The stability of travelling fronts for general scalar viscous balance law

Transcription

1 J. Math. Anal. Appl ) The stability of travelling fronts for general scalar viscous balance law Yaping Wu, Xiuxia Xing Department of Mathematics, Capital Normal University, Beijing 137, China Received 29 April 23 Available online 2 January 25 Submitted by M.D. Gunzburger Abstract This paper is concerned with the existence and stability of travelling front solutions for some general scalar viscous balance law. By shooting methods we prove the existence of some class of travelling fronts for any positive viscosity. Further by analytic semigroup theory and detailed spectral analysis, we show that the travelling fronts obtained are asymptotically stable in some appropriate exponentially weighted space. Especially for all sufficiently small viscosity, the travelling waves are proved to be uniformly exponentially stable in the same weighted space. 24 Elsevier Inc. All rights reserved. Keywords: Balance law; Travelling front; Existence; Phase plane method; Asymptotic stability; Spectral analysis 1. Introduction This paper is concerned with the travelling front solutions of the viscous balance law u t + fu) x = εu xx + gu), x R, 1.1) here ε> is called the viscosity parameter. Research is supported by the National Nature Science Foundation of China , Beijing Natural Science Foundation 1523) and FBEC KZ231281). * Corresponding author. address: yaping_wu@hotmail.com Y. Wu) X/$ see front matter 24 Elsevier Inc. All rights reserved. doi:1.116/j.jmaa

2 Y. Wu, X. Xing / J. Math. Anal. Appl ) When f =, 1.1) is the standard reaction diffusion equation, it is well known that the existence and stability of travelling waves for this type of equations has been widely investigated [1,11]. On the other hand, when g =, 1.1) becomes the standard viscous conservation law, to which a lot of important research work have been devoted in the past several decades, such as the admissibility of the shock waves of hyperbolic conservation law, the existence and stability of viscous shock profiles, etc. Especially by spectral analysis and semigroup methods [4,5,9,1,12], the asymptotic stability including exponential stability and algebraic stability of the viscous shock profiles has been extensively studied. In recent years, much work [6 8] has been devoted to the following hyperbolic balance laws: u t + fu) x = gu), x R, 1.2) where the reaction term gu) describing some chemical reactions, combustion or other interactions [2] can dramatically change the long time behavior of solutions of 1.2), compared to hyperbolic conservation laws. For example, C. Mascia [6] showed that for the case f is convex there exist many types of travelling waves for 1.2), including the continuous monotone waves and many kinds of discontinuous waves shock waves, sub-shocks, etc.), while for hyperbolic conservation laws the only travelling waves are shock waves. More recently, when the convection term f is convex, by invariant manifold theory J. Härterich [2] showed that several types of waves of 1.2) obtained in [6] admit the viscous profiles of travelling waves of 1.1), and they are close in weighted space L 1 β R) when the viscosity ε is small enough. In particular, the author proved that for every continuous wave u,c x ct) connecting u k to u j of 1.2) in the following four cases: i) j = k + 1, u k Ag), 1 c>f u k+1 ), ii) j = k 1, u k Ag), c>f u k ), iii) j = k + 1, u k Rg), c<f u k ), iv) j = k 1, u k Rg), c<f u k 1 ), when ε is small enough 1.1) possesses the smooth monotone waves u ε,c x ct), which converge to u,c x ct) in the above mentioned weighted space as ε. In fact, the methods and proof in [2] are still valid for the nonconvex convection cases, although the author did not mention that. In fact, as explicitly shown in [7], the convexity hypothesis on f is too restrictive to understand the qualitative behavior of 1.2) in the general cases. Recently, the existence of travelling waves of hyperbolic balance law 1.2) for some nonconvex convection cases was investigated in [8], where the extremal speeds for the existence of continuous travelling waves were also obtained. For the cases with convex convection terms and small compact initial perturbations, the asymptotic stability of continuous travelling waves for hyperbolic balance law 1.2) were obtained in [6]. As far as we know, until now there is no result on the stability of travelling waves for viscous balance law 1.1). 1 For Ag) and Rg), the reader may refer to the notations in 1.3).

