On the Nonlinear Instability of Traveling Waves for a Sixth-Order Parabolic Equation

Abstract

We study the instability of the traveling waves of a sixth-order parabolic equation which arises naturally as a continuum model for the formation of quantum dots and their faceting. We prove that some traveling wave solutions are nonlinear unstable under 𝐻⁴ perturbations. These traveling wave solutions converge to a constant as 𝑥→∞.

1. Introduction

In this paper, we consider the following sixth-order parabolic equation 𝜕𝑢𝜕𝑡=𝐷⁶𝑢+𝐷⁴𝑢−𝑢³+𝑔(𝑢),(𝑥,𝑡)∈𝑅×(0,𝑇),(1.1) where 𝑔(𝑢)=𝑎(1−𝑢²),𝑎>0.

Equation (1.1) arises naturally as a continuum model for the formation of quantum dots and their faceting; see [1]. Here 𝑢(𝑥,𝑡) denotes the surface slope. The high-order derivatives are a result of the additional regularization energy which is required to form an edge between two-plane surfaces with different orientations.

During the past years, only a few works have been devoted to the sixth-order parabolic equation [2–7]. Barrett et al. [2] considered the above equation with 𝑚=2. A finite element method is presented which proves to be well posed and convergent. Numerical experiments illustrate the theory.

Recently, Jüngel and Milišić [5] studied the sixth-order nonlinear parabolic equation 𝜕𝑢𝜕𝑡=𝑢1𝑢𝑢(ln𝑢)^𝑥𝑥_𝑥𝑥+12(ln𝑢)^𝑥𝑥²_𝑥_𝑥.(1.2) They proved the global-in-time existence of weak nonnegative solutions in one space dimension with periodic boundary conditions.

Evans et al. [3, 4] considered the sixth-order thin film equation containing an unstable (backward parabolic) second-order term 𝜕𝑢𝜕𝑡=div|𝑢|^𝑛∇Δ²𝑢−Δ|𝑢|^𝑝−1𝑢,𝑛>0,𝑝>1.(1.3) By a formal matched expansion technique, they show that, for the first critical exponent 𝑝=𝑝₀=𝑛+1+4/𝑁 for 𝑛∈(0,5/4), where 𝑁 is the space dimension, the free-boundary problem with zero-height, zero-contact-angle, zero-moment, and zero-flux conditions at the interface admits a countable set of continuous branches of radially symmetric self-similar blow-up solutions 𝑢_𝑘(𝑥,𝑡)=(𝑇−𝑡)^{(−𝑁/(𝑛𝑁+6))}𝑓_𝑘(𝑦), 𝑦=𝑥/((𝑇−𝑡)^{(1/(𝑛𝑁+6))}), where 𝑇>0 is the blow-up time.

Korzec et al. [8] considered the sixth-order equation 𝑢_𝑡−𝜈𝑢𝑢_𝑥−𝑢−𝑢³+𝜀²𝑢_𝑥𝑥_{𝑥𝑥𝑥𝑥}=0.(1.4) New type of stationary solutions is derived by an extension of the method of matched asymptotic expansion.

In this paper, we study instability of the traveling waves of (1.1). Our main result is as follows.

Theorem 1.1. All the traveling waves 𝜑(𝑥−𝑐𝑡) of (1.1) satisfying 𝜑∈𝐿^∞(𝑅), 𝜑^(𝑛)∈𝐿^∞(𝑅)∩𝐿²(𝑅)(𝑛=1,2,…,6) are nonlinearly unstable in the space 𝐻⁴(𝑅), where 𝜑^(𝑛) denotes 𝑛𝑡ℎ derivative of 𝜑.

The stability and instability of special solutions for the higher-order parabolic equation are very important in the applied fields. Carlen et al. [9] proved the nonlinear stability of fronts for the Cahn-Hilliard, under 𝐿¹ perturbations. Gao and Liu [10] prove that it is nonlinearly unstable under 𝐻² perturbations, for some traveling wave solution of the convective Cahn-Hilliard equation. The relevant equations have also been studied in [11, 12]. The main difficulties for treating (1.1) are caused by the principal part and the lack of the Lyapunov functional. Our proof is based on the principle of linearization. We invoke a general theorem that asserts that linearized instability implies nonlinear instability.

This paper is organized as follows. In the next section, we find an exact traveling wave solution for (1.1). In Section 3, we give the proof of our main result.

2. Exact Traveling Wave Solutions

In this section, we construct an exact traveling wave which satisfies all conditions of Theorem 1.1.

If 𝜑(𝑥−𝑐𝑡)=𝜑(𝑧) is a traveling wave solution of (1.1), then 𝜑 satisfies the ordinary differential equation −𝑐𝜑′=𝜑⁽⁶⁾+1−3𝜑²𝜑⁽⁴⁾−36𝜑^2𝜑^−18𝜑𝜑^2−24𝜑𝜑′𝜑^+𝑎1−𝜑².(2.1) Let 𝜑′=𝜕𝜑/𝜕𝑧=𝑘(1−𝜑²). Then 𝜑^=𝜕𝜕𝑧𝑘1−𝜑²=−2𝑘²𝜑_𝜑1−𝜑²,=𝜕𝜕𝑧−2𝑘²𝜑₍₄₎𝜑1−𝜑²=2𝑘³−1+3𝜑²1−𝜑²,=𝜕𝜑₍₅₎𝜕𝑧2𝑘³−1+3𝜑²1−𝜑²=2𝑘⁴8𝜑−12𝜑³1−𝜑²,=𝜕𝜕𝑧2𝑘⁴8𝜑−12𝜑³1−𝜑²=8𝑘⁵1−𝜑²2−15𝜑²𝜑₍₆₎+15𝜑⁴,=𝜕𝜕𝑧8𝑘⁵1−𝜑²2−15𝜑²+15𝜑⁴=16𝑘⁶𝜑1−𝜑²60𝜑²−45𝜑⁴−17.(2.2) Substituting the above equations into (2.1), we have −𝑐𝑘−𝑎=360𝑘⁴−720𝑘⁶𝜑⁵+960𝑘⁶−480𝑘⁴𝜑³+136𝑘⁴−272𝑘⁶𝜑.(2.3) Then comparing the order of 𝜑, we obtain 360𝑘₄−𝑐𝑘=𝑎,960𝑘₆−720𝑘⁶=0,136𝑘₄−480𝑘⁴=0,−272𝑘⁶=0.(2.4) A simple calculation shows that 𝑘=1/√2, 𝑐=−√2𝑎. Hence, we get 𝜑′=1√21−𝜑²,(2.5) that is, 12ln11−+𝜑𝜑=√12𝑧,(2.6) that is, 𝜑(𝑧)=𝑒^√(1/2)𝑧−𝑒^{√−(1/2)𝑧}𝑒^√(1/2)𝑧+𝑒^{√−(1/2)𝑧}=tanh1√2𝑧.(2.7) We easily proved that lim_𝑧→+∞𝜑(𝑧)=1,lim_𝑧→+∞𝜑(𝑧)=−1(2.8) and 𝜑(𝑧) satisfies the conditions of the Theorem 1.1.

