k + ) + k + 8k one can easily verify that ε and c satisfy 3.). What follows will be a series of lemmas contribting to or desired estimates. Lemma 3.3. Sppose that the assmptions in Theorem 3. hold and x, t), vx, t)) is a soltion to 3.), 3.3) obtained in Lemma 3., then it holds that for any ν<4α α), t + v )dx+ t + v )dxdτ + ), x + vx ) dxdτ Cδ + δ ), 3.) provided that δ and δ are sfficiently small.

7 . Dan, C. Zh / J. Math. Anal. Appl. 33 5) 5 35 Proof. Mltiplying the first eqation of 3.) by and the second eqation of 3.) by c v and integrating the reslting identity over,t), we arrive at by Cachy Schwarz ineqality t + c v )dx+ α) t = + c v t + 4c t θ x dxdτ c t + c v )dxdτ + α v x dxdτ + νc t θ x vdxdτ + c t + c )δ + ε α) t + εα + t δ α) + c θ x L + c δ t x dxdτ + c ν εα t t t x v dxdτ c dxdτ + t vfx,τ)dx dτ v x dxdτ t t ε α) t v dxdτ + δ α) θ x dxdτ + c ) t x L + ψ x L t + v )dxdτ + c δ x + c vx ) dxdτ ψ x v dxdτ v x dxdτ v dxdτ dxdτ v dxdτ F x, τ) dx dτ. 3.) Employing Lemma. and 3.9), we have from the above ineqality + c v )dx+ ε δ) α) Cc δ } t t + ε)α x dxdτ + α) c ν dxdτ εα Cδ + δ) } t c v dxdτ

8 . Dan, C. Zh / J. Math. Anal. Appl. 33 5) 5 35 } t + c α ε α) Cδ + Cδ + c δ t v x dxdτ where we have sed the following ineqality: t δ α) F x, τ) dx dτ, 3.3) θ x dxdτ = t θ x t dτ δ α) δ α) Cδ e α)τ dτ Cδ. Now we estimate the last term in the right side of ineqality 3.3). In fact from Lemma., we have by Cachy Schwarz ineqality c δ t F x, τ) dx dτ = c δ C δ C δ t t t ν ψ x + ψ θ x ) dxdτ ψ x + ψ θ x ) dxdτ ψ x + θ x ) dxdτ Cδ. 3.4) Ths, 3.3) and 3.4) give + c v )dx+ ε δ) α) Cδ } t dxdτ t + ε)α + c α ε α) x dxdτ + α) c ν } t v x dxdτ εα Cδ + δ ) } t c v dxdτ Cδ + δ), 3.5) which implies, with the help of 3.), t t + v )dx+ + v )dxdτ + x + vx ) dxdτ Cδ + δ), 3.6)

9 . Dan, C. Zh / J. Math. Anal. Appl. 33 5) provided that δ and δ are sfficiently small. This proves Lemma 3.3. Lemma 3.4. Let the assmptions in Theorem 3. hold. Then the soltion x, t), vx, t)) of 3.), 3.3) obtained in Lemma 3. satisfies for any ν<4α α), t x + vx ) dx + provided that δ and δ are sfficiently small. xx + vxx ) dxdτ Cδ + δ ), 3.7) Proof. First, we mltiply the first eqation of 3.) by xx ) and the second eqation of 3.) by c v xx ) respectively, and add the reslting eqations together. After all these, we take integration over x, t),t)and reach t x + c vx ) dx + α) x + c vx ) dxdτ t + α xx + c vxx ) dxdτ = x + c v x t 4c t t x θ xx dxdτ + c + c )δ + ε α) t + εα t + x v xx dxdτ + νc t θ x v xx dxdτ c t xx dxdτ + c ν εα t t x v x dxdτ + c t Fx,τ)v xx dxdτ x dxdτ + ε α) t t t v x xx dxdτ ψ x v x dxdτ v xx dxdτ vx dxdτ + x dxdτ θ xx dxdτ + c ) t x L + ψ x L vx dxdτ + c θ x L t + vxx ) t dxdτ + c δ vxx dxdτ

