Convergence to diffusion waves for nonlinear evolution equations with different end states

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1 J. Math. Anal. Appl ) Convergence to diffsion waves for nonlinear evoltion eqations with different end states Walter Allegretto, Yanping Lin, Zhiyong Zhang Department of Mathematics and Statistics, University of Alberta, Edmonton, Alberta T6G G1, Canada eceived 17 December 6 Available online 16 May 7 Sbmitted by Goong Chen Abstract In this paper, we consider the global existence and the asymptotic decay of soltions to the Cachy problem for the following nonlinear evoltion eqations with ellipticity and dissipative effects: ψt = 1 α)ψ + ψψ x + fθ) ) x + αψ xx, E) θ t = 1 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<1, sν < 4α1 α) s is defined in 1.14)). 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.1), 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 similar problem was stdied by Tang and Zhao [S.Q. Tang, H.J. Zhao, Nonlinear stability for dissipative nonlinear evoltion eqations with ellipticity, J. Math. Anal. Appl ) ] for the case of ψ ±,θ ± ) =, ). 7 Elsevier Inc. All rights reserved. Keywords: Evoltion eqations; Diffsion waves; Decay rate; Energy method; A priori estimates 1. Introdction In paper [1], Jian, Wang, and Hsieh stdied the dissipative nonlinear evoltion eqations with divergence form ψt = αψ + λψψ x + fθ) ) x + ε 1ψ xx, θ t = βθ + νψ x + ψθ) x + ε θ xx, <x<l, t>, 1.1) with initial data Work spported by NSEC Canada). * Corresponding athor. address: zzhang@math.alberta.ca Z. Zhang). -47X/$ see front matter 7 Elsevier Inc. All rights reserved. doi:1.116/j.jmaa

2 W. Allegretto et al. / J. Math. Anal. Appl ) ψx,), θx, ) ) = ψ x), θ x) ), and bondary condition ψ, θ)x, t) =, ), x,l}, t, 1.3) where L,ε 1, and ε are all constants, while α, β, λ, and ν are given real constants. The nonlinear term f Cloc ) satisfies f z z) k, z. Global smooth soltion ψ, θ) C[, ), H 1[,L]) C,L)), ) and the global attractor for the above dissipative nonlinear system is stdied in the abstract theory method on evoltion eqations as in Henry [4]. There are also papers abot related model stdied by Hsiao and Jian in [5] and Jian and Chen in [9], ψt = σ α)ψ σθ x + αψ xx, 1.4) θ t = 1 β)θ + νψ x + ψθ) x + βθ xx, where α, β, σ and ν are all positive constants satisfying the relations α<σ and β<1. We refer to Hsieh [7] and Tang [15] for the physical backgrond of 1.4). If we ignore the damping and diffsion terms temporarily, system 1.1) is simplified as ) ) ) ψ σ ψ =. 1.5) θ t ν + θ ψ θ x It is easy to get the two characteristic vales of the system 1.3) are λ ± = 1 ψ ± ψ 4σν+ θ) ). This means the system 1.5) is elliptic for ψ < σν+ θ), and hyperbolic for ψ > σν+ θ). Arond the zero eqilibrim, system 1.5) sbject to initial small distrbance is nstable owing to the ellipticity and ψ will grow since the inherent instability of system 1.5). When the growth of ψ leads to ψ > σν, system 1.5) 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 1.5) qite complicated, even occr chaos. Bt we may predict the damping and diffsion terms joining to system 1.5) will prevent ψ from growing and make the soltions stable. As to the stdy of system 1.1), there are only a few rigoros reslts available so far de to the complexity of system 1.1) as we mentioned above, cf. [5,9]. In paper [5], Hsiao and Jian discssed the initial bondary vale problem of 1.4) with initial data ) ψx,), θx, ) = ψ x), θ x) ), 1.6) and bondary condition ψ, θ),t)= ψ, θ)1, t), ψ x,θ x ),t)= ψ x,θ x )1,t), t T. 1.7) When initial data ψ x), θ x)) C,δ [, 1]) C,δ [, 1]), <δ<1, they established the global existence and niqeness of classical soltions for the corresponding problem by applying the energy method and Leray Schader fixed point theorem. In paper [9], Jian and Chen considered the Cachy problem for system 1.4) with the initial data 1.6). When initial data ψ x), θ x) ) H 1 ) L 1 ), 1.8) they sed the abstract theory concerning the initial vale problem in a Banach space to obtain the existence and niqeness of local soltions to 1.4) and 1.7). Then, by a priori estimates of H 1 -norm, the global existence reslts were obtained. On the other hand, the system with the same complicated properties was also stdied by Tang and Zhao [16]. Precisely, Tang and Zhao considered the following Cachy problem that was proposed by Hsieh [7]: ψt = 1 α)ψ θ x + αψ xx, 1.9) θ t = 1 α)θ + νψ x + ψθ x + αθ xx, with initial data 1.6). 1.)