3. Proof of The Result

To prove the Theorem 1.1, we first consider an evolution equation 𝜕𝑢𝜕𝑡=𝐿𝑢+𝐹(𝑢),(3.1) where 𝐿 is a linear operator that generates a strongly continuous semigroup 𝑒^𝑡𝐿 on a Banach space 𝑋, and 𝐹 is a strongly continuous operator such that 𝐹(0)=0. In [13], authors considered the whole problem only on space X, that is, the nonlinear operator maps 𝑋 to 𝑋. However, many equations posses nonlinear terms that include derivatives and therefore, 𝐹 maps into a large Banach space 𝑍. Hence, they again got the following lemma.

Lemma 3.1 (see [14]). Assume the following.(i)X, Z are two Banach spaces with 𝑋⊂𝑍 and ‖𝑢‖_𝑍≤𝑐₁‖𝑢‖_𝑋 for 𝑢∈𝑋.(ii)L generates a strongly continuous semigroup 𝑒^𝑡𝐿 on the space Z, and the semigroup 𝑒^𝑡𝐿 maps Z into X for 𝑡>0 and ∫¹⁰‖𝑒^𝑡𝐿‖_𝑍→𝑋𝑑𝑡=𝐶₄<∞.(iii)The spectrum of 𝐿 on 𝑋 meets the right half-plane, {𝑅𝑒𝜆>0}.(iv)𝐹∶𝑋→𝑍 is continuous and ∃  𝜌₀>0,𝐶₃>0,𝛼>1 such that ‖𝐹(𝑢)‖_𝑍<𝐶₀‖𝑢‖^𝛼𝑋, for ‖𝑢‖_𝑋<𝜌₀.
Then the zero solution of (3.1) is nonlinearly unstable in the space 𝑋.

In this paper, we are going to use Lemma 3.1 for the proof of Theorem 1.1.

Definition 3.2. A traveling wave solution 𝜑(𝑥−𝑐𝑡) of (1.1) is said to be nonlinearly unstable in the space X, if there exist positive 𝜀₀ and 𝐶₀, a sequence {𝑢_𝑛} of solutions of (1.1) and a sequence of time 𝑡_𝑛>0 such that ‖𝑢_𝑛(0)−𝜑(𝑥)‖_𝑋→0 but ‖𝑢_𝑛(𝑡_𝑛)−𝜑(⋅−𝑐𝑡_𝑛)‖_𝑋≥𝜀₀.

If 𝜑(𝑥−𝑐𝑡)∈𝐻⁴(𝑅) is a traveling wave solution of (1.1), then letting 𝜔(𝑥,𝑡)=𝑢(𝑥,𝑡)−𝜑(𝑥−𝑐𝑡), we have (𝜔+𝜑)_𝑡==𝜕𝜕^6𝑥_6𝑥(𝜔+𝜑)+𝜕^4𝑥(𝜔+𝜑)−(𝜔+𝜑)³+𝑎1−(𝜔+𝜑)²+𝜔𝑎+1𝜑−𝜔⁽⁶⁾+₂𝜕^4𝑥𝜔+𝜑−𝜔³−𝜑³−3𝜔²𝜑−3𝜔𝜑²−𝜑²−2𝜔𝜑,(3.2) that is, 𝜔_𝑡=𝜕^6𝑥𝜔+𝜕^4𝑥𝜔−𝜔³−3𝜔²𝜑−3𝜔𝜑²+𝑎−𝜔²−2𝜔𝜑,(3.3) that is, 𝜔_𝑡−+𝜕^6𝑥𝜔24−𝜑𝜑1−_3𝜑²𝜕^4𝑥𝜔+24𝜑𝜑′𝜕^3𝑥𝜔+36𝜑𝜑^+36𝜑^2𝜕^2𝑥𝜔=𝐹(𝜔),+72𝜑^𝜑^𝜕_𝑥𝜔+18𝜑^2+24𝜑^𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑𝜔(3.4) where 𝐹(𝜔)=+33𝜑6⁽⁴⁾𝜕−_𝑥𝑎𝜔²+24𝜑^𝜔𝜕_𝑥𝜔+36𝜑^(𝜕_𝑥𝜔)²+72𝜑′𝜕_𝑥𝜔𝜕^2𝑥𝜔+(18𝜑+𝜔1²8𝜕𝜔^2𝑥𝜔)+𝜕_2𝑥(24𝜑+24𝜔)𝜕_𝑥𝜔𝜕^3𝑥𝜔+36𝜑^𝜔𝜕^2𝑥𝜔𝜔²+24𝜑′𝜔𝜕^3𝑥𝜔+6𝜑𝜔𝜕^4𝑥𝜔+3𝜔²𝜕^4𝑥𝜔,(3.5) with the initial value 𝜔(𝑥,0)=𝜔₀(𝑥)≡𝑢₀(𝑥)−𝜑(𝑥).(3.6) So the stability of traveling wave solutions of (1.1) is translated into the stability of the zero solution of (3.4). In order to prove Theorem 1.1, taking 𝑍=𝐿²(𝑅), 𝑋=𝐻⁴(𝑅), we need to prove that the four conditions of Lemma 3.1 are satisfied by the associated equation (3.4). The condition (i) is satisfied by our choice of 𝑍 and 𝑋.

Denote the linear partial differential operator in (3.4) by 𝐿=(𝜕^6𝑥+𝜕^4𝑥)−[3𝜑²𝜕^4𝑥+24𝜑𝜑′𝜕^3𝑥+(36𝜑𝜑^+36𝜑′²)𝜕^2𝑥+(24𝜑𝜑^+72𝜑′𝜑^)𝜕_𝑥+(18𝜑^2+24𝜑′𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑)] =𝐿₀−[3𝜑²𝜕^4𝑥+24𝜑𝜑′𝜕^3𝑥+(36𝜑𝜑^+36𝜑′²)𝜕^2𝑥+(24𝜑𝜑^+72𝜑′𝜑^)𝜕_𝑥+(18𝜑^2+24𝜑′𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑)] with 𝐿₀=𝜕^6𝑥+𝜕^4𝑥. Then (3.4) may be rewritten in the form of (3.1) 𝜔_𝑡=𝐿𝜔+𝐹(𝜔).(3.7) Note the F maps 𝐻⁴(𝑅) into 𝐿²(𝑅), using the Sobolev embedding theorem, we have ‖𝐹(𝜔)‖_𝐿²≤𝐶‖𝜔‖^2𝐻₄,𝐶>0,for‖𝜔‖_𝐻⁴<1.(3.8) So, the condition (iv) is satisfied.

To prove condition (ii) in Lemma 3.1, we need the following two lemmas.