3 7 Y. Wu, X. Xing / J. Math. Anal. Appl ) In this paper, we restrict our attention to the existence and stability of the above four types of travelling waves for 1.1), the existence and stability for the other kinds of travelling waves for 1.1) will be investigated in our another paper. First, using shooting methods [1] we prove that in the general nonlinearity cases no convexity condition on f ), for any positive viscosity, there exist smooth monotone waves of 1.1) connecting adjacent zeroes of g. As in [8,11], the estimates on the extremal speeds for 1.1) are also obtained. Furthermore, based on our existence results, by semigroup theory and detailed spectral analysis we show that for every fixed viscosity ε the travelling fronts for general 1.1) are locally asymptotically stable in some suitable weighted spaces, and especially for all sufficiently small viscosity, we can choose the same weighted spaces in which the travelling fronts are uniformly exponentially stable, which further implies the stability of the corresponding waves of hyperbolic balance law 1.2) in some suitable exponential weighted space. These also extend the stability results on the travelling waves of 1.2) in [6] to the cases with general nonlinearity and with noncompact initial perturbations. Finally, it should be remarked that the weights we imposed on are both necessary and sufficient for shifting the essential spectrum to the left to obtain exponential stability, so the stability results obtained in this paper are optimal in the sense of exponential stability. This paper is planed as follows: The existence of travelling fronts of 1.1) for the more general cases is proved in Section 2, and the asymptotic stability of the travelling waves in the weighted spaces is obtained in Section Notations The zeroes of g are denoted by u k, where k {1, 2,...,n}, and the set of all zeroes of g is called Zg). Depending on the sign of g the zeroes of g are divided into two sets: Ag) := { u k Zg): g u k )< }, Rg) := { u k Zg): g u k )> }. 1.3) 2. The existence of travelling fronts In this and the following sections, we always assume the following two assumptions hold: I) g C 1 R), and g has finitely many simple zeroes, II) f C 2 R). In this section, by shooting methods we shall prove the existence of continuous travelling fronts of 1.1) for every fixed ε>, here we also give the estimates on the maximal or minimum wave speed. Theorem 2.1. Let I) and II) hold, then for any fixed ε> there exist travelling fronts U ε,c x ct) connecting u k to u j for 1.1), where u k, u j, and c are in one of the following cases:

4 Y. Wu, X. Xing / J. Math. Anal. Appl ) i) j = k + 1, u k Ag), c c, where f u j ) + 2 εg u j ) c max f u) + 2 ε sup gu)/u u j )), [u k,u j ] u k,u j ) ii) j = k 1, u k Ag), c c, where f u j ) + 2 εg u j ) c max f u) + 2 ε sup gu)/u u j )), [u j,u k ] u j,u k ) iii) j = k + 1, u k Rg), c c, where min f u) 2 ε sup gu)/u u k )) c f u k ) 2 εg u k ), [u k,u j ] u k,u j ) iv) j = k 1, u k Rg), c c, where min f u) 2 ε sup gu)/u u k )) c f u k ) 2 εg u k ). [u j,u k ] u j,u k ) Proof of Theorem 2.1. Without loss of generality, here we only give the proof for case iii), the proof for the other cases is similar. For case iii), let U ε,c z) z = x ct) be the travelling wave of 1.1) connecting u k to u j, then U ε,c z) satisfies { εu ε,c + c f U ε,c ))U ε,c + gu ε,c) =, 2.1) U ε,c ) = u k,u ε,c + ) = u k+1. Let q = U ε,c u k )/[u], with [u]=u k+1 u k, then 2.1) is equivalent to q = p, εp = g 1 q) + f 1 q) c)p, 2.2) q, p) ) =, ), q, p)+ ) = 1, ), where f 1 q) = f q[u]+u k ), g 1 q) = gq[u]+u k )/[u]. Similar to [1,11], by linearizing the system 2.2) around 1, ) and, ) respectively, one can easily check that if c<f 1 ) 2 ), 2.3) εg 1 then, ) is an unstable node and 1, ) is a saddle point of 2.2), so there exists an unique trajectory Γ c of 2.2) hitting 1, ) from the left and above as in Fig. 1. For some µ> and small µ >, consider the lines p = µq and p = µ q as indicated in Fig. 1. Label the points A, B, C and the radius ɛ of sufficiently small neighborhood of 1, ) as shown in Fig. 1. Following the trajectory Γ c backward into the triangle OCA, we know that it must either lead to the other critical point, ) in the closed triangle or else cross the boundary of the triangle. By the standard phase plane analysis as in [11], it is easy to check that Γ c once enters the triangle, it will neither leave the triangle from side OC nor from AB. On the segment OA, p = µq, ε dp min dq f 1q) c 1 [,1] µ sup g1 q)/q ) = τ c ν [,1] µ,