3 46 W. Allegretto et al. / J. Math. Anal. Appl ) Under the assmption ν<4α1 α) and the initial data ψ x), θ x) ) L W 1,, ), 1.1) they proved the global existence of the soltions to Cachy problem 1.6), 1.4) and obtained the decay rates of the soltions by the Forier analysis and the energy method. However, the assmption 1.6) in [9] or 1.1) in [16] implies ψ x), θ x) ), ), as x ±, 1.11) which is a rigoros restriction on the initial data ψ x), θ x)). Very recently, Zh and Wang [], Dan and Zh [] extended Tang and Zhao s reslt to the case of more general initial data ) ψx,), θx, ) = ψ x), θ x) ) ψ ±,θ ± ), as x ±, 1.1) where ψ ±, θ ± are constant states and ψ + ψ,θ + θ ), ). And the existence and the decay rates of the soltions of the corresponding problem was also obtained. For other reslts on this direction refer to [8,11,1,15]. Based on the idea in [,,1], we will consider the existence and asymptotic behavior of the soltions to Cachy problem of 1.1). To make the analysis easier for reading, we take simple coefficients in 1.1). Now we consider the Cachy problem ψt = 1 α)ψ + ψψ x + fθ) ) x + αψ xx, 1.13) θ t = 1 α)θ + νψ x + ψθ) x + αθ xx, with initial data 1.1). The nonlinear term f Cloc ) satisfies f n z) s 1, z, n = 1,, ) To establish the global existence and get the decay rates of the corresponding soltions by applying the energy method, or plan is arranged as follows: First of all, we need to find the asymptotic profile, linear diffsion waves, defined by the linear diffsion eqations, which are obtained by approximating the system 1.13), cf..1). This will be done in Section. Secondly, in Section 3, the global existence and asymptotic behavior are obtained from the local existence and a priori estimates. Finally, in Section 4, we get decay rates to the diffsion waves obtained in Section for the soltions to 1.13) and 1.1). Notations. Throghot this paper, we denote positive constants by C. Moreover, the character C may differ in different places. L p = L p ) 1 p ) denotes the sal Lebesge space on =, ) with its norm f L p = fx) p dx) p 1,1 p<, f L = sp fx), and when p =, 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 ) 1. 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 [,6,14,19], we expect the soltions of 1.13) time-asymptotically behave as those of the following linear system: ψ t = 1 α) ψ + α ψ xx,.1) θ t = 1 α) θ + α θ xx. Since both eqations in.1) are independent, it sffices to solve.1) 1. By setting the transformation ψx,t) = φx,t)e 1 α)t, we can derive a heat eqation from.1) 1, that is φ t = α φ xx..) We hope to find the soltion φx,t) of the following form: x pξ) = p ), <ξ<,.3) 1 + t satisfying bondary conditions p± ) = ψ ±, where ξ = x 1+t.

4 W. Allegretto et al. / J. Math. Anal. Appl ) It follows from.) and.3) 1 ξp ξ) = αp ξ), p± ) = ψ ±. We get by the direct calclation φx,t) = pξ) = ψ + ψ 4πα1 + t) which gives the soltions of.1) x ψx,t)= e ψ 1 α)t + ψ ) θx,t)= e θ 1 α)t + θ ) exp x x y 4α1 + t) ) dy + ψ, Gy, t + 1)dy+ ψ ), Gy, t + 1)dy+ θ ), where Gx, t) = 1 exp x 4παt 4αt ) is the heat kernel fnction. It is easy to show ψx,t) ψ ± e 1 α)t, x ±, θx,t) θ ± e 1 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.1. When 1 p +, l, k<+, we have l t k x Gx, t) 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.1) satisfy the following properties: i) t l ψt) L Ce 1 α)t, t l θt) L Ce 1 α)t, l =, 1,,...; ii) for any p with 1 p +, it holds that l t x k ψt) L p C ψ + ψ e 1 α)t 1 + t) p 1 k, k= 1,,..., l=, 1,,..., t l k x θt) L p C θ + θ e 1 α)t 1 + t) p 1 k, k= 1,,..., l=, 1,, Global existence and asymptotic behavior 3.1. eformlation of the problem.4).5) Let x, t) = ψx,t) ψx,t), vx,t) = θx,t) θx,t). Then from.1), we can rewrite problem 1.3) and 1.7) as follows t = 1 α) + α xx + x + ψ) x + f θ v x + θ x ) + Ex,t), v t = 1 α)v + αv xx + ν x + v) x + θ) x + ψv) x + Fx,t), 3.1) 3.)