10 4. Dan, C. Zh / J. Math. Anal. Appl. 33 5) c δ t F x, τ) dx dτ. 3.8) By Lemmas. and 3.3, we have from 3.4), t x + c vx ) dx + ε)α xx dxdτ + c α ε α) Cδ } t v xx dxdτ Cδ + δ). 3.9) Ths the proof of Lemma 3.4 is completed by 3.) and 3.9) provided that δ and δ are sfficiently small. Lemma 3.5. Sppose that x, t), vx, t)) is a soltion to 3.), 3.3) obtained in Lemma 3. nder the assmptions in Theorem 3., then for any ν<4α α), we have t xx + vxx ) dx + provided that δ and δ are sfficiently small. xxx + vxxx ) dxdτ Cδ + δ ), 3.) Proof. Differentiating 3.) twice with respect to x, mltiplying the reslts by xx and c v xx, respectively, integrating the reslting eqation with respect to x, t) over,t), we get t xx + c vxx ) dx + α) xx + c vxx ) dxdτ t + α xxx + c vxxx ) dxdτ = xx + c v xx t + 4c t t xx θ xxx dxdτ + 4c xx v xxx dxdτ + νc t ψv x ) xx v xx dxdτ + 4c v x ) xx v xx dxdτ t t θ x ) xx v xx dxdτ v xx xxx dxdτ

11 + c t. Dan, C. Zh / J. Math. Anal. Appl. 33 5) F xx x, τ)v xx dxdτ + c )δ + ε α) t + εα t + 4c c t t t xxx dxdτ + c ν εα θ xxx dxdτ 4c t xx dxdτ + t ε α) t ψv x ) x v xxx dxdτ 4c t v xxx dxdτ vxx dxdτ + xx dxdτ v x ) x v xxx dxdτ t θ x ) x v xxx dxdτ F x x, τ)v xxx dxdτ. 3.) Shffling the terms, we get by Lemmas. and 3.4, t xx + c vxx ) dx + ε)α xxx dxdτ } t + c α vxxx ε α) dxdτ t Cδ + δ) 4c v x ) x v xxx dxdτ 4c 4c t θ x ) x v xxx dxdτ c t t ψv x ) x v xxx dxdτ F x x, τ)v xxx dxdτ. 3.) Next we are devoted to estimate the terms in the right side of 3.) as follows. First, we obtain from 3.9), Lemmas 3.3 and 3.4 by Cachy Schwarz ineqality 4c t c δ t v x ) x v xxx dxdτ v xxx dxdτ + 8c δ t x vx + vxx ) dxdτ

12 6. Dan, C. Zh / J. Math. Anal. Appl. 33 5) 5 35 c δ c δ t t vxxx dxdτ + 8c x δ ) t L + L v x + v ) xx dxdτ v xxx dxdτ + Cδ + δ ). 3.3) Similarly, we have from Lemmas. and 3.4, and 4c t = 4c c δ c δ 4c t t t t ψv x ) x v xxx dxdτ ψ x v x v xxx dxdτ 4c t v xxx dxdτ + 4c δ ψ x L t ψv xx v xxx dxdτ v x dxdτ + c t ψ x v xx dxdτ v xxx dxdτ + Cδ + δ ), 3.4) t θ x ) x v xxx dxdτ c δ v xxx dxdτ + Cδ + δ ). 3.5) In addition, applying Lemmas., 3.3 and 3.4, we derive by Cachy Schwarz ineqality c t c δ c δ t + C δ c δ t F x x, τ)v xxx dxdτ = c t t v xxx dxdτ + c δ v xxx dxdτ + C δ ψ θ xx dxdτ t t t ν ψ x + ψ θ x ) x v xxx dxdτ ν ψ xx + ψ θ x ) x ) dxdτ ψ xx dxdτ + C δ t ψ x θ x dxdτ vxxx dxdτ + Cδ + δ ). 3.6)

13 . Dan, C. Zh / J. Math. Anal. Appl. 33 5) Sbstitting 3.3) 3.6) into 3.), we get t xx + c vxx ) dx + ε)α + c α Cδ + δ ), ε α) C δ + δ) xxx dxdτ } t v xxx dxdτ which proves 3.) provided that δ and δ are sfficiently small. Ths the combination of Lemmas implies 3.5). Finally, we have to show that the a priori assmption 3.8) can be closed. Since, nder this a priori assmption 3.8), we have dedced that 3.5) holds provided that δ and δ are sfficiently small. Therefore the assmption 3.8) is always tre provided that δ and δ are sfficiently small. Now we trn to show that 3.6) is tre. To do this, we introdce the following lemma. Lemma 3.6. If gt), gt) L, ) and g t) L, ),thengt) as t. The proof of Lemma 3.6 can be fond in [8,] and the details are omitted. Taking gt) = x t) in Lemma 3.6, we can conclde from 3.5) that gt) L, ). Denote L -inner prodct by,. By sing the definition of L -inner prodct and integrating by parts, we have g t) = x, xt = t, xx. It is easy to verify from the proof of Lemma 3.4 that t, xx = t, xx = α) + v x α xx + θ x, xx L, ). Hence, g t) L, ) which implies, by Lemma 3.6, x t) ast. 3.7) Applying Sobolev ineqality, we have from 3.7) and 3.5), sp x, t) t) / x t) / ast. 3.8) The same process is applied to vx,t) so that sp vx,t) vt) / vx t) / ast. 3.9) Similarly, taking gt) = xx t) and gt) = v xx t), we have from 3.5) and Lemma 3.6, xx t) and vxx t) ast. 3.3) So Sobolev ineqality and 3.3) give sp x x, t) x t) / xx t) / ast, 3.3)