5 72 Y. Wu, X. Xing / J. Math. Anal. Appl ) Fig. 1. where g1 q)/q ) g 1 ν = sup [,1] Thus, if µ> satisfies ), τ = min [,1] f 1q) = min [,1] f u). τ c µ 1 ν εµ, 2.4) then Γ c will not leave the triangle through OA. It is easy to check that if c satisfies c τ 2 εν, 2.5) then there exists µ > satisfying 2.4). Therefore for any fixed c satisfying 2.5), there exists an unique orbit of travelling front connecting, ) to 1, ). Further, by a similar argument as in [11] we can prove that there exists a maximal speed c satisfying τ 2 εν c f 1 ) 2 εg 1 ), such that for every c c, there exists a travelling front of 2.2). This completes the proof of Theorem 2.1 for case iii). In fact, from the above proof of Theorem 2.1, we notice that for any c<τ 2 εν, there exists µ> lying between the two distinct roots 2ε) 1 τ c ± τ c) 2 4εν ) such that the orbit Γ c of travelling front satisfies dp dq µ<2ε) 1 τ c + τ c) 2 4εν ),) 2ε) 1 f 1 ) c + f 1 ) c) 2 ) 4εg 1 ) then by 2.4), Γ c must satisfy

6 Y. Wu, X. Xing / J. Math. Anal. Appl ) dp dq = 2ε) 1 f 1 ) c,) = 2ε) 1 f u k ) c f 1 ) c) 2 ) 4εg 1 ) f u k ) c) 2 4εg u k ) ). 2.6) Furthermore, the continuous dependence of Γ c on c implies 2.6) are valid for any c<c. Thus additionally we have proved the following proposition. Proposition 2.1. For the waves iii) in Theorem 2.1, ifc<c, then the trajectory Γ c must satisfy dp dq = 2ε) 1 f u k ) c f u k ) c) 2 4εg u k ) ).,) Remark 2.1. i) For the other waves i), ii), and iv) in Theorem 2.1, similar results to Proposition 2.1 can be obtained. ii) Proposition 2.1 will be very useful in the proof of the stability theorem in the next section. Remark 2.2. Considering the cases with sufficiently small viscosity and comparing our results with the results for inviscid case in [2,6,8], one can easily understand the following fact: There is a critical speed separating the continuous profiles from the discontinuous ones of 1.2), as was shown in [8], where the critical speed c cri = min [uk,u j ] f u). When ε =, it is easy to check that in nonconvex convection cases f u k ) may not equal to min [uk,u j ] f u)), only for speeds c<c cri there exist smooth monotone waves iii) of 1.2), no smooth waves exist for the case c cri c c with c satisfying min [u k,u j ] f u) c f u k ). In fact, for every speed c between c cri and c, with c = f u ) for some u u k,u j ), 1.2) possesses a sub-shock solution. The above phase plane analysis and Theorem 2.1 imply that some sub-shock profiles of 1.2) admit viscous profiles of travelling fronts of 1.1). Combining the results in [2], we can immediately obtain Corollary 2.1. Let I) and II) hold, for any wave U,c x ct) connecting u k and u j of 1.2), where u k, u j, and c are in one of the following cases: i) j = k + 1, u k Ag), c>max [uk,u j ] f u), ii) j = k 1, u k Ag), c>max [uj,u k ] f u), iii) j = k + 1, u k Rg), c<min [uk,u j ] f u), iv) j = k 1, u k Rg), c<min [uj,u k ] f u), then there exists sufficiently small ε > such that for any <ε ε there exist viscous profiles of travelling fronts U ε,c x ct) connecting u k to u j of 1.1), which converge to U,c x ct) as ε in L 1 β R), where β<min{ g u k ) / c f u k ), g u j ) / c f u j ) }, and L 1 β R) ={u L1 R) R 1 + eβ ξ ) uξ) dx < }.