5 48 W. Allegretto et al. / J. Math. Anal. Appl ) with initial data x, ) = x) = ψ x) ψx,), x ±, vx,) = v x) = θ x) θx,), x ±, where Ex,t) = ψ ψ x, Fx,t)= ν ψ x + ψ θ x + θ ψ x. We seek the soltions of 3.), 3.3) in the set of fnction X,T)defined by 3.3) 3.4) X,T)=, v), v L,T; H ) L,T; H 3)}. Now we state or first main reslts as follows. Theorem 3.1. Let x), v x)) H, ). Frthermore, sppose that both δ = ψ + ψ + θ + θ and δ = + v are sfficiently small. Then for any <α<1, ν<4α1 α), the Cachy problem 3.), 3.3) admits a niqe global soltion x, t), vx, t)) X,T)satisfying t) + t vt) + 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 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: t x, t) = Gx, t) x) 1 α) Gx, t s) x, s) ds t + t G x x, t s) ) x, s) ds Gx, t s) ψ ψ x )x, s) ds + vx,t) = Gx, t) v x) 1 α) t ν t t G x x, t s) x, s) ds G x x, t s) θ)x, s) ds t t G x x, t s) ψ)x, s) ds Gx, t s) f x v + θ)x, s) ds, Gx, t s) vx,s)ds t t G x x, t s) v)x, s) ds t G x x, t s) ψv)x, s) ds + 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 [1].

6 W. Allegretto et al. / J. Math. Anal. Appl ) 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 and asymptotic behavior 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 t) + vt) 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 <t<t k= k x t) + } x k vt) δ1, 3.8) k= where <δ 1 1. By Sobolev ineqality f L f 1 f x 1,wehave, x,v,v x ) L δ 1, 3.9) which will be sed later. Moreover, if sν < 4α1 α), we can find ε, ), c > sch that s c α 1 α)ε >, 1 α) c 3.1) ν αε >. In fact, from sν < 4α1 α), we know that there exists a constant k, 1), sch that sν = 4kα1 α). Choosing ε = k + 1 and c = α1 α) s 1 k+1) + k+1 ), one can easily verify that ε and c 8k satisfy 3.1). What follows will be a series of lemmas contribting to or desired estimates. Lemma 3.3. Sppose that the assmptions in Theorem 3.1 hold and x, t), vx, t)) is a soltion to 3.), 3.3) obtained in Lemma 3., then it holds that for any ν<4α1 α), + v ) t dx + + v ) t dxdτ + x + vx ) dxdτ Cδ + δ ), 3.11) provided δ and δ 1 are sfficiently small. 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 Schwartz ineqality + c v ) t dx + 1 α) + c v ) t dxdτ + α x + c vx ) dxdτ t = + c v + + c t x v dxdτ + c f θ v x dxdτ + νc t t ψ x v dxdτ c v x dxdτ t t θv x dxdτ + c f θ θ x dxdτ t vfx,τ)dx dτ

7 5 W. Allegretto et al. / J. Math. Anal. Appl ) t + t x dxdτ c )δ + ε1 α) + c ν εα t t t ψ x dxdτ + dxdτ + v dxdτ + δ1 α) } t 1 t s ε1 α) t dxdτ + ) t + c x L + ψ x L v dxdτ c c α s ε1 α) + ) t x L + ψ x L Ex, τ) dx dτ c α t θ dxdτ + c δ dxdτ + δ t v x t dxdτ + εα x dxdτ s δ1 α) t s ε1 α) dxdτ + 1 δ } t θ x dxdτ v dxdτ + c δ t v x dxdτ t F x, τ) dx dτ E x, τ) dx dτ. 3.1) Employing Lemma. and 3.9), we have from the above ineqality + c v ) dx + ε δ)1 α) Cδ + δ 1 ) } t dxdτ + ε)α Cδ + δ 1 ) } t x dxdτ + 1 α) c ν } t εα Cδ 1 + δ) Cδ + Cδ + c + 1 δ t c α E x, τ) dx dτ, s ε1 α) } t 1 where we have sed the following ineqality: s t θ x δ1 α) dxdτ = s t θ x τ) dτ δ1 α) s δ1 α) Cδ Cδ. t c v dxdτ + 1 e 1 α)τ dτ θ dxdτ + c δ s } t c α ε1 α) t F x, τ) dx dτ v x dxdτ 3.13) 3.14) Now we estimate the last term in the right side of ineqality 3.13). In fact from Lemma., we have by Cachy Schwartz ineqality c δ t F x, τ) dx dτ = c δ t ν ψ x + ψ θ x + ψ x θ) dxdτ