14 8. Dan, C. Zh / J. Math. Anal. Appl. 33 5) 5 35 and sp vx x, t) vx t) / vxx t) / ast. 3.3) Ths, 3.6) is proved by 3.8), 3.9), 3.3) and 3.3). The proof of Theorem 3. is completed. 4. Decay rates of soltions In this section, we will stdy the decay rates of soltions to the Cachy problem 3.), 3.3) nder a priori assmption x k t) + x k vt) e lt, <t<t, 4.) k= k= with l = min ε) α), α) c ν }, 4.) εα where ε and c are defined by 3.). By Sobolev ineqality, we have from 4.),, x,v,v x ) L e lt/, 4.3) which will be sed later. Moreover, we list the following Gronwall s ineqality which is sed in the following text. Lemma 4. Gronwall s ineqality). Let η ) be a nonnegative continos fnction on [, ), which satisfies the differential ineqality η t) + ληt) ωt), where λ is a positive constant and ωt) is a nonnegative continos fnction on [, ). Then t ) ηt) η) + e λτ ωτ)dτ e λt. Now we can state the main reslts of decay rates of soltions. Theorem 4.. Sppose that x,t),vx,t)) is a soltion to problem 3.), 3.3) nder the assmptions imposed in Theorem 3., then when ν<4α α), we have for any t [,T], k x t) + k x vt) Cδ + δ ) /4 e lt, k=,,, 4.4) provided that δ and δ are sfficiently small, where l is defined by 4.).

15 . Dan, C. Zh / J. Math. Anal. Appl. 33 5) Proof. The proofs are divided into three steps. First, 3.) + 3.) c v and integrating the reslting identities over x, we reach by Cachy Schwarz ineqality from Lemma., d dt + c v )dx+ α) + c v )dx+ α x + c vx ) dx = v x dx + νc v x dx θ x dx c x + ψ x )v dx + 4c θ x v dx + c vfx,t)dx ε α) dx + vx ε α) dx + εα c x dx + c ν εα v dx θ x dx + c θ x L ) dx + c ψ x L + θ x L v dx x v dx + c vfx,t)dx. 4.5) After shffling the terms, we have by Lemma., d + c v )dx+ ε) α) dx + α) c ν } c v dx dt εα } + ε)α x c dx + α vx ε α) dx Cδe α)t +c + c v )dx θ x dx c x v dx vfx,t)dx. 4.6) Next, we will estimate the terms in the right side of 4.6). In fact, we have from the assmption 4.) and Lemma. by employing Cachy Schwarz ineqality θ x dx δe l/ α)}t dx + δ e l/ α)}t θ x dx δe l/ α)}t e lt + Cδe l/ α)}t e α)t Cδe l/+ α)}t. 4.7)

16 3. Dan, C. Zh / J. Math. Anal. Appl. 33 5) 5 35 Moreover, we get from 3.5) by Sobolev ineqality, x,v,v x ) L Cδ + δ ) /. 4.8) Ths we derive by Cachy Schwarz ineqality from 4.), 4.3) and 4.8), ) c x v dx c x v dx + v 3 dx c v L x + v ) dx = c v / L v / L x + v ) dx Cδ + δ ) /4 e lt/4 e lt = Cδ + δ ) /4 e 5lt/4. 4.9) In addition, we dedce from Lemma. and Cachy Schwarz ineqality c vfx,t)dx δe l/ α)}t v dx + c δ e l/ α)}t F x, t) dx δe l/ α)}t e lt + C δ e l/ α)}t ψ x + θ x ) dx δe l/ α)}t e lt + C δ e l/ α)}t δ e α)t Cδe l/+ α)}t. 4.) Ths, we get from 4.6), 4.7), 4.9), 4.) and 4.), d + c v )dx+ ε) α) dx + α) c ν } c v dx dt εα Cδe l+ α)}t + Cδe l/+ α)}t + Cδ + δ ) /4 e 5lt/4 Cδ + δ ) /4 e 5lt/4 + Cδe l/+ α)}t. 4.) ecalling the definition 4.) of l, we have from 4.), d + c v )dx+ l + c v )dx dt Cδ + δ ) /4 e 5lt/4 + Cδe l/+ α)}t. 4.) Noticing that l/ < α, we obtain from Lemma 4., + c v )dx Cδ + Cδ + δ ) /4 t t e 5lτ/4 e lτ dτ + Cδ e l/+ α)}τ e lτ dτ Cδ + δ ) /4 e lt, 4.3) } e lt