Lemma 3.3. Let 𝐿₀=𝜕^6𝑥+𝜕^4𝑥. Then ‖‖𝑒‖‖𝑒^𝑡𝐿0‖‖_{𝐻𝑚→𝐻𝑚}≤𝑒^(4/27)𝑡,for𝑚∈𝑅⁺,0≤𝑡<∞,(3.9)_𝑡𝐿0‖‖_{𝐿2→𝐻4}≤𝑎(𝑡)≡5𝑡^−2/3,for0<𝑡≤1.(3.10)

Proof. We write 𝑢(𝑥,𝑡)=𝑒^𝑡𝐿0𝑢₀(𝑥). By Fourier transformation ‖𝑢‖_2𝐻𝑚𝑢̂(𝜉,𝑡)=𝑒^{−𝑡(𝜉6−𝜉4)}𝑢₀̂(𝜉),≡=^∞−∞1+𝜉²^𝑚||𝑢̂(𝜉,𝑡)||²𝑑𝜉_∞−∞≤sup_𝜉∈𝑅1+𝜉²^𝑚𝑒^{−2𝑡(𝜉6−𝜉4)}||𝑢₀̂(𝜉)||²𝑑𝜉𝑒^{−2𝑡(𝜉6−𝜉4)}^∞−∞1+𝜉²^𝑚=𝑒_(8/27)𝑡||𝑢₀̂(𝜉)||²𝑑𝜉‖‖𝑢₀‖‖^2𝐻_𝑚.(3.11) Hence, ‖‖𝑒^𝑡𝐿0‖‖_{𝐻𝑚→𝐻𝑚}≤𝑒^(4/27)𝑡.(3.12) On the other hand, letting 𝑠=𝜉², we have ‖𝑢‖^2𝐻₄≤sup_𝑠∈𝑅𝑓(𝑠)∫^∞−∞||𝑢₀̂(𝜉)||²𝑑𝜉,(3.13) with 𝑓(𝑠)=(1+𝑠)⁴𝑒^{−2𝑡(𝑠3−𝑠2)}, 𝑡>0. Elementary computation shows that sup_𝑠>0𝑓(𝑠)≤43+61𝑡^−4/3𝑒^(8/27)𝑡.(3.14) Thus, ‖𝑢(𝑥,𝑡)‖_𝐻⁴≤43+16𝑡^−4/3^1/2‖‖𝑒_𝑡𝐿0𝑒^(4/27)𝑡‖‖𝑢₀‖‖_𝐿2,‖‖_{𝐿2→𝐻4}≤43+16𝑡^−4/3^1/2𝑒^(4/27)𝑡≤5𝑡^−2/3,for0<𝑡≤1,(3.15) since 𝑒^(4/27)𝑡≤𝑒^4/27<2. Thus, Lemma 3.3 has been proved.

Lemma 3.4. Let 𝐿=(𝜕^6𝑥+𝜕^4𝑥)−[3𝜑²𝜕^4𝑥+24𝜑𝜑^𝜕3𝑥+(36𝜑𝜑^+36𝜑^2)𝜕^2𝑥+(24𝜑𝜑^+72𝜑^{𝜑′′})𝜕_𝑥+(18𝜑^2+24𝜑′𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑)]=𝐿₀−[3𝜑²𝜕^4𝑥+24𝜑𝜑′𝜕^3𝑥+(36𝜑𝜑^+36𝜑′²)𝜕^2𝑥+(24𝜑𝜑^+72𝜑′𝜑^)𝜕_𝑥 + (18𝜑^2+24𝜑′𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑)] with 𝐿₀=𝜕^6𝑥+𝜕^4𝑥, 𝜑^(𝑖)∈𝐿^∞(𝑅),𝑖=0,1,2,3,4. Then ‖‖‖‖ee^tL‖‖_L2→H4≤C₁t^−2/3,for0<t≤1,(3.16)_tL‖‖_H4→H4≤C₂<∞,for0<t≤1.(3.17)