7 74 Y. Wu, X. Xing / J. Math. Anal. Appl ) The asymptotic stability of travelling waves In this section, we shall prove the asymptotic stability of travelling waves obtained in Section 2. Consider the following initial value problem of 1.1): { ut + fu) x = εu xx + gu), x R, t>, 3.1) u t= = u x). Introducing new variable z = x ct, 3.1) can be written as { ut cu z + f u)u z = εu zz + gu), 3.2) u t= = u z). Define wz,t) = uz, t) U ε,c z) with U ε,c z) the travelling front i), ii), iii), or iv) respectively in Theorem 2.1, then wz,t) satisfies with w t = L ε,c w + hw, w z ) 3.3) L ε,c = ε 2 z 2 + c f U ε,c ) ) z + g U ε,c ) f U ε,c )U ε,c, 3.4) and hw, w z ) L 2 R) = O w 2 H 1 R) ) as w H 1 R) is sufficiently small. Now we state the main stability results in this paper: Theorem 3.1. Let U ε,c z) be the travelling waves i) or ii) iii) or iv) respectively) in Theorem 2.1, and define weight function V α z) = 1 + e αz V α z) = 1 + e αz respectively). Given ε>, for any fixed α> satisfying α 1 ε) < α < α 2 ε) with and α 1 ε) = 2g u j ) f u j ) c + f u j ) c) 2 4εg u j ) 2g u k ) f u k ) c + f u k ) c) 2 4εg u k ) α 2 ε) = 2g u j ) f u j ) c f u j ) c) 2 4εg u j ) 2g u k ) f u k ) c f u k ) c) 2 4εg u k ) then there exists small δ> such that if Vα z) u z) U ε,c z) ) H 1 δ, R) ) respectively 3.5) 3.5 ) 3.6) ) respectively, 3.6 )

8 Y. Wu, X. Xing / J. Math. Anal. Appl ) then V α z) ut, z) U ) ε,cz) H 1 R) Ce σt, with σ>and C depends on c, ε, and α. Note that, in sufficiently small viscosity case, we can choose the uniform bound for the weight parameter α both from bottom and above such that for any α between them an uniform decay rate for all positive small enough ε can be obtained. Theorem 3.2. Let U ε,c z) be the travelling waves i) or ii) iii) or iv) respectively) of 1.1) in Corollary 2.1, for <ε ε with ε > small enough. Define weight function V α z) = 1 + e αz V α z) = 1 + e αz respectively), for any fixed α> satisfying α 1 ε )<α<α 2 ε ) 3.7) with α 1 ε ) and α 2 ε ) defined in 3.5) and 3.6) respectively. There exists small δ> such that if Vα z) u z) U ε,c z) ) H 1 δ, R) then V α z) ut, z) U ) ε,cz) H 1 R) Ce σt, with σ>and C depends only on c, ε, and α. As in Section 2, without losing generality, in the following we only give the proof of Theorem 3.1 for case iii) in Theorem 2.1 and the proof of Theorem 3.2 for case iii) in Corollary 2.1, the proof for other cases is similar The distribution of essential spectra of A ε,α Denote weighted space X α = { wz) V α z)wz) X } and the norm w Xα = V α z)wz) X. Obviously, with weight function V α z) = 1 + e αz for some α>, L 2 α R) = { wz) V α z)wz) L 2 R) }, Hα k R) = { wz) V α z)wz) H k R) }. For any fixed ε> and c<c, where c is given in Theorem 2.1 iii), i.e. min f u) 2 ε sup gu)/u u k )) c f u k ) 2 εg u k ), [u k,u j ] u k,u j ) define an operator A ε,α : Hα 2 R) L2 α R), A ε,αw = L ε,c w, w Hα 2 R),

9 76 Y. Wu, X. Xing / J. Math. Anal. Appl ) thus with A ε,α w = εw zz + az)w z + bz)w, az) = c f U ε,c z) ), bz)= g U ε,c z) ) f U ε,c z) ) U ε,c z). 3.8) Define the map ψ : X X α, ψw= w V α, w X, and further, we define an operator  ε,α : H 2 R) L 2 R),  ε,α = ψ 1 A α ψ, i.e. for w H 2 R), 1  ε,α w = εw zz + 2εV α V α 1 + εv α V α ) + c f U ε,c)) w z ) + c f U ε,c) ) V α 1 + g U ε,c ) f U ε,c )U ε,c V α ) ) w. 3.9) Obviously, A ε,α : H 2 α R) L2 α R) is equivalent to  ε,α : H 2 R) L 2 R), with σ p A ε,α ) = σ p  ε,α ), ρa ε,α ) = ρâ ε,α ), 3.1) and λi A 1 ε,α) L 2 = λi  ε,α ) 1 α L2 α L 2 L ) Obviously 3.9) 3.11) imply A ε,α generates an analytic semigroup on L 2 α. In the following, we consider the essential spectra of A ε,α. By 3.1), obviously σ ess A ε,α ) = σ ess  ε,α ). 3.12) By 3.9), we can define  ε,α w = εw zz + 2εα + c f u k ) ) w z + εα 2 + α c f u k ) ) + g u k ) ) w, 3.13)  + ε,α w = εw zz + c f u j ) ) w z + g u j )w. 3.14) Define Sα = { λ C λ = εiτ) 2 + 2εα + c f u k ) ) iτ + εα 2 + α c f u k ) ) + g u k ) for some τ R }, 3.15) S α + = { λ C λ = εiτ) 2 + c f u j ) ) iτ + g u j ) for some τ R }. 3.16)