8 C δ W. Allegretto et al. / J. Math. Anal. Appl ) t Cδ. We have similar estimates as follows, 1 t E x, τ) dx dτ Cδ. δ Ths, 3.13) and 3.15) give by Lemma. ψ x + θ x ) dxdτ + c v ) dx + ε δ)1 α) Cδ + δ 1 ) } t + 1 α) c ν } t εα Cδ + δ 1) t Cδ + δ) + C which implies by Lemma., e 1 α)τ t dx Cδ + δ) + C e 1 α)τ c v dxdτ + 1 dxdτ + ε)α Cδ 1 + δ) } t s } t c α ε1 α) v x dxdτ 3.15) 3.16) x dxdτ dxdτ, 3.17) dxdτ. 3.18) We easily dedce from 3.18) by Gronwall s ineqality t ) dx Cδ + δ)exp C e 1 α)τ dτ Cδ + δ). 3.19) Sbstitting 3.19) into 3.17), we have + c v ) dx + ε δ)1 α) Cδ + δ 1 ) } t + 1 α) c ν } t εα Cδ + δ 1) t Cδ + δ) + Cδ + δ) which implies by 3.1) + v ) t dx + e 1 α)τ dτ, + v ) t dxdτ + c v dxdτ + 1 dxdτ + ε)α Cδ 1 + δ) } t s } t c α ε1 α) v x dxdτ x dxdτ x + vx ) dxdτ Cδ + δ), 3.) provided δ and δ 1 are sfficiently small. This proves Lemma 3.3.

9 5 W. Allegretto et al. / J. Math. Anal. Appl ) Lemma 3.4. Let the assmptions in Theorem 3.1 hold. Then the soltion x, t), vx, t)) of 3.), 3.3) obtained in Lemma 3. satisfies for any ν<4α1 α), t x + vx ) dx + xx + vxx ) dxdτ Cδ + δ ), 3.1) provided δ and δ 1 are sfficiently small. 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 + 1 α) t x + c vx ) dxdτ + α xx + c vxx ) dxdτ t = x + c v x + + νc c c t t t t v x xx dxdτ + c x vv xx dxdτ c t f θ x v xx + θ xx )dxdτ + t t t Fx,τ)v xx dxdτ Ex,τ) xx dxdτ t 1 + c )δ + ε1 α) + c ν εα t t vx dxdτ + ) t + c x L + ψ x L + c v L + 1 t x v x dxdτ + c t ψ x vv xx dxdτ c t t x xx dxdτ x dxdτ + s t ε1 α) s } t c α ε1 α) x dxdτ + s t v x dxdτ + c θ x L x + vxx ) dxdτ + c ψ x L v xx dxdτ + c t v xx f θθ x θ x + v x ) dxdτ ψ x v x dxdτ c θ x v xx dxdτ t t ψ x xx dxdτ + t dxdτ + εα xx dxdτ θ xx dxdτ + Cδ + δ) t + vxx ) dxdτ v + vxx ) dxdτ s } 1 t c α ε1 α) θ x v xx dxdτ θ x dxdτ ψ x x dxdτ

10 t + c δ + ψ x L + 1 δ t t v xx dxdτ + c δ W. Allegretto et al. / J. Math. Anal. Appl ) t F x, τ) dx dτ + L + ) xx dxdτ + ψ x L E x, τ) dx dτ, where we have sed the following ineqality: t t f θ θ)θ x xx dxdτ = = t Cδ + δ) + t t x + ) xx dxdτ t x dxdτ + δ xx dxdτ fθθ θ)θ x ) + f θ θ)θ xx ) x dxdτ fθθ θ)v x + θ x ) + f θ θ)v xx + θ xx ) ) x dxdτ t 3.) f θ θ)v xx + θ xx ) x dxdτ. 3.3) By Lemmas. and 3.3, we have from 3.15) and the above ineqalities x + c vx ) } t dx + ε)α Cδ xx dxdτ + 1 } t 1 c α ε1 α) Cδ + δ 1) vxx dxdτ Cδ + δ), which implies by 3.1) t x + vx ) dx + 3.4) xx + vxx ) dxdτ Cδ + δ ), 3.5) provided δ and δ 1 are sfficiently small. This proves Lemma 3.4. 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 3.1, then for any ν<4α1 α), we have t xx + vxx ) dx + provided δ and δ 1 are sfficiently small. xxx + vxxx ) dxdτ Cδ + δ ), 3.6) Proof. Differentiating 3.) twice with respect to x, mltiplying the reslts by xx and c v xx, respectively, integrating the reslting eqations with respect to x, t) over,t), we get t xx + c vxx ) dx + 1 α) t xx + c vxx ) dxdτ + α xxx + c vxxx ) dxdτ t t = xx + c v xx + f θ xx v xxx dxdτ + νc v xx xxx dxdτ

11 54 W. Allegretto et al. / J. Math. Anal. Appl ) t + + c t + t f θ xx θ xxx dxdτ + c t θ) xxx v xx dxdτ + c ) xxx xx dxdτ c )δ + ε1 α) + c ν εα c t t t t t t vxx dxdτ + t v) xxx v xx dxdτ + c F xx x, τ)v xx dxdτ ψ) xxx xx dxdτ + xx dxdτ + s ψv) xx v xxx dxdτ c ) xx v xxx dxdτ where we have sed the following ineqality: t t f xxx θ) xx = = t t ε1 α) xx dxdτ + s t t t t v xxx t ψv) xxx v xx dxdτ E xx x, τ) xx dxdτ t dxdτ + εα xxx dxdτ θ xxx dxdτ c θ) xx v xxx dxdτ c ψ) xx xxx dxdτ t fθθθ θ x ) 3 + 3f θ θ x θ xx + f θ θ xxx ) xx t t v) xx v xxx dxdτ F x x, τ)v xxx dxdτ E x x, τ)v xxx dxdτ, 3.7) fθθθ v x + θ x ) 3 + 3f θ v x + θ x )v xx + θ xx ) xx + f θ v xxx + θ xxx ) ) xx Cδ + δ) + f θ v xxx + θ xxx ) xx. 3.8) Shffling the terms, we get by Lemmas. and 3.4 t xx + c vxx ) dx + ε)α Cδ + δ) c c t t t xxx dxdτ + c α v) xx v xxx dxdτ c θ) xx v xxx dxdτ c ) xx v xxx dxdτ t t t } t 1 ε1 α) ψv) xx v xxx dxdτ F x x, τ)v xxx dxdτ ψ) xx xxx dxdτ Next we are devoted to estimate the terms in the right side of 3.9) as follows. t v xxx dxdτ E x x, τ)v xxx dxdτ. 3.9)

12 W. Allegretto et al. / J. Math. Anal. Appl ) First, we obtain from 3.9), Lemmas 3.3 and 3.4 by Cachy Schwartz ineqality c t We also have t v) xx v xxx dxdτ δ 1 ) xx xxx dxdτ δ 1 δ 1 t t t Similarly, we have from Lemmas., 3.3 and 3.4, and Also c c c t t t ψv) xx v xxx dxdτ = c = c c δ v xxx dxdτ + C δ 1 t v xx + x v x + vxx ) dxdτ v xxx dxdτ + Cδ + δ ). 3.3) xxx dxdτ + Cδ + δ ). 3.31) t t t t t θ) xx v xxx dxdτ δ t ψ) xx xxx dxdτ δ ψv xx + ψ x v x + ψ xx v)v xxx dxdτ ψ x v xx dxdτ c t ψ x v x + ψ xx v)v xxx dxdτ t ψ x vxx dxdτ + δ vxxx dxdτ + C δ t ψ x v x + ψ xx v) dxdτ v xxx dxdτ + Cδ + δ ) 3.3) v xxx dxdτ + Cδ + δ ). 3.33) xxx dxdτ + Cδ + δ ). 3.34) In addition, applying Lemmas., 3.3 and 3.4, we derive by Cachy Schwartz ineqality c t = c t δ t F x x, τ)v xxx dxdτ ν ψ x + ψ θ x + ψ x θ) x v xxx dxdτ v xxx dxdτ + c δ t ) ν ψ xx + ψ θ x ) x + ψ x θ) x dxdτ

13 56 W. Allegretto et al. / J. Math. Anal. Appl ) and t δ δ t t v xxx dxdτ + C δ t v xxx dxdτ + Cδ + δ ) t E x x, τ) xxx dxdτ δ ψ xx dxdτ + C δ Sbstitting 3.6) 3.9) into 3.5), we get xx + c vxx ) dx + ε)α δ 1 + 3δ )} t xxx dxdτ t ψ x θ x dxdτ + C δ t ψ θ xx dxdτ 3.35) xxx dxdτ + Cδ + δ ). 3.36) s + c α ε1 α) } t δ1 + 3δ) vxxx dxdτ Cδ + δ ), 3.37) which proves 3.6) from 3.1) provided δ and δ 1 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 dedced that 3.5) holds provided δ and δ 1 are sfficiently small. Therefore the assmption 3.8) is always tre provided 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 1, ) and g t) L 1, ), then gt) as t. The proof of Lemma 3.6 can be fond in [13,18] and the details are omitted. Taking gt) = x t) in Lemma 3.6, we can conclde from 3.5) that gt) L 1, ). 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 estimates 3.5) and Cachy Schwartz ineqality that t, xx = t, xx L 1, ). Hence, g t) L 1, ) which implies by Lemma 3.6 x t) as t. Applying Sobolev ineqality, we have from 3.38) and 3.5) 3.38) sp 1 x, t) t) x t) 1 ast. 3.39) The same process applied to vx,t) dedces that sp vx,t) vt) 1 v x t) 1 as t. 3.4) Similarly, taking gt) = xx t) and gt) = v xx t), we have from 3.5) and Lemma 3.6 xx t) and vxx t), as t. 3.41) So Sobolev ineqality and 3.41) give sp x x, t) x t) 1 xx t) 1 ast, 3.4)

14 W. Allegretto et al. / J. Math. Anal. Appl ) and sp v x x, t) v x t) 1 v xx t) 1 as t. 3.43) Ths, 3.6) is proved by 3.39), 3.4), 3.4) and 3.43). The proof of Theorem 3.1 is completed. 4. Decay rates 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, 4.1) k= k= with l = min ε)1 α),1 α) c ν }, 4.) εα where ε and c are defined by 3.1). By Sobolev ineqality, we have from 4.1), x,v,v x ) L e l t, 4.3) which will be sed later. Moreover, we list the following Gronwall s ineqality which is sed in the following text. Lemma 4.1 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 3.1, then when ν<4α1 α), we have for any t [,T], k x t) + k x vt) Cδ + δ ) 1 4 e lt, k=, 1,, 4.4) provided δ and δ are sfficiently small, where l is defined by 4.). Proof. The proof is divided into three steps. First, 3.) 1 +3.) c v and integrating the reslting identities over x, we reach by Cachy Schwartz ineqality from Lemma. d dt + c v ) dx + 1 α) + c v ) dx + α x + c vx ) dx =+ f θ v x dx + νc v x dx + f θ θ x dx c vv x dx c θv x dx

15 58 W. Allegretto et al. / J. Math. Anal. Appl ) c ψvv x dx + c vfx,t)dx ε1 α) s dx + vx ε1 α) dx + εα + c + x v dx c x dx + θv x dx + c x dx x dx + c ν εα ψ x v dx + c ψ x dx+ Ex, t) dx v dx f θ θ x dx vfx,t)dx ψ x dx + Ex, t) dx. 4.5) After shffling the terms, we have by Lemma. d + c v ) dx + ε)1 α) dx + 1 α) c ν } c v dx dt εα + ε)α x c dx + s } α vx ε1 α) dx Cδe l+1 α)}t + f θ θ x dx + c x v dx c + x dx + θv x dx + c vfx,t)dx ψ x dx + Ex, t) dx. 4.6) Next, we will estimate the terms in the right side of 4.6). In fact, we have from the assmption 4.1) and Lemma. by employing Cachy Schwartz ineqality f θ θ x dx δe l 1 α)}t dx + s δ e l 1 α)}t δe l 1 α)}t e lt + Cδe l 1 α)}t e 1 α)t Cδe l +1 α)}t. Moreover, we get from 3.5) by Sobolev ineqality, x,v,v x ) L Cδ + δ ) 1. θ x dx Ths we derive by Cachy Schwartz ineqality from 4.1), 4.3) and 4.8) ) c x v dx c x v dx + v 3 dx c v L = c v 1 L v 1 L x + v ) dx Cδ + δ ) 1 4 e l 4 t e lt = Cδ + δ ) 1 4 e 5l 4 t x + v ) dx 4.7) 4.8) 4.9)

16 W. Allegretto et al. / J. Math. Anal. Appl ) and x dx Cδ + δ ) 1 4 e 5l 4 t. Similarly, we get the following ineqality: c θv x dx 1 s } c α vx ε1 α) dxdτ + C 1 s } c α ε1 α) θ dx v x dxdτ + Ce 1 α)t In addition, we dedce from Lemma. and Cachy Schwartz ineqality c vfx,t)dx δe l 1 α)}t v dx + c δ e l 1 α)}t F x, t) dx δe l 1 α)}t e lt + C δ e l 1 α)}t ψ x + θ x ) dx δe l 1 α)}t e lt + C δ e l 1 α)}t δ e 1 α)t 4.1) dx. 4.11) Cδe l +1 α)}t. 4.1) Similarly, we get Ex, t) dx Cδe l +1 α)}t. 4.13) Ths, we get from 4.6), 4.7), 4.9), 4.11), 4.1) and 4.1) d + c v ) dx + ε)1 α) dx + 1 α) c ν } c v dx dt εα Cδe l+1 α)}t + Cδe l +1 α)}t + Cδ + δ ) 1 4 e 5l 4 t + Ce 1 α)t dx Cδ + δ ) 1 4 e 5l 4 t + Cδe l +1 α)}t + Ce 1 α)t dx. 4.14) ecalling the definition 4.) of l, we have from 4.14) d + c v ) dx + l + c v ) dx Cδ + δ ) 1 4 e 5l 4 t + Cδe l +1 α)}t + Ce 1 α)t dt Noticing that l < 1 α, we obtain from Lemmas 4.1 and 3.3 t } + c v )dx Cδ + Cδ + δ ) 1 4 e 5l 4 τ e lτ dτ + Cδ t e l +1 α)}τ e lτ dτ + C t e lt e lτ e 1 α)τ v dxdτ } e lt v dx. 4.15) Cδ + δ ) 1 4 e lt, 4.16) which implies 4.4) for k =.

17 6 W. Allegretto et al. / J. Math. Anal. Appl ) Similarly, for k = 1, if we mltiply 3.) 1 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 4.14) x + c vx ) dx Cδ + δ ) 1 4 e lt, 4.17) which proves 4.4) for k = 1. Finally, we shall show 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 arrive at d dt xx + c vxx ) dx + 1 α) xx + c vxx ) dx + α xxx + c vxxx ) dx = xx v xxx dx + νc f θ v xx xxx dx + f θ xx θ xxx dx + c v) xxx v xx dx + c ψv) xxx v xx dx + c θ) xxx v xx dx + c F xx x, t)v xx dx + E xx x, t) xx dx + ) xxx v xx dx + ψ) xxx xx dx ε1 α) xx dx + s vxxx ε1 α) dx + εα xxx dx + c ν vxx εα dx f θ xx θ xxx dx c v) xx v xxx dx c ψv) xx v xxx dx c θ) xx v xxx dx + c F xx x, t)v xx dx + E xx x, t) xx dx ) xx xxx dx ψ) xx xxx dx, 4.18) where we have sed the following ineqality: f xxx θ) xx = fθθθ θ)θ x ) 3 ) + 3f θ θ)θ x θ xx + f θ θ)θ xxx xx = fθθθ θ)v x + θ x ) 3 + 3f θ θ)v x + θ x )v xx + θ xx ) xx + f θ θ)v xxx + θ xxx ) ) xx Cδe l +1 α)}t + Cδ + δ )e 1 α)t + Ce lt 4.18) implies d dt xx + c vxx ) dx + ε)1 α) xx dx + 1 α) c ν xx dx + εα } c vxx dx f θ θ)v xxx + θ xxx ) xx. 4.19)

18 + ε)α + c W. Allegretto et al. / J. Math. Anal. Appl ) xxx c dx + s } α vxxx ε1 α) dx v) xx v xxx dx c ψv) xx v xxx dx f θ xx θ xxx dx c θ) xx v xxx dx + c F xx x, t)v xx dx + E xx x, t) xx dx ) xx xxx dx ψ) xx xxx dx. 4.) Next, we are going to give the estimates of the terms in the right side of 4.). Indeed, sing Cachy Schwartz ineqality, we have from 4.1) and Lemma. f θ xx θ xxx dx δe 1 1 α)}t s xx dx + δ e l 1 α)}t θ xxx dx δe 1 1 α)}t e lt + s δ e l 1 α)}t Cδ e 1 α)t Cδe l +1 α)}t. 4.1) For the simplicity of notation, we define λ = c α ε1 α) 1, μ = α ε). Then we derive from 4.3) by Cachy Schwartz ineqality c v) xx v xxx dx 1 16 λ 1 16 λ 1 16 λ v xxx dx + C v xxx dx + C x L v xxx dx + Ce lt x vx + vxx + v ) xx dx v x dx + C L We also have estimate ) xx xxx dx 1 16 μ xxx dx + Ce lt Similarly, we have from Lemma. with Cachy Schwartz ineqality c ψv) xx v xxx dx v xx dx + C v L xx dx v x + vxx + ) xx dx. 4.) x + ) xx dx. 4.3) 1 16 λ vxxx dx + C ψ x v x + ψ vxx + ψ xx v) dx 1 16 λ vxxx dx + C ψ x L vx dx + C ψ L vxx dx + C ψ xx L v dx 1 16 λ vxxx dx + Ce 1 α)t v + vx + ) v xx dx, 4.4)

19 6 W. Allegretto et al. / J. Math. Anal. Appl ) also and c c ψ) xx xxx dx 1 16 μ θ) xx v xxx dx 1 8 λ xxx dx + Ce 1 α)t v xxx dx + Ce 1 α)t Finally, we have by Lemma., Cachy Schwartz ineqality and 3.4) c F xx x, t)v xx dx δe l 1 α)}t vxx dx + c δ e l 1 α)}t δe l +1 α)}t + C δ e l 1 α)}t δe l +1 α)}t + C δ e l 1 α)}t δ e 1 α)t Cδe l +1 α)}t. Fxx x, t) dx + x + ) xx dx, 4.5) + x + ) xx dx. 4.6) ψ xxx + ψ xx θ x + ψ x θ xx + ψ θ xxx + θ ψ xxx ) dx 4.7) We get a similar estimate, E xx x, t)v xx dx Cδe l +1 α)}t. 4.8) Sbstitting 4.18) 4.8) into 4.), we get from 3.5) d dt xx + c vxx ) dx + ε)1 α) xx dx + 1 α) c ν } εα ε)α xxx dx λ vxxx dx Cδe l +1 α)}t + Ce 1 α)t Cδe l +1 α)}t + Cδ + δ )e 1 α)t + Ce lt c v xx dx + v + x + v x + v xx + xx) dx + Ce lt v x + vxx + ) xx dx v x + vxx + ) xx dx. 4.9) ecalling the definition 4.) of l, we obtain from 3.1) and 4.9) d dt xx + c vxx ) dx + l xx + c v ) xx dx Cδe l +1 α)}t + Cδ + δ )e 1 α)t + Ce lt v x + vxx + ) xx dx. 4.3) Noticing l < 1 α, we easily dedce from 4.4) with the help of Lemma 4.1 and 3.5) xx + c vxx ) t t dx Cδ + Cδ e l +1 α)}τ e lτ dτ + Cδ + δ ) e 1 α)τ e lτ dτ

20 t + C W. Allegretto et al. / J. Math. Anal. Appl ) e lτ v x + vxx + ) xx dxe lτ dτ Cδ + Cδ + Cδ + δ ) + C t ) e lt v x + vxx + ) xx dxdτ )e lt Cδ + δ )e lt. 4.31) The proof of Theorem 4. is completed by 4.16), 4.17) and 4.31). Finally, we verify the a priori assmption 4.1) is reasonable. Indeed, nder this a priori assmption, we show 4.4) holds. Therefore, the assmption 4.1) is always tre provided δ and δ are sfficiently small. 5. Frther discssion We may also establish the L p decay properties of the system 1.1), for the decay rates of the soltion is dictated by the kernel fnction sed for the integral form of the soltion representation in Section 3.. For related discssion and conclsion of the L p decay properties of the soltion, we refer to [3,16,17]. Acknowledgment The athors wold like to thank the anonymos referee for many helpfl sggestions. eferences [1] X.X. Ding, J.H. Wang, Global soltion for a semilinear parabolic system, Acta Math. Sci ) [].J. Dan, C.J. Zh, Asymptotics of dissipative nonlinear evoltion eqations with ellipticity: Different end states, J. Math. Anal. Appl. 33 1) 5) [3].J. Dan, S.Q. Tang, C.J. Zh, Asymptotics in nonlinear evoltion system with dissipation and ellipticity on qadrant, J. Math. Anal. Appl. 33 ) 6) [4] D. Henry, Geometric Theory of Semilinear Parabolic Eqations, Lectre Notes in Math., vol. 84, Springer-Verlag, New York, 198. [5] L. Hsiao, H.Y. Jian, Global smooth soltions to the spatically periodic Cachy problem for dissipative nonlinear evoltion eqations, J. Math. Anal. Appl ) [6] L. Hsiao, T.-P. Li, Convergence to nonlinear diffsion waves for soltions of a system of hyperbolic conservation laws with damping, Comm. Math. Phys ) [7] D.Y. Hsieh, On partial differential eqations related to Lorenz system, J. Math. Phys ) [8] D.Y. Hsieh, S.Q. Tang, X.P. Wang, On hydrodynamic instability, chaos, and phase transition, Acta Mech. Sinica ) [9] H.Y. Jian, D.G. Chen, On the Cachy problem for certain system of semilinear parabolic eqations, Acta Math. Sinica N.S.) ) [1] H.Y. Jian, X.P. Wang, D.Y. Hsieh, The global attractor of a dissipative nonlinear evoltion system, J. Math. Anal. Appl ) [11] L.. Keefe, Dynamics of pertrbed wavetrain soltions to the Ginzberg Landa eqation, Std. Appl. Math ) [1] Y. Kramoto, T. Tszki, On the formation of dissipative strctres in reaction diffsion systems, Progr. Theoret. Phys ) [13] A. Matsmra, K. Nishihara, Asymptotics toward the rarefaction waves of the soltions of a one-dimensional model system for compressible viscos gas, Japan J. Appl. Math ) [14] K. Nishihara, Convergence rates to nonlinear diffsion waves for soltions of system of hyperbolic conservation laws with damping, J. Differential Eqations ) [15] S.Q. Tang, Dissipative nonlinear evoltion eqations and chaos, PhD thesis, Hong Kong Univ. Sci. Tech., [16] S.Q. Tang, H.J. Zhao, Nonlinear stability for dissipative nonlinear evoltion eqations with ellipticity, J. Math. Anal. Appl ) [17] Z.A. Wang, Optimal decay rates to diffsion wave for nonlinear evoltion eqations with ellipticity, J. Math. Anal. Appl ) [18] H.J. Zhao, Nonlinear stability of strong planar rarefaction waves for the relaxation approximation of conservation laws in several space dimensions, J. Differential Eqations 163 ) 198. [19] C.J. Zh, Asymptotic behavior of soltions for p-system with relaxation, J. Differential Eqations 18 ) [] C.J. Zh, Z.A. Wang, Decay rates of soltions to dissipative nonlinear evoltion eqations with ellipticity, Z. Angew. Math. Phys. 55 6) 4) [1] C.J. Zh, Z.Y. Zhang, H. Yin, Convergence to diffsion waves for nonlinear evoltion eqations with ellipticity and damping, and with different end states, Acta Math. Sin. Engl. Ser.) 5) 6)