Proof. Consider the initial value problem 𝑢_𝑡=𝐿−𝑢=24𝐿𝜑₀𝜑𝑢−_3𝜑²𝜕^4𝑥𝑢−24𝜑𝜑′𝜕^3𝑥𝑢−36𝜑𝜑^+36𝜑^2𝜕^2𝑥𝑢+72𝜑^𝜑^𝑢𝜕_𝑥(𝑥𝑢,−0)=18𝑢𝜑^2₀+24𝜑^𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑𝑢,(𝑥).(3.18) Then 𝑢(𝑥,𝑡)=𝑒^𝑡𝐿𝑢₀(𝑥), 𝑡≥0, 𝑥∈𝑅, thus 𝑢(𝑥,𝑡)=𝑒^𝑡𝐿0𝑢₀−^𝑡0𝑒^{(𝑡−𝜏)𝐿0}3𝜑+²𝜕2^4𝑥𝑢4𝜑+𝜑24_𝜑𝜑′𝜕^3𝑥𝑢+36𝜑𝜑^+36𝜑^2𝜕^2𝑥𝑢+18𝜑_2+72𝜑^𝜑^𝜕_𝑥𝑢+24𝜑^𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑𝑢𝑑𝜏.(3.19) Denote 𝐴=‖𝜑‖_𝐿^∞, 𝐵=‖𝜑′‖_𝐿^∞, 𝐶=‖𝜑^‖_𝐿^∞, 𝐷=‖𝜑^‖_𝐿^∞, 𝐸=‖𝜑⁽⁴⁾‖_𝐿^∞ and 𝑀=3𝐴²+24𝐴𝐵+36𝐴𝐶+36𝐵²+24𝐴𝐷+72𝐵𝐶+18𝐶²+24𝐵𝐷+6𝐴𝐸+2𝑎𝐴.(3.20) Then, we have ‖𝑢(𝑡)‖_𝐻⁴≤‖‖+𝑒24^𝑡𝐿0‖‖_{𝐻4→𝐻4𝑡0}‖‖𝑢₀‖‖_𝐻4+^𝑡0‖‖𝑒^{(𝑡−𝜏)𝐿0}‖‖_{𝐻4→𝐻4}3‖𝜑‖^2𝐿_∞‖‖𝜕^4𝑥𝑢‖‖_𝐿2𝑑𝜏+‖‖𝑒^{(𝑡−𝜏)𝐿0}‖‖_{𝐻4→𝐻4}‖𝜑‖_𝐿^∞‖𝜑′‖_𝐿^∞‖‖𝜕^3𝑥𝑢‖‖_𝐿2𝑑𝜏_𝑡0‖‖𝑒^{(𝑡−𝜏)𝐿0}‖‖_{𝐻4→𝐻4}36‖𝜑‖^𝐿∞‖‖𝜑^‖‖_𝐿∞+36‖‖𝜑^2‖‖_𝐿∞+‖‖𝜕^2𝑥𝑢‖‖_𝐿2𝑑𝜏_𝑡0+‖‖𝑒^{(𝑡−𝜏)𝐿0}‖‖_{𝐻4→𝐻4}24‖𝜑‖_𝐿^∞‖‖𝜑^‖‖_𝐿∞+72‖𝜑′‖_𝐿^∞‖‖𝜑^‖‖_𝐿∞‖‖𝜕_𝑥𝑢‖‖_𝐿2𝑑𝜏_𝑡0‖‖𝑒^{(𝑡−𝜏)𝐿0}‖‖_{𝐻4→𝐻4}18‖‖𝜑^‖‖^2𝐿_∞+24‖𝜑′‖_𝐿^∞‖‖𝜑^‖‖_𝐿∞≤𝑒+_(4/27)𝑡6‖𝜑‖_𝐿^∞‖‖𝜑⁽⁴⁾‖‖_𝐿∞+2𝑎‖𝜑‖_𝐿^∞‖𝑢‖^𝐿2𝑑𝜏‖‖𝑢₀‖‖_𝐻4+𝑀^𝑡0𝑒^{(4/27)(𝑡−𝜏)}‖𝑢(𝜏)‖_𝐻⁴𝑑𝜏,(3.21) where we use 𝑢(𝑡) to denote 𝑢(⋅,𝑡).
By iteration, ‖𝑢(𝑡)‖_𝐻⁴≤𝑒^(4/27)𝑡‖‖𝑢₀‖‖_𝐻4+𝑀^𝑡0𝑒^{(4/27)(𝑡−𝜏)}𝑒^(4/27)𝜏=𝑒_(4/27)𝑡‖‖𝑢₀‖‖_𝐻4+𝑀^𝜏0𝑒^{(4/27)(𝜏−𝑠)}‖𝑢(𝑠)‖_𝐻⁴𝑑𝑠𝑑𝜏‖‖𝑢₀‖‖_𝐻4+𝑀^𝑡0𝑒^(4/27)𝑡‖‖𝑢₀‖‖_𝐻4𝑑𝜏+𝑀²^𝑡0^𝜏0≤𝑒_(4/27)𝑡𝑒^{(4/27)(𝑡−𝑠)}‖𝑢(𝑠)‖_𝐻⁴𝑑𝑠𝑑𝜏‖‖𝑢₀‖‖_𝐻4+𝑀𝑡𝑒^(4/27)𝑡‖‖𝑢₀‖‖_𝐻4+𝑀²^𝑡0^𝑡𝑠≤𝑒_(4/27)𝑡𝑒^{(4/27)(𝑡−𝑠)}‖𝑢(𝑠)‖_𝐻⁴𝑑𝜏𝑑𝑠‖‖𝑢₀‖‖_𝐻4+𝑀𝑡𝑒^(4/27)𝑡‖‖𝑢₀‖‖_𝐻4+𝑀²𝑡𝑒^(4/27)𝑡^𝑡0≤𝑒_4/27‖𝑢(𝑠)‖_𝐻⁴𝑑𝑠‖‖𝑢₀‖‖_𝐻4+𝑀𝑒^4/27‖‖𝑢₀‖‖_𝐻4+𝑒^4/27𝑀²^𝑡0‖𝑢(𝑠)‖_𝐻⁴𝑑𝑠,for0<𝑠≤𝜏≤𝑡≤1.(3.22) Let 𝑣(𝑡)=∫^𝑡0‖𝑢(𝑠)‖_𝐻⁴𝑑𝑠. Then 𝑑𝑣(𝑡)𝑑𝑡≤𝑒^4/27+𝑒^4/27𝑀‖‖𝑢₀‖‖_𝐻4+𝑒^4/27𝑀²𝑣(𝑡),for0<𝑡≤1.(3.23) Multiplying both sides of the above inequality by 𝑒^{−𝑒(4/27)𝑀2𝑡}, we have 𝑑𝑒^{−𝑒4/27𝑀2𝑡}𝑣(𝑡)𝑑𝑡≤𝑒^{−𝑒4/27𝑀2𝑡}𝑒^4/27(1+𝑀)‖‖𝑢₀‖‖_𝐻4,where0<𝑡≤1.(3.24) Integrating the above inequality with respect to 𝑡 over (0,𝑡), we obtain 𝑒^{−𝑒4/27𝑀2𝑡}𝑣(𝑡)≤^𝑡0𝑒^{−𝑒4/27𝑀2𝑠}𝑒^4/27(1+𝑀)‖‖𝑢₀‖‖_𝐻4𝑑𝑠,(3.25) that is, 𝑣(𝑡)≤𝑒^{𝑒4/27𝑀2𝑡}^𝑡0𝑒^{−𝑒4/27𝑀2𝑠}𝑒^4/27(1+𝑀)‖‖𝑢₀‖‖_𝐻4𝑑𝑠.(3.26) Observing that 𝑣(𝑡)=∫^𝑡0‖𝑢(𝑠)‖_𝐻⁴𝑑𝑠 is bounded and substituting the above inequality into (3.22), we get ‖𝑢(𝑡)‖_𝐻⁴≤𝑒^4/27‖‖𝑢₀‖‖_𝐻4+𝑀𝑒^4/27‖‖𝑢₀‖‖_𝐻4+𝑒^4/27𝑀²^𝑡0≤𝑐₂‖𝑢(𝑠)‖_𝐻⁴𝑑𝑠<∞,for0<𝑡≤1,𝑐₂>0,(3.27) thus (3.17) has been proven.
Next, we prove the inequality (3.16). Clearly, we have ‖𝑢(𝑡)‖_𝐻⁴≤‖‖+𝑒^𝑡𝐿0‖‖_{𝐿2→𝐻4}‖‖𝑢₀‖‖_𝐿2+^𝑡0‖‖𝑒^{(𝑡−𝜏)𝐿0}‖‖_{𝐿2→𝐻4}3‖𝜑‖^2𝐿_∞‖‖𝜕^4𝑥𝑢‖‖_𝐿2𝑑𝜏_𝑡0+‖‖𝑒^{(𝑡−𝜏)𝐿0}‖‖_{𝐿2→𝐻4}24‖𝜑‖_𝐿^∞‖𝜑′‖_𝐿^∞‖‖𝜕^3𝑥𝑢‖‖_𝐿2𝑑𝜏_𝑡0‖‖𝑒^{(𝑡−𝜏)𝐿0}‖‖_{𝐿2→𝐻4}36‖𝜑‖^𝐿∞‖‖𝜑^‖‖_𝐿∞+36‖‖𝜑^2‖‖_𝐿∞+‖‖𝜕^2𝑥𝑢‖‖_𝐿2𝑑𝜏_𝑡0‖‖𝑒^{(𝑡−𝜏)𝐿0}‖‖_{𝐿2→𝐻4}24‖𝜑‖_𝐿^∞‖‖𝜑^‖‖_𝐿∞++72‖𝜑′‖_𝐿^∞‖‖𝜑^‖‖_𝐿∞‖‖𝜕_𝑥𝑢‖‖_𝐿2𝑑𝜏_𝑡0‖‖𝑒^{(𝑡−𝜏)𝐿0}‖‖_{𝐿2→𝐻4}18‖‖𝜑^‖‖^2𝐿_∞+24‖𝜑′‖_𝐿^∞‖‖𝜑^‖‖_𝐿∞≤𝑎(𝑡)‖‖+𝑢6₀‖𝜑‖_𝐿^∞‖‖𝜑⁽⁴⁾‖‖_𝐿∞+2𝑎‖𝜑‖_𝐿^∞‖𝑢‖^𝐿2𝑑𝜏‖‖_𝐿2+𝑀^𝑡0𝑎(𝑡−𝜏)‖𝑢(𝜏)‖_𝐻⁴𝑑𝜏,(3.28) where 𝑎(𝑡) is defined in Lemma 3.3, and we use 𝑢(𝑡) to denote 𝑢(⋅,𝑡).
By iteration, ‖𝑢(𝑡)‖_𝐻⁴≤=𝑎𝑎((𝑡𝑡))‖‖‖‖𝑢𝑢₀₀‖‖_𝐿2+𝑀^𝑡0𝑎(𝑡−𝜏)𝑎(𝜏)‖‖𝑢_𝑜‖‖_𝐿2+𝑀^𝜏0𝑎(𝜏−𝑠)‖𝑢(𝑠)‖_𝐻⁴𝑑𝑠𝑑𝜏+𝑀₂‖‖_𝐿2+𝑀^𝑡0𝑎(𝑡−𝜏)𝑎(𝜏)‖‖𝑢₀‖‖_𝐿2𝑑𝜏^𝑡0^𝜏0𝑎(𝑡−𝜏)𝑎(𝜏−𝑠)‖𝑢(𝑠)‖_𝐻⁴𝑑𝑠𝑑𝜏.(3.29) The second term on the right of (3.29) is 𝑀^𝑡0𝑎(𝑡−𝜏)𝑎(𝜏)‖‖𝑢₀‖‖_𝐿2𝑑𝜏=𝑀‖‖𝑢₀‖‖_𝐿2^𝑡0=25𝑀‖‖𝑢₀5(𝑡−𝜏)^−2/35𝜏^−2/3𝑑𝜏‖‖_𝐿2^𝑡0𝑡^−4/31−𝜏𝑡^−2/3𝜏𝑡^−2/3=25𝑀𝐶₃𝑑𝜏𝑡^−1/3‖‖𝑢₀‖‖_𝐿2,0<𝑡<1,(3.30) where 𝐶₃=∫¹⁰(1−𝑟)^−1/4𝑟^−1/4𝑑𝑟. By exchanging the order of integration, we get from the third term on the right side of (3.29), ^𝑡0^𝜏0𝑎(𝑡−𝜏)𝑎(𝜏−𝑠)‖𝑢(𝑠)‖_𝐻⁴𝑑𝑠𝑑𝜏=^𝑡0^𝑡𝑠𝑎(𝑡−𝜏)𝑎(𝜏−𝑠)𝑑𝜏‖𝑢(𝑠)‖_𝐻⁴𝑑𝑠,(3.31) then ^𝑡𝑠𝑎(𝑡−𝜏)𝑎(𝜏−𝑠)𝑑𝜏==2255𝐶^𝑡𝑠₃(𝑡−𝜏)^−2/3(𝜏−𝑠)^−2/3𝑑𝜏(𝑡−𝑠)^−1/3,0<𝑠≤𝑡≤1.(3.32) Therefore (3.28)–(3.32) imply ‖𝑢(𝑡)‖_𝐻⁴≤𝑎(𝑡)+25𝐶₃𝑀𝑡^−1/3‖‖𝑢₀‖‖_𝐿2+25𝐶₃𝑀²^𝑡0(𝑡−𝑠)^−1/3‖𝑢(𝑠)‖_𝐻⁴𝑑𝑠,0<𝑡≤1.(3.33) From (3.17), we know ‖𝑢(𝑡)‖_𝐻⁴≤𝐶₂‖𝑢₀‖_𝐻⁴,0<𝑡≤1. Then ‖𝑢(𝑡)‖_𝐻⁴≤𝑎(𝑡)+25𝐶₃𝑀𝑡^−1/3‖‖𝑢₀‖‖_𝐿2+752𝐶³𝐶₂𝑀²‖‖𝑢₀‖‖_𝐻4𝑡^2/3,0<𝑡≤1.(3.34) Therefore, there exists a 𝑡^∗, such that ‖𝑢(𝑡)‖_𝐻⁴≤𝐶₁𝑡^−2/3‖‖𝑢₀‖‖_𝐿2,0<𝑡≤𝑡^∗≤1,𝐶₁>0.(3.35) So, we proved the inequality (3.16).
Hence (3.17) is proven and proof of Lemma 3.4 is finished.