10 Y. Wu, X. Xing / J. Math. Anal. Appl ) For travelling wave iii) with g u k )>, g u j )<, if εα 2 + α c f u k ) ) + g u k )<, 3.17) then sup { Re { Sα }} { S+ α max εα 2 + α c f u k ) ) + g u k ), g u j ) } <. 3.18) By [3] and 3.12) 3.18), we have shown that Lemma 3.1. For each fixed ε>, c<c, and α> satisfying α 1 ε) < α < α 2 ε) 3.19) with α 1 ε) and α 2 ε) defined in 3.5 ) and 3.6 ) respectively, there exists some constant C α > such that sup { Re { σ ess A ε,α ) }} C α <. Remark 3.1. From 3.19) it is easy to see that for any <ε ε with ε > small enough), we can choose the uniform lower bound α 1 ε ) and upper bound α 2 ε ) such that for any weight parameter α lying between them all the essential spectrum of the linearized operator A ε,α have negative real parts. Furthermore, for each fixed c<min [uk,u k+1 ] f u), it is easy to check that there exists constant C ε,α > such that for any <ε<ε, sup { Re { σ ess A ε,α ) }} C ε,α < The distribution of eigenvalues of A ε,α In this section, we always assume α> satisfies 3.19). First, we consider the asymptotic behavior of eigenfunction w λ of A ε,α w λ = λw λ, w λ Hα 2 R). 3.2) By standard asymptotic analysis, it is easy to prove that w λ z) exp { 2ε) 1 f u k ) c ± Re f u k ) c) 2 + 4ελ g u k )) ) z }, as z. 3.21) Let λ = Re λ + i Im λ and f u k ) c) 2 + 4ελ g u k )) = c 1 + ic 2, with c 1, and c 2 R, then { f u k ) c) 2 + 4εRe λ g u k )) = c1 2 c2 2, 3.22) 2ε Im λ = c 1 c 2. For any fixed ε> and α> satisfying 3.19), if Re λ σ 1,σ 1 > small enough, then thus c 2 1 f u k ) c ) 2 4ε σ1 + g u k ) ),

11 78 Y. Wu, X. Xing / J. Math. Anal. Appl ) ε) 1 f u k ) c + c 1 ) 2g u k ) f u k ) c f u k ) c) 2 4εσ 1 + g u k )) >α, 2ε) 1 f u k ) c c 1 ) 2g u k ) f u k ) c + f u k ) c) 2 4εσ 1 + g u k )) <α, which implies w λ z) exp { 2ε) 1 f u k ) c + Re f u k ) c) 2 + 4ελ g u k )) ) z }, as z. 3.23) A similar but simpler analysis tells that for any fixed ε>, if λ is an eigenvalue of A ε,α with Re λ σ 2, σ 2 > small enough, then the corresponding eigenfunction w λ z) must satisfy w λ z) exp { 2ε) 1 f u j ) c Re f u j ) c) 2 + 4ελ g u j )) ) z }, as z +. Thus we obtain 3.24) Lemma 3.2. For ε>, c<c, α> satisfying 3.19) and Re λ σ, with σ > small enough, if λ is an eigenvalue of A ε,α, then the eigenfunction w λ z) Hα 2 R) must decay to zero exponentially as z ±, satisfying 3.23) and 3.24). Let { ŵ λ z) = exp 2ε) 1 z c f U ε,c s) )) } ds w λ z) 3.25) for Re λ σ, the eigenvalue problem 3.2) can be written as ˆLŵ λ εŵ λ ) a 2 ) z) + a z) bz) ŵ λ = λŵ λ, 3.26) 4ε 2 with ˆL : H 2 R) L 2 R), and az) and bz) defined in 3.8). By Lemma 3.2, 3.25) 3.26), it is easy to check that Lemma 3.3. For ε>, c<c, α> satisfying 3.19) and Re λ σ, with σ > small enough, there exists a nonzero function w λ H 2 α R) satisfying A ε,αw λ = λw λ if and only if there exists a nonzero function ŵ λ H 2 R) such that ˆLŵ λ = λŵ λ. Obviously, operator ˆL is a self-adjoint operator on H 2 R), then Lemma 3.3 implies that under the condition of Lemma 3.3, for Re λ σ, all the eigenvalues of A ε,α are real. In fact, in the following we can further show that all the eigenvalues of A ε,α are negative. Lemma 3.4. For any ε>, c<c, α> satisfying 3.19) and σ > small enough, the eigenvalues of A ε,α in Ω ={λ C, Re λ σ } are real and sup { Re { σ p A ε,α ) }} <.

