# A Class of Shock Wave Solution to Singularly Perturbed Nonlinear Time-Delay Evolution Equations

Authors: Yi-Hu Feng; Lei Hou
Journal: Shock and Vibration
 (2020)

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## Abstract

Nonlinear singularly perturbed problem for time-delay evolution equation with two parameters is studied. Using the variables of the multiple scales method, homogeneous equilibrium method, and approximation expansion method from the singularly perturbed theories, the structure of the solution to the time-delay problem with two small parameters is discussed. Under suitable conditions, first, the outer solution to the time-delay initial boundary value problem is given. Second, the multiple scales variables are introduced to obtain the shock wave solution and boundary layer corrective terms for the solution. Then, the stretched variable is applied to get the initial layer correction terms. Finally, using the singularly perturbed theory and the fixed point theorem from functional analysis, the uniform validity of asymptotic expansion solution to the problem is proved. In addition, the proposed method possesses the advantages of being very convenient to use.


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## Body

## 1. Introduction

There are many important applications of nonlinear singular perturbation in applied mathematics, engineering, and physics [1, 2]. Recently, some scholars have done a great deal of work, for example, Nicaise and Pignotti [3] considered a stabilization problem for abstract second-order evolution equations with dynamic boundary feedback laws with a time delay and distributed structural damping. They proved an exponential stability result under a suitable condition between the internal damping and the boundary layers. The proof of the main result is based on an identity with multipliers that allows to obtain a uniform decay estimate for a suitable energy functional. Some concrete examples are detailed. Some counterexamples suggest that this condition is optimal.Jeong et al. [4] considered a quasilinear wave equation utt−Δut−div∇Uα−2∇u−div∇utβ−2∇ut+autm−2+μ1utx,t+μ2utx,t−τ=bup−2u, associated with initial and Dirichlet boundary conditions at one part and acoustic boundary conditions at another part, respectively. They proved under suitable conditions and for negative initial energy, a global nonexistence of solutions.Feng [5] studied the following Cauchy problem with a time-delay term in the internal feedback:(1)uttx,t−φxΔux,t−∫0tgt−sΔux,sds+μ1utx,t+μ2utx,t−τ=0,ux,0=u0x,utx,0=u1x,x∈Rn,utx,t−τ=f0x,t−τ,x∈Rn,0<t<τ,and in order to solve the problem in the noncompactness of some operators, they introduced some weighted spaces. Under suitable assumptions on the relaxation function, they established a general decay result of solution for the initial-value problem by using the energy perturbation method and their result extends earlier results.Nicaise and Valein [6] considered abstract second-order evolution equations with unbounded feedback with time delay. Existence results are obtained under some realistic assumptions. Sufficient and explicit conditions are derived that guarantee the exponential or polynomial stability. Some new examples that enter into our abstract framework are presented.Weidenfeld and Frankel [7] focused on the early evolution of small (linear) perturbations following the sudden (step function) exposure of a liquid layer to a cold adjacent atmosphere. On a short time scale relative to that characterizing thermal relaxation across the liquid layer, the temperature distribution is nonlinear and highly transient. Thus, the conduction reference state may not be regarded quasisteady. They accordingly considered the initial-value problem and obtained a Volterra-type integral equation governing the evolution of surface-temperature perturbations.Many approximate methods have been improved, such as Graef and Kong [8], Hovhannisyan and Vulanovic [9], Bonfoh et al. [10], Barbu and Cosma [11], Faye et al. [12], Samusenko [13], Mo [14], Das et al. [15, 16], and so on [17–26]. By using the singular perturbation theories, Feng et al. also studied a class of nonlinear singular perturbation problems [27–34].For instance, Feng and Mo [32] in 2015 considered