By Lemma 3.4, the condition (ii) is proved.

We now proceed to verify condition (iii) of Lemma 3.1. Observing that if 𝑢(𝑥,𝑡) satisfies 𝜕𝑢(𝑥,𝑡)𝜕𝑡=𝜕⁶𝑢𝜕𝑥⁶+𝜕⁴𝑢𝜕𝑥⁴−3𝜑²𝜕⁴𝑢𝜕𝑥⁴−24𝜑𝜑′𝜕³𝑢𝜕𝑥³−36𝜑𝜑^+36𝜑^2𝜕²𝑢𝜕𝑥²−24𝜑𝜑^+72𝜑^𝜑^𝜕𝑢𝜕𝑥−18𝜑^2+24𝜑^𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑𝑢,(3.36) then 𝑢(𝑥,𝑠+𝑡) also satisfies the above equation. By uniqueness of solution, we know that 𝐿 generates a strongly continuous semigroup on the Banach space 𝐻⁴(𝑅)(see [15] p.344). By Fourier transformation, the essential spectrum of 𝐿₀ on 𝐻⁴(𝑅) is 𝜎𝐿₀⊃−𝜉⁶+𝜉⁴∣𝜉∈𝑅.(3.37) The curve 𝜆=−𝜉⁶+𝜉⁴ meets the vertical lines 𝑅𝑒𝜆=𝛼 for −∞<𝛼≤4/27 because −∞<−𝜉⁶+𝜉⁴≤4/27.

We now prove that the same curve belongs to the essential spectrum of 𝐿.

Lemma 3.5. The essential spectrum of 𝐿 on 𝐻⁴(𝑅) contains that of 𝐿₀.