17 . Dan, C. Zh / J. Math. Anal. Appl. 33 5) which implies 4.4) for k =. Similarly, for k =, if we mltiply 3.) by xx ) and 3.) by c v xx ),add and take integration of the reslting identities over x, then we get by condcting the same procedre to the proofs of ineqality 4.3), x + c vx ) dx Cδ + δ ) /4 e lt, 4.4) which proves 4.4) for k =. Finally, we shall show that 4.4) is tre for k =. In fact, differentiating 3.) twice with respect to x, mltiplying the reslts by xx and c v xx, respectively, integrating the reslting eqation over x, we arriveat d dt Hence d dt xx + c vxx ) dx + α) xx + c vxx ) dx + α = xx v xxx dx + νc v xx xxx dx xx θ xxx dx + 4c + c v x ) xx v xx dx + 4c F xx x, t)v xx dx ψv x ) xx v xx dx + 4c xxx + c vxxx ) dx θ x ) xx v xx dx ε α) xx dx + vxxx ε α) dx + εα xxx dx + c ν εα xx θ xxx dx 4c v x ) x v xxx dx 4c ψv x ) x v xxx dx 4c θ x ) x v xxx dx + c xx + c vxx ) dx + ε) α) v xx dx F xx x, t)v xx dx. 4.5) xx dx + α) c ν } c vxx εα dx + ε)α xxx dx } + c α vxxx ε α) dx xx θ xxx dx 4c v x ) x v xxx dx 4c ψv x ) x v xxx dx

18 3. Dan, C. Zh / J. Math. Anal. Appl. 33 5) c θ x ) x v xxx dx + c F xx x, t)v xx dx. 4.6) Next, we are going to give the estimates of the terms in the right side of 4.6). Indeed, sing Cachy Schwarz ineqality, we have from 4.) and Lemma., xx θ xxx dx δe l/ α)}t xx dx + δ e l/ α)}t θ xxx dx δe l/ α)}t e lt + δ e l/ α)}t Cδ e α)t Cδe l/+ α)}t. 4.7) For the simplicity of notation, we define λ = c α /ε α)). Then we derive from 4.3) by Cachy Schwarz ineqality 4c v x ) x v xxx dx 4 λ vxxx dx + 3c λ x vx + vxx ) dx 4 λ vxxx dx + 3c λ x L vx dx + 3c λ L vxx dx 4 λ vxxx dx + Ce lt v x + vxx ) dx. 4.8) Moreover, we have from Lemma. with Cachy Schwarz ineqality 4c ψv x ) x v xxx dx 4 λ vxxx dx + 3c ψ x λ v x + ψ vxx ) dx 4 λ 4 λ vxxx dx + 3c λ ψ x L vxxx dx + Ce α)t Similarly, we have 4c θ x ) x v xxx dx 4 λ vx dx + 3c λ ψ L v xx dx v x + vxx ) dx. 4.9) v xxx dx + 3c λ θ xx + θ x ) x dx 4 λ vxxx dx + 3c λ θ xx L dx + 3c λ θ x L x dx 4 λ vxxx dx + Ce α)t + ) x dx. 4.)

19 . Dan, C. Zh / J. Math. Anal. Appl. 33 5) On the other hand, we have by Lemma., Cachy Schwarz ineqality and 4.), c F xx x, t)v xx dx δe l/ α)}t vxx dx + c δ e l/ α)}t Fxx x, t) dx δe l/ α)}t e lt + C δ e l/ α)}t δe l/+ α)}t + C δ e l/ α)}t ψ xxx + [ ψ θ x ) xx ] } dx ψ xxx + ψ xx θ x + ψ x θ xx + ψ θ xxx ) dx δe l/+ α)}t + C δ e l/ α)}t δ e α)t Cδe l/+ α)}t. 4.) Sbstitting 4.7) 4.) into 4.6), we get from 3.5), d dt xx + c vxx ) dx + ε) α) xx dx + α) c ν } c vxx εα dx + ε)α xxx dx + 4 λ vxxx dx Cδe l/+ α)}t + Ce α)t + x + v x + ) v xx dx + Ce lt v x + vxx ) dx Cδe l/+ α)}t + Cδ + δ )e α)t + Ce lt ecalling the definition 4.) of l, we obtain from 3.) and 4.), d dt xx + c vxx ) dx + l xx + c vxx ) dx Cδe l/+ α)}t + Cδ + δ )e α)t + Ce lt v x + vxx ) dx. 4.) v x + vxx ) dx. 4.3) Noticing l/ < α, we easily dedce from 4.3) with the help of Lemma 4. and 3.5), xx + c vxx ) dx t Cδ + Cδ e l/+ α)}τ e lτ dτ