the nonlinear elliptic boundary value problem with two parameters:(2)ε2mLmu+μ2kLku=fx,u,x∈Ω,∂ru∂nr=grx,x∈∂Ω,where L is the uniform elliptic operator which can be expressed as follows:(3)L≡∑i,j=1naijx∂2∂xi∂xj,∑i,j=1naijxξiξj≥λ∑i=1nξi2,ξi∈R,λ>0,and by introducing stretched variables, setting local coordinate systems, and using the differential inequalities, the authors proved the existence of the shock solution for boundary value problem and studied the asymptotic behavior of the solution.Feng et al. [33] in 2017 considered the singularly perturbed boundary value problem for a class of nonlinear integral-differential elliptic equation:(4)ε2mLmu−Tu=fx,u,x∈Ω,∂lu∂nl=glx,l=0,1,…,m−1,x∈∂Ω,where L denotes the uniform elliptic type:(5)L=∑i,j=1nαijx∂2∂xi∂xj+∑i,j=1nβix∂∂xi,x∈R,Tu=∫ΩKxux,x∈Ω,and by using the multiple scales variable, the method of component expansion, and the singular perturbation theory, we proved the existence of solution to the problem and the uniformly valid asymptotic estimation.Feng et al. [34] in 2018 considered a class of nonlinear differential-integral singular perturbation problem for the disturbed evolution equations. Using the singular perturbation method, the structure of solution to problem is discussed in the cases of two small parameters under suitable conditions.The same authors considered the singular perturbation differential-integral initial boundary value problem of the form(6)μ∂2y∂t2−εLy+Ty=ft,x,εy,t,x∈0,T0×Ω,y=gt,x,x∈∂Ω,yt=0=h1x,ε∂y∂tt=0=h2x,where(7)L=∑i,j=1naijx∂2∂xi∂xj+∑i,j=1nbix∂∂xi,∑i,j=1naijxξiξj≥λi∑i=1nξi2,ξi∈R,λi>0,Ty≡ϕx+∫∂ΩKxyxdx,Kx≥0.First, the outer solution to the boundary value problem is given. Second, by constructing the nonsingular coordinate system near the boundary, the variables of multiple scales are introduced to obtain the boundary layer corrective term for the solution. Then, the stretched variable is applied to get the initial layer correction term. Finally, using the fixed point theorem, the uniformly valid asymptotic expansion of the solution to problem is proved. The proposed method possesses the advantages of being convenient to use.By introducing stretched variables, setting local coordinate systems, and using the differential inequalities, we proved the existence of the shock solution for boundary value problem and studied the asymptotic behavior of the solution.In this paper, using the variables of the multiple scales method from the singularly perturbed theory, after simplifying the method, we consider a class of shock wave solution to the nonlinear singularly perturbed time-delay evolution equations initial boundary value problem with two parameters as follows, and the structure of the solution to the problem is discussed. In addition, the proposed method possesses the advantages of being very convenient to use.(8)ε2∂2wt,x∂t2−μ2Lwt,x+cwt−κ,x=Ft,x,wt,x,t,x∈−κ,T0×Ω,(9)w=gt,x,x∈∂Ω,(10)w−k≤t≤0=h1x,ε∂w∂t−k≤t≤0=h2x,x∈Ωμ,where L signifies a uniformly elliptic operator:(11)L=∑i,j=1naijx∂2∂xi∂xj+∑i=1nbix∂∂xi,∑i,j=1naijxζiζj≥λi∑i=1nζi2,∀ζi∈R,λ>0.ε and μ are small parameters and k=bε is a time-delay parameter, b>0 and c are constants, x≡x1,x2,…,xn∈Ω,Ω is a bounded convex region in Rn,∂Ω denotes boundary of Ω for class C1+α, where α∈0,1 is Hölder exponent, T0 is a large enough positive constant, and F is a disturbed term.We have the hypotheses thatH1σ=μ/ε⟶0 as ε⟶0.H2aij and bi with regard to x are Hölder continuous functions and g and hi are sufficiently smooth functions in correspondence ranges.H3F is a sufficiently smooth function in correspondence ranges except x0∈Ω, Fwt,x,w≤−c¯<0x≠x0, where c¯>0 is a constant.H4 For cw−Ft,x,w=0, there exists a solution W00t,x and limx⟶x0W00t,x≠W00t,x0.The rest of this paper is organized as follows. In Section2, we construct the outer solution to the initial boundary value problem (8)–(10). In Sections 3 and 4, we set up a local coordinate system, and then we construct the spike layer corrective term and boundary layer corrective term. In Section 5, we obtain the formal asymptotic expansion solution for the nonlinear singular perturbation time-delay evolution equation initial boundary value problem (8)–(10) with two parameters. At last, in Section 6, we prove that the formal asymptotic expansion solution is uniformly valid.


## 2. Outer Solution

Now, we construct the outer solution to the problem (8)–(10). First, we develop wt−k,x,y in small parameter k=bε:(12)wt−k,x=wt,x+∑l=1∞−bll!∂lw∂klk=0εl.The degradation of problem (8)–(10) is(13)cw=Ft,x,w,x,y∈Ω.From the hypotheses, there is a solutionW00t,xx≠x0 to equation (13).We setWt,x as the outer solution to problem (8)–(10), and we have(14)Wt,x=∑i.j=1∞Wijt,xεiμj.Substitute equation (14) into equation (8), develop the nonlinear term F in ε and μ, and equate coefficients of the powers εiμji,j=0,1,…,i+j≠0, respectively.(15)cWij=Fwt,x,W00Wij−c∑l=1i−bεll!∂lWi−lj∂klk=0+Fij,x,y∈Ω,where(16)Fij=1i!j!∂i+jF∂εi∂μjε=μ=0,i,j=0,1,2,…,i+j≠0.From equation (15), we can obtain Wijt,x,i,j=0,1,…,i+j≠0. Substituting W00t,x and Wijt,x,i,j=0,1,…,i+j≠0 into equation (14), we have the outer solution Wt,x to the original problem. But it does not continue at t,x0 and may not satisfy the boundary and initial conditions (9) and (10), so we need to construct the spike layer, boundary layer, and initial layer corrective functions.


## 3. Spike Layer Corrective Term

Set up a local coordinate systemρ,ϕ near x0∈Ω. Define the coordinate of every point Q in the neighborhood of x0 in the following way: the coordinate ρ≤ρ0 is the distance from the point Q to x0, where ρ0 is small enough. ϕ=ϕ1,ϕ2,…,ϕn−1 is a nonsingular coordinate.In the neighborhood ofx0: 0≤ρ≤ρ0⊂Ω, we have(17)L=a˜nn∂2∂ρ2+∑i=1n−1a˜ni∂2∂ρ∂ϕi+∑i,j=1n−1a˜ij∂2∂ϕi∂ϕj+b˜n∂∂ρ+∑i=1n−1b˜i∂∂ϕi,where(18)a˜nn=∑i,j=1naij∂ρ∂xi∂ρ∂xj,a˜ni=2∑j,k=1najk∂ρ∂xj∂ϕi∂xk,a˜ij=∑k,l=1nakl∂ϕi∂xk∂ϕj∂xl,b˜n=∑i,j=1naij∂2ρ∂xi∂xj,b˜i=∑j,k=1najk∂2∂xj∂ϕi∂xk.We introduce the variables of multiple scales [1, 2] on 0≤ρ≤ρ0⊂Ω:(19)σ˜=hρ,ϕμ,ρ˜=ρ,ϕ˜=ϕ,where hρ,ϕ is a function to be determined. For convenience, we still substitute ρ,ϕ for ρ˜,ϕ˜ below, respectively. From equation (17), we have(20)L=1μ2K0+1μK1+K2,while K0=a˜nnhρ2∂2/∂σ˜2 and K1,K2 are determined operators and their constructions are omitted.Lethρ=1/a˜nn, and the solution w to the original problem (8)–(10) is(21)w=Wt,x+W1t,ρ,ϕ,where W1 is the spike layer corrective term, and(22)W1∼∑i,j=0∞w1ijt,ρ,ϕεiμj.Substituting equations (17)–(22) into equation (8), expanding nonlinear terms in ε and μ, and equating the coefficients of εiμji,j=0,1,…, respectively, we obtain(23)K10w100=0,0≤ρ≤ρ0,(24)w100ρ=0=W00t,x0,(25)K10w1ij=Gij,0≤ρ≤ρ0,i=0,1,…,i+j≠0,(26)w1ijρ=0=Wijt,x0,i=0,1,…,i+j≠0,where Gij,i=0,1,…,i+j≠0 are determined functions and their constructions are omitted.From problems (23)-(24), we can have w100, From w100 and equations (25)-(26), we can obtain solutions w1iji,j=0,1,…,i+j≠0 successively.From the hypotheses, it is easy to see thatw1iji,j=0,1,… possesses spike layer behavior:(27)w1ij=Oexp−δijρμ,i,j=0,1,…,where δij>0i,j=0,1,2,… are constants.Letw¯1ij=ψρw1ij, where ψρ is a sufficiently smooth function in 0≤ρ≤ρ0 and satisfies(28)ψρ=1,0≤ρ≤13ρ0,0,ρ≥23ρ0.For convenience, we still substitutew1ij for w¯1ij below. Then from equation (22), we have the spike layer corrective term W1 near 0≤ρ≤ρ0⊂Ω.


## 4. Boundary Layer Corrective Term

Now, we set up a local coordinate systemρ¯,ϕ¯ in the neighborhood near ∂Ω:0≤ρ¯≤ρ¯0 as Ref. [14], where ϕ¯=ϕ¯1,ϕ¯2,…,ϕ¯n−1. In the neighborhood of ∂Ω:0≤ρ¯≤ρ¯0,(29)L=a¯nn∂2∂ρ¯2+∑i=1n−1a¯ni∂2∂ρ¯∂ϕ¯i+∑i,j=1n−1a¯ij∂2∂ϕ¯i∂ϕ¯j+b¯n∂∂ρ¯+∑i=1n−1b¯i∂∂ϕ¯i,where(30)a¯nn=∑i,j=1naij∂ρ¯∂xi∂ρ¯∂xj,a¯ni=2∑j,k=1najk∂ρ¯∂xj∂ϕ¯i∂xk,a¯ij=∑k,l=1nakl∂ϕ¯i∂xk∂ϕ¯j∂xl,b¯n=∑i,j=1naij∂2ρ¯∂xi∂xj,b¯i=∑j,k=1najk∂2ϕ¯i∂xj∂xk.We introduce the variables of multiple scales, see the Ref [1, 2], on 0≤ρ¯≤ρ¯0⊂Ω:(31)σ¯=h¯ρ¯,ϕ¯μ,ρ˜=ρ¯,ϕ˜=ϕ¯,where h¯ρ¯,ϕ¯ is a function to be determined. For convenience, we still substitute ρ¯,ϕ¯ for ρ˜,ϕ˜ below, respectively. From (29), we have(32)L=1μ2K¯0+1μK¯1+K¯2,where K¯0=a¯nnh¯ρ¯2∂2/∂σ¯2 and K¯1,K¯2 are determined operators and their constructions are omitted too.We set the solutionw to original boundary value problem (8)–(10), where(33)w=W+U,where U is a boundary layer corrective function.Seth¯ρρ¯,ϕ¯=1/a¯nn. Thus, we have K¯0=∂2/∂σ¯2. And let(34)U∼∑i,j=0∞uijt,ρ¯,ϕ¯εiμj.Substituting equation (34) into equations (8) and (9) and expanding nonlinear terms in ε,μ, we equate the coefficients of the same powers for εiμji,j=0,1,…. And we obtain(35)K¯0u00=0,0≤ρ¯≤ρ¯0,(36)u00ρ¯=0=gt,x−W00t,x,(37)K¯0uij=G¯ij,0≤ρ¯≤ρ¯0,i,j=0,1,…,i+j≠0,(38)uijρ¯=0=−Wijt,x,i,j=0,1,…,i+j≠0,where G¯iji,j=0,1,…,k−1,i+j≠0 are determined functions successively, and their constructions are omitted too.From equations (35) and (36), we can have solution u00. And from equations (37) and (38), we can obtain uij,i,j=0,1,2,…,i+j≠0 successively. Substituting uij,i,j=0,1,2,… into equation (34), we obtain U.From the hypotheses, it is easy to see thatuij,i,j=0,1,… possesses boundary layer behavior:(39)uij=Oexp−δ¯ijρ¯μ,i,j=0,1,…,where δ¯ij>0,i=1,2,… are constants.Letu¯ij=ψ¯ρ¯uij, where ψ¯ρ¯ is a sufficiently smooth function in 0≤ρ¯≤ρ¯0 and satisfies(40)ψ¯ρ¯=1,0≤ρ¯≤13ρ¯0,0,ρ¯≥23ρ¯0.For convenience, we still substituteuij for u¯ij below. Then, from equation (34), we have the boundary corrective term U near 0≤ρ¯≤ρ¯0.


## 5. Initial Layer Corrective Term

The solutionw to the original problem (8)–(10) is(41)w=·W+W1+U+V,where V is an initial layer corrective term. Substituting (41) into equations (8)–(10), we have(42)ε2Vttt,x−μ2LVt,x+cVt−k,x=Ft,x,Wt,x+W1t,x+Ut,x+Vt,x−Ft,x,Wt,x+W1t,x+Ut,x−ε2Wt,x+W1t,x+Ut,x−μ2LWt,x+W1t,x+Ut,x+cWt−k,x+W1t−k,x+Ut−k,x,(43)V=gt,x−Wt,x−W1t,x−Ut,x,x∈∂Ω,(44)V−k≤t≤0=h1x−W−k≤t≤0−W1−k≤t≤0−U−k≤t≤0,x∈