Proof. Let 𝜉∈𝑅 and let 𝜆=𝑃(𝜉)=−𝜉⁶+𝜉⁴. Following Schechter [16], 𝜆∈𝜎(𝐿) if there exists a sequence {𝜉_𝑛}⊂𝐻⁴(𝑅) with ‖‖𝜉_𝑛‖‖_𝐻4=1,‖‖(𝐿−𝜆)𝜉_𝑛‖‖_𝐻4⟶0,(3.38) and {𝜉_𝑛} does not have a strongly convergent subsequence in 𝐻⁴(𝑅). Here we use the definition 𝜆∉𝜎(𝐿) if and only if 𝐿−𝜆 is Fredholm with index zero. Now let 𝜉₀≢0 be a 𝐶^∞ function with compact support in (0,∞). Define 𝜉_𝑛(𝑥)=𝑐^𝑛𝑒^𝑖𝜉𝑥𝜉₀(𝑥/𝑛)√𝑛,𝑛=1,2,…,(3.39) where 𝑐_𝑛 is chosen so that ‖𝜉_𝑛‖_𝐻⁴=1. In fact, ‖‖𝜉_𝑛‖‖_𝐿2=𝑐_𝑛‖‖𝜉₀‖‖_𝐿2,1=‖‖𝜉_𝑛‖‖_𝐻4≤𝑘𝑐_𝑛,(3.40) for some positive constant 𝑘. Hence 𝑐_𝑛≥1/𝑘>0. Since ‖𝜉_𝑛‖_𝐿^∞→0 but ‖𝜉_𝑛‖_𝐿² is bounded away from zero, {𝜉_𝑛} can have no convergent subsequence in 𝐿²(𝑅).
It remains to show that ‖(𝐿−𝜆)𝜉_𝑛‖_𝐻⁴→0. We write 𝐿−𝜆=𝐿₀−𝜆−3𝜑²𝜕^4𝑥+24𝜑𝜑′𝜕^3𝑥+36𝜑𝜑^+36𝜑^2𝜕^2𝑥+24𝜑𝜑^+72𝜑^𝜑^𝜕_𝑥+18𝜑^2+24𝜑^𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑.(3.41) A simple calculation shows that 𝐿₀−𝜆𝜉_𝑛(𝑥)=𝑐_𝑛𝑒^𝑖𝜉𝑥_1≤𝑠≤6(−𝑖)^𝑠𝑃^(𝑠)(𝜉)𝜉^0(𝑠)(𝑥/𝑛)𝑠!𝑛^(1/2)+𝑠𝜕𝐿₀,−𝜆𝜉_𝑛(𝑥)=𝑖𝜉𝐿₀−𝜆𝜉_𝑛(𝑥)+𝑐_𝑛𝑒^𝑖𝜉𝑥₁_≤𝑠≤6(−𝑖)^𝑠𝑃^(𝑠)(𝜉)𝜉^0(𝑠+1)(𝑥/𝑛)𝑠!𝑛^(3/2)+𝑠𝜕₂,𝐿₀−𝜆𝜉_𝑛(𝑥)=−𝜉²𝐿₀−𝜆𝜉_𝑛(𝑥)+2𝑖𝜉𝑐_𝑛𝑒^𝑖𝜉𝑥_1≤𝑠≤6(−𝑖)^𝑠𝑃^(𝑠)(𝜉)𝜉^0(𝑠+1)(𝑥/𝑛)𝑠!𝑛^(3/2)+𝑠+𝑐_𝑛𝑒^𝑖𝜉𝑥₁_≤𝑠≤6(−𝑖)^𝑠𝑃^(𝑠)(𝜉)𝜉^0(𝑠+2)(𝑥/𝑛)𝑠!𝑛^(5/2)+𝑠𝜕₃,𝐿₀−𝜆𝜉_𝑛(𝑥)=−𝑖𝜉³𝐿₀−𝜆𝜉_𝑛(𝑥)−3𝜉²𝑐_𝑛𝑒^𝑖𝜉𝑥_1≤𝑠≤6(−𝑖)^𝑠𝑃^(𝑠)(𝜉)𝜉^0(𝑠+1)(𝑥/𝑛)𝑠!𝑛^(3/2)+𝑠+3𝑖𝜉𝑐_𝑛𝑒^𝑖𝜉𝑥₁_≤𝑠≤6(−𝑖)^𝑠𝑃^(𝑠)(𝜉)𝜉^0(𝑠+2)(𝑥/𝑛)𝑠!𝑛^(5/2)+𝑠+𝑐_𝑛𝑒^𝑖𝜉𝑥₁_≤𝑠≤6(−𝑖)^𝑠𝑃^(𝑠)(𝜉)𝜉^0(𝑠+3)(𝑥/𝑛)𝑠!𝑛^(7/2)+𝑠𝜕₄,𝐿₀−𝜆𝜉_𝑛(𝑥)=𝜉⁴𝐿₀−𝜆𝜉_𝑛(𝑥)−4𝑖𝜉³𝑐_𝑛𝑒^𝑖𝜉𝑥_1≤𝑠≤6(−𝑖)^𝑠𝑃^(𝑠)(𝜉)𝜉^0(𝑠+1)(𝑥/𝑛)𝑠!𝑛^(3/2)+𝑠−6𝜉²𝑐_𝑛𝑒^𝑖𝜉𝑥₁_≤𝑠≤6(−𝑖)^𝑠𝑃^(𝑠)(𝜉)𝜉^0(𝑠+2)(𝑥/𝑛)𝑠!𝑛^(5/2)+𝑠+4𝑖𝜉𝑐_𝑛𝑒^𝑖𝜉𝑥₁_≤𝑠≤6(−𝑖)^𝑠𝑃^(𝑠)(𝜉)𝜉^0(𝑠+3)(𝑥/𝑛)𝑠!𝑛^(7/2)+𝑠+𝑐_𝑛𝑒^𝑖𝜉𝑥₁_≤𝑠≤6(−𝑖)^𝑠𝑃^(𝑠)(𝜉)𝜉^0(𝑠+4)(𝑥/𝑛)𝑠!𝑛^(9/2)+𝑠.(3.42) Thus, ‖‖𝐿₀−𝜆𝜉_𝑛(𝑥)‖‖_𝐻4≤1+||𝜉||+||𝜉||²+||𝜉||³+||𝜉||⁴_1≤𝑠≤6||𝑃^(𝑠)(𝜉)||𝑐_𝑛‖‖𝜉^0(𝑠)(𝑥/𝑛)‖‖_𝐿2𝑠!𝑛^(1/2)+𝑠+1+2||𝜉||+3||𝜉||²+4||𝜉||³_1≤𝑠≤6||𝑃^(𝑠)(𝜉)||𝑐_𝑛‖‖𝜉^0(𝑠+1)(𝑥/𝑛)‖‖_𝐿2𝑠!𝑛^(3/2)+𝑠+1+3||𝜉||+6||𝜉||²_1≤𝑠≤6||𝑃^(𝑠)(𝜉)||𝑐_𝑛‖‖𝜉^0(𝑠+2)(𝑥/𝑛)‖‖_𝐿2𝑠!𝑛^(5/2)+𝑠+1+4||𝜉||_1≤𝑠≤6||𝑃^(𝑠)(𝜉)||𝑐_𝑛‖‖𝜉^0(𝑠+3)(𝑥/𝑛)‖‖_𝐿2𝑠!𝑛^(7/2)+𝑠+_1≤𝑠≤6||𝑃^(𝑠)(𝜉)||𝑐_𝑛‖‖𝜉^0(𝑠+4)(𝑥/𝑛)‖‖_𝐿2𝑠!𝑛^(9/2)+𝑠⟶0,as𝑛⟶∞.(3.43) Moreover, for any positive integer 𝑚, ‖𝜕^𝑚𝑥𝜉_𝑛‖_𝐿^∞→0, as 𝑛→∞, we have ‖‖𝜕‖‖3𝜑²𝜕^4𝑥𝜉_𝑛‖‖^2𝐿₂≤‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖3𝜑²‖‖^2𝐿₂⟶0,_𝑥‖‖𝜕3𝜑²𝜕^4𝑥𝜉_𝑛‖‖^2𝐿2≤‖‖𝜕^5𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖3𝜑²‖‖^2𝐿₂+‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖6𝜑𝜑′‖^2𝐿₂⟶0,_2𝑥‖‖𝜕3𝜑²𝜕^4𝑥𝜉_𝑛‖‖^2𝐿2≤‖‖𝜕^6𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖3𝜑²‖‖^2𝐿₂+2‖‖𝜕^5𝑥𝜉_𝑛‖‖^2𝐿_∞‖6𝜑𝜑′‖^2𝐿₂+‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖6𝜑′²+6𝜑𝜑^‖‖^2𝐿₂⟶0,_3𝑥3𝜑²𝜕^4𝑥𝜉_𝑛‖‖^2𝐿2≤‖‖𝜕^7𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖3𝜑²‖‖^2𝐿₂+3‖‖𝜕^6𝑥𝜉_𝑛‖‖^2𝐿_∞‖6𝜑𝜑′‖^2𝐿₂‖‖𝜕+3‖‖𝜕^5𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖6𝜑′²+6𝜑𝜑^‖‖^2𝐿₂+‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖18𝜑′𝜑^+6𝜑𝜑^‖‖^2𝐿₂⟶0,_4𝑥3𝜑²𝜕^4𝑥𝜉_𝑛‖‖^2𝐿2≤‖‖𝜕^8𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖3𝜑²‖‖^2𝐿₂+4‖‖𝜕^5𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖18𝜑′𝜑^+6𝜑𝜑^‖‖^2𝐿₂+4‖‖𝜕^7𝑥𝜉_𝑛‖‖^2𝐿_∞‖6𝜑𝜑′‖^2𝐿₂+6‖‖𝜕^6𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖6𝜑′²+6𝜑𝜑^‖‖^2𝐿₂+‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖18𝜑^2+24𝜑′𝜑^+6𝜑𝜑⁽⁴⁾‖‖^2𝐿₂⟶0.