20 34. Dan, C. Zh / J. Math. Anal. Appl. 33 5) 5 35 t + Cδ + δ ) t e α)τ e lτ dτ + C Cδ + Cδ + Cδ + δ ) + C Cδ + Cδ + Cδ + δ ) + Cδ + δ ) ) e lt t e lτ v x + vxx ) dxe lτ dτ v x + vxx ) dxdτ )e lt Cδ + δ )e lt. 4.4) The proof of Theorem 4. is completed by 4.3), 4.4) and 4.4). Finally, we verify that the a priori assmption 4.) is reasonable. Indeed, nder this a priori assmption, we have showed that 4.4) holds. Therefore, the assmption 4.) is always tre provided that δ and δ are sfficiently small. We remark that we can obtain the L p p ) optimal decay rate with the help of Lemma. and Theorem 4.. Its proof is completely similar to the one in [] and the details are omitted. Precisely, we have the following reslt. Corollary 4.3. Sppose that x,t),vx,t)) is a soltion to problem 3.), 3.3) nder the assmptions imposed in Theorem 3.. If the initial data x), v x)) L, ), ν<4α α), and δ and δ are sfficiently small, then the soltion x, t), vx, t)) has the following optimal decay rate: k x t) L p + k x vt) L p C + t) /+/p) e α ν/4α))t. 4.5) k= k= emark 4.4. It is noticed that l < α ν 4α. 4.6) In fact, let k = ν/4α α)).sinceα<4α α), thenk, ). Hence 4.6) is eqivalent to l < α k α). 4.7) In what follows we prove 4.7). If k<ε/, then l α) ε α) < α k α) de to 4.). If k ε/, we have from 3.), c ν = 8c kα α) 4c εα α) > > k ε. This leads to l α) c ν αε = α) c νk α) < α) k α) ε again de to 4.). Ths 4.7) holds, which implies 4.6). ) e lt

21 . Dan, C. Zh / J. Math. Anal. Appl. 33 5) Acknowledgments The research was spported by three grants from the National Natral Science Fondation of China #737, the Key Project of Chinese Ministry of Edcation #48 and the National Key Program for Basic esearch of China #CCA37, respectively. Particlar thanks go to the anonymos referees for their helpfl sggestions and comments. eferences [] X.X. Ding, J.H. Wang, Global soltion for a semilinear parabolic system, Acta Math. Sci ) [] L. Hsiao, T.-P. Li, Convergence to nonlinear diffsion waves for soltions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys ) [3] D.Y. Hsieh, On partial differential eqations related to Lorenz system, J. Math. Phys ) [4] D.Y. Hsieh, S.Q. Tang, X.P. Wang, On hydrodynamic instability, chaos, and phase transition, Acta Mech. Sinica 996) 4. [5] D.Y. Hsieh, S.Q. Tang, X.P. Wang, L.X. W, Dissipative nonlinear evoltion eqations and chaos, Std. Appl. Math. 998) [6] L.. Keefe, Dynamics of pertrbed wavetrain soltions to the Ginzberg Landa eqation, Std. Appl. Math ) [7] Y. Kramoto, T. Tszki, On the formation of dissipative strctres in reaction diffsion systems, Progr. Theoret. Phys ) [8] K. Nishihara, Convergence rates to nonlinear diffsion waves for soltions of system of hyperbolic conservation laws with damping, J. Differential Eqations 3 996) [9] G. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames, I. Derivation of basic eqations, Acta Astronat ) [] S.Q. Tang, H.J. Zhao, Nonlinear stability for dissipative nonlinear evoltion eqations with ellipticity, J. Math. Anal. Appl ) [] Z.A. Wang, C.J. Zh, Stability of the rarefaction wave for the generalized KdV Brgers eqation, Acta Math. Sci. ) [] C.J. Zh, Asymptotic behavior of soltions for p-system with relaxation, J. Differential Eqations 8 )