12 Y. Wu, X. Xing / J. Math. Anal. Appl ) Proof. By contradiction, assume λ is an eigenvalue of 3.2) with eigenfunction w λ Hα 2 R). Let w λ z) = U z)u λ z) 3.27) with U z) = U z), then u λ z) satisfies εu λ + 2ε U ) + az) u λ U = λu λ. 3.28) Let p λ = u λ, then 3.28) becomes εp λ + 2ε U ) + az) p λ = λu λ, 3.29) U thus { { z 2U exp + as) ) } } { z ds p λ z) = λu λ 2U ε ε exp + as) ) } ds. 3.3) ε U First, we shall deduce a contradiction for the case λ =. Let λ =, then from 3.3) we have { z p z) = C exp 2 U + as) ) } ds, 3.31) U ε with C some constant. Note that U z) >, and satisfies A ε U =. By Proposition 2.1 we have { U z) exp 2ε) 1 ) } a ± + a± 2 4εb ± z, as z ± 3.32) with thus a = c f u k ), b = g u k ), a + = c f u j ), b + = g u j ), 2 U + az) ) 2ε) 1 a+ 2 U ε 4εb + >, as z ) By 3.24), 3.27), and 3.32), it is easy to check that p z) = u z) must be bounded as z ) and 3.33) implies that p z) is bounded at + iff p z) forz R, which further means w z) = C U z) with C some constant. Comparing 3.23) with 3.32), it is easy to see that U z) is not an eigenfunction of A ε,α with respect to λ =, thus w z), z R, which contradicts λ = is an eigenvalue of A ε,α. In the following, we consider that case λ>. Let λ 1 > be the first eigenvalue of ˆL. By Sturm Liouville theorem, the eigenfunction ŵ λ1 of ˆL corresponding to λ 1 does not change sign. Let ŵ λ1 z) >, Lemma 3.4 implies there exists eigenfunction w λ1 H 2 α R) corresponding to the first eigenvalue λ 1 of A ε,α, with w λ1 >. U

13 71 Y. Wu, X. Xing / J. Math. Anal. Appl ) From 3.23), 3.24), 3.27), and 3.32), u λ1 z) satisfies u λ 1 z), u λ1 z) { C exp 2ε) 1 ) } a± 2 + 4ελ 1 b + ) + a± 2 4εb ± z, as z ±. 3.34) From 3.29), it is easy to see p λ1 z), z R. By 3.34) and the fact λ 1 >, w λ1 >, there exists z R such that p λ1 z )>. Without losing generality, let p λ1 )>, then 3.3) implies { z 2U exp + as) ) } ds p λ1 z) > p λ1 ), z >, ε i.e. U p λ1 z) > p λ1 ) exp { z 2U + as) ) } ds, z>. U ε Again, by 3.33), we have p λ1 z) + as z + ), which contradicts 3.34). So the first eigenvalue λ 1 of A ε,α must be negative, which completes the proof of Lemma 3.4. By Lemmas 3.1 and 3.4, we obtain Lemma 3.5. For each fixed ε>, c<c, and α> satisfying 3.19), sup { Re { σa ε,α ) }} <. Applying the standard stability theory to the viscous equation 1.1), we complete the proof of Theorem 3.1 for case iii) in Theorem 2.1. By 3.8) and 3.26) it is easy to prove that Proposition 3.1. For each fixed c satisfying c<min [uk,u k+1 ] f u), there exists small ε >, and constant σ> such that for any <ε<ε, sup { Re { σ p ˆL) }} σ <. Thus by Lemma 3.3, Proposition 3.1, and Remark 3.1, we have Lemma 3.6. For each fixed c<min [uk,u k+1 ] f u), α> satisfying 3.7), and ε small enough, there exists constant σ ε,α > such that for any <ε<ε, sup { Re { σa ε,α ) }} σ ε,α <. By Lemma 3.6 and the standard semigroup theory, we complete the proof of Theorem 3.2 for case iii) in Corollary 2.1. The proof for other cases in Theorems 3.1 and 3.2 can be similarly obtained.

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