(3.44) From the assumptions on 𝜑, we obtain ‖‖𝜕‖‖24𝜑𝜑′𝜕^3𝑥𝜉_𝑛‖‖^2𝐿₂≤‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖24𝜑𝜑′‖^2𝐿₂⟶0,_𝑥‖‖𝜕24𝜑𝜑^𝜕^3𝑥𝜉_𝑛‖‖^2𝐿2≤‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖24𝜑𝜑′‖^2𝐿₂+‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖24𝜑′²+24𝜑𝜑^‖‖^2𝐿₂⟶0,_2𝑥24𝜑𝜑^𝜕^3𝑥𝜉_𝑛‖‖^2𝐿2≤‖‖𝜕^5𝑥𝜉_𝑛‖‖^2𝐿_∞‖24𝜑𝜑′‖^2𝐿₂+2‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖24𝜑′²+24𝜑𝜑^‖‖^2𝐿₂‖‖𝜕+‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖72𝜑′𝜑^+24𝜑𝜑^‖‖^2𝐿₂⟶0,_3𝑥24𝜑𝜑^𝜕^3𝑥𝜉_𝑛‖‖^2𝐿2≤‖‖𝜕^6𝑥𝜉_𝑛‖‖^2𝐿_∞‖24𝜑𝜑′‖^2𝐿₂+3‖‖𝜕^5𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖24𝜑′²+24𝜑𝜑^‖‖^2𝐿₂+3‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖72𝜑′𝜑^+24𝜑𝜑^‖‖^2𝐿₂‖‖𝜕+‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖72𝜑^2+96𝜑′𝜑^+24𝜑𝜑⁽⁴⁾‖‖^2𝐿₂⟶0,_4𝑥24𝜑𝜑^𝜕^3𝑥𝜉_𝑛‖‖^2𝐿2≤‖‖𝜕^7𝑥𝜉_𝑛‖‖^2𝐿_∞‖24𝜑𝜑′‖^2𝐿₂+4‖‖𝜕^6𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖24𝜑′²+24𝜑𝜑^‖‖^2𝐿₂+6‖‖𝜕^5𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖72𝜑′𝜑^+24𝜑𝜑^‖‖^2𝐿₂+4‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖72𝜑^2+96𝜑′𝜑^+24𝜑𝜑⁽⁴⁾‖‖^2𝐿₂+‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖240𝜑^𝜑^+120𝜑′𝜑⁽⁴⁾+24𝜑𝜑⁽⁵⁾‖‖^2𝐿₂⟶0.(3.45) Similarly, we have ‖‖36𝜑𝜑^+36𝜑^2𝜕^2𝑥𝜉_𝑛‖‖𝜕‖‖^2𝐿₂≤‖‖𝜕^2𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖36𝜑𝜑^+36𝜑′²‖‖^2𝐿₂⟶0,_𝑥36𝜑𝜑^+36𝜑^2𝜕^2𝑥𝜉_𝑛‖‖^2𝐿₂≤‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖36𝜑𝜑^+36𝜑′²‖‖^2𝐿₂‖‖𝜕+‖‖𝜕^2𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖108𝜑′𝜑^+36𝜑𝜑^‖‖^2𝐿₂⟶0,_2𝑥36𝜑𝜑^+36𝜑^2𝜕^2𝑥𝜉_𝑛‖‖^2𝐿₂≤‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖36𝜑𝜑^+36𝜑′²‖‖^2𝐿₂+2‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖108𝜑′𝜑^+36𝜑𝜑^‖‖^2𝐿₂‖‖𝜕+‖‖𝜕^2𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖108𝜑^2+144𝜑′𝜑^+36𝜑𝜑⁽⁴⁾‖‖^2𝐿₂⟶0,_3𝑥36𝜑𝜑^+36𝜑^2𝜕^2𝑥𝜉_𝑛‖‖^2𝐿₂≤‖‖𝜕^5𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖36𝜑𝜑^+36𝜑′²‖‖^2𝐿₂+3‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖108𝜑′𝜑^+36𝜑𝜑^‖‖^2𝐿₂+3‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖108𝜑^2+144𝜑′𝜑^+36𝜑𝜑⁽⁴⁾‖‖^2𝐿₂‖‖𝜕+‖‖𝜕^2𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖360𝜑^𝜑^+180𝜑′𝜑⁽⁴⁾+36𝜑𝜑⁽⁵⁾‖‖^2𝐿₂⟶0,_4𝑥36𝜑𝜑^+36𝜑^2𝜕^2𝑥𝜉_𝑛‖‖^2𝐿₂≤‖‖𝜕^6𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖36𝜑𝜑^+36𝜑′²‖‖^2𝐿₂+4‖‖𝜕^5𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖108𝜑′𝜑^+36𝜑𝜑^‖‖^2𝐿₂+6‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖108𝜑^2+144𝜑′𝜑^+36𝜑𝜑⁽⁴⁾‖‖^2𝐿₂+4‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖360𝜑^𝜑^+180𝜑′𝜑⁽⁴⁾+36𝜑𝜑⁽⁵⁾‖‖^2𝐿₂‖‖24𝜑+𝜑‖‖𝜕^2𝑥_𝜉_𝑛‖‖^2𝐿_∞‖‖360𝜑^2+540𝜑^𝜑⁽⁴⁾+216𝜑′𝜑⁽⁵⁾+36𝜑𝜑⁽⁶⁾‖‖²_L2⟶0,‖‖𝜕+72𝜑^𝜑^𝜕_𝑥𝜉_𝑛‖‖^2𝐿₂≤‖‖𝜕_𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖24𝜑𝜑^+72𝜑′𝜑^‖‖^2𝐿₂⟶0,_𝑥24𝜑𝜑^+72𝜑^𝜑^𝜕_𝑥𝜉_𝑛‖‖^2𝐿2≤‖‖𝜕^2𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖24𝜑𝜑^+72𝜑′𝜑^‖‖^2𝐿₂‖‖𝜕+‖‖𝜕_𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖96𝜑′𝜑^+72𝜑^2+24𝜑𝜑⁽⁴⁾‖‖^2𝐿₂⟶0,_2𝑥24𝜑𝜑^+72𝜑^𝜑^𝜕_𝑥𝜉_𝑛‖‖^2𝐿2≤‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖24𝜑𝜑^+72𝜑′𝜑^‖‖^2𝐿₂+2‖‖𝜕^2𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖96𝜑′𝜑^+72𝜑^2+24𝜑𝜑⁽⁴⁾‖‖^2𝐿₂‖‖𝜕+‖‖𝜕_𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖240𝜑^𝜑^+120𝜑′𝜑⁽⁴⁾+24𝜑𝜑⁽⁵⁾‖‖^2𝐿₂⟶0,_3𝑥24𝜑𝜑^+72𝜑^𝜑^𝜕_𝑥𝜉_𝑛‖‖^2𝐿2≤‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖24𝜑𝜑^+72𝜑^𝜑^‖‖^2𝐿₂+3‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖96𝜑′𝜑^+72𝜑^2+24𝜑𝜑⁽⁴⁾‖‖^2𝐿₂+3‖‖𝜕^2𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖240𝜑^𝜑^+120𝜑′𝜑⁽⁴⁾+24𝜑𝜑⁽⁵⁾‖‖^2𝐿₂‖‖𝜕+‖‖𝜕_𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖240𝜑^2+360𝜑^𝜑⁽⁴⁾+144𝜑′𝜑⁽⁵⁾+24𝜑𝜑⁽⁶⁾‖‖^2𝐿₂⟶0,_4𝑥24𝜑𝜑^+72𝜑^𝜑^𝜕_𝑥𝜉_𝑛‖‖^2𝐿2≤‖‖𝜕^5𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖24𝜑𝜑^+72𝜑′𝜑^‖‖^2𝐿₂+4‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖96𝜑′𝜑^+72𝜑^2+24𝜑𝜑⁽⁴⁾‖‖^2𝐿₂+6‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖240𝜑^𝜑^+120𝜑′𝜑⁽⁴⁾+24𝜑𝜑⁽⁵⁾‖‖^2𝐿₂+4‖‖𝜕^2𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖240𝜑^2+360𝜑^𝜑⁽⁴⁾+144𝜑′𝜑⁽⁵⁾+24𝜑𝜑⁽⁶⁾‖‖^2𝐿₂+‖‖𝜕_𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖840𝜑^𝜑⁽⁴⁾+504𝜑^𝜑⁽⁵⁾+168𝜑′𝜑⁽⁶⁾+24𝜑𝜑⁽⁷⁾‖‖^2𝐿₂⟶0.(3.46) In addition, ‖‖18𝜑^2+24𝜑^𝜑‖‖𝜕^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑𝜉_𝑛‖‖^2𝐿₂≤‖‖𝜉_𝑛‖‖^2𝐿_∞‖‖18𝜑^2+24𝜑′𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑‖‖^2𝐿₂⟶0,_𝑥18𝜑^2+24𝜑^𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑𝜉_𝑛‖‖^2𝐿₂≤‖‖𝜕_𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖18𝜑^2+24𝜑′𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑‖‖^2𝐿₂‖‖𝜕+‖‖𝜉_𝑛‖‖^2𝐿_∞‖‖60𝜑^𝜑^+30𝜑′𝜑⁽⁴⁾+6𝜑𝜑⁽⁵⁾+2𝑎𝜑′‖‖^2𝐿₂⟶0,_2𝑥18𝜑^2+24𝜑^𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑𝜉_𝑛‖‖^2𝐿₂≤‖‖𝜕^2𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖18𝜑^2+24𝜑′𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑‖‖^2𝐿₂+2‖‖𝜕_𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖60𝜑^𝜑^+30𝜑′𝜑⁽⁴⁾+6𝜑𝜑⁽⁵⁾+2𝑎𝜑′‖‖^2𝐿₂‖‖𝜕+‖‖𝜉_𝑛‖‖^2𝐿_∞‖‖60𝜑^2+90𝜑^𝜑⁽⁴⁾+36𝜑′𝜑⁽⁵⁾+6𝜑𝜑⁽⁶⁾+2𝑎𝜑^‖‖^2𝐿₂⟶0,_3𝑥18𝜑^2+24𝜑^𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑𝜉_𝑛‖‖^2𝐿₂≤‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖18𝜑^2+24𝜑′𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑‖‖^2𝐿₂+3‖‖𝜕^2𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖60𝜑^𝜑^+30𝜑′𝜑⁽⁴⁾+6𝜑𝜑⁽⁵⁾+2𝑎𝜑′‖‖^2𝐿₂+3‖‖𝜕_𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖60𝜑^2+90𝜑^𝜑⁽⁴⁾+36𝜑′𝜑⁽⁵⁾+6𝜑𝜑⁽⁶⁾+2𝑎𝜑^‖‖^2𝐿₂‖‖𝜕+‖‖𝜉_𝑛‖‖^2𝐿_∞‖‖210𝜑^𝜑⁽⁴⁾+126𝜑^𝜑⁽⁵⁾+42𝜑′𝜑⁽⁶⁾+6𝜑𝜑⁽⁷⁾+2𝑎𝜑^‖‖^2𝐿₂⟶0,_4𝑥18𝜑^2+24𝜑^𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑𝜉_𝑛‖‖^2𝐿₂≤‖‖𝜕^4𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖18𝜑^2+24𝜑′𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑‖‖^2𝐿₂+4‖‖𝜕^3𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖60𝜑^𝜑^+30𝜑′𝜑⁽⁴⁾+6𝜑𝜑⁽⁵⁾+2𝑎𝜑′‖‖^2𝐿₂+6‖‖𝜕^2𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖60𝜑^2+90𝜑^𝜑⁽⁴⁾+36𝜑′𝜑⁽⁵⁾+6𝜑𝜑⁽⁶⁾+2𝑎𝜑^‖‖^2𝐿₂+4‖‖𝜕_𝑥𝜉_𝑛‖‖^2𝐿_∞‖‖210𝜑^𝜑⁽⁴⁾+126𝜑^𝜑⁽⁵⁾+42𝜑′𝜑⁽⁶⁾+6𝜑𝜑⁽⁷⁾+2𝑎𝜑^‖‖^2𝐿₂+‖‖𝜉_𝑛‖‖^2𝐿_∞‖‖210𝜑⁽⁴⁾²+12𝜑^𝜑⁽⁵⁾+168𝜑^𝜑⁽⁶⁾+48𝜑′𝜑⁽⁷⁾+6𝜑𝜑⁽⁸⁾+2𝑎𝜑⁽⁴⁾‖‖^2𝐿₂⟶0.(3.47) Thus, ‖‖3𝜑²𝜕^4𝑥𝜉_𝑛+24𝜑𝜑^𝜕^3𝑥𝜉_𝑛+36𝜑𝜑^+36𝜑^2𝜕^2𝑥𝜉_𝑛+24𝜑𝜑^+72𝜑^𝜑^𝜕_𝑥𝜉_𝑛+18𝜑^2+24𝜑^𝜑^+6𝜑𝜑⁽⁴⁾+2𝑎𝜑𝜉_𝑛‖‖_𝐻4⟶0,as𝑛⟶∞.(3.48) So from the estimates above, ‖‖(𝐿−𝜆)𝜉_𝑛‖‖_𝐻4⟶0,as𝑛⟶∞.(3.49) The proof of Lemma 3.5 is completed.

Therefore all the four conditions of Lemma 3.1 are satisfied by the linearized equation (3.4) and Theorem 1.1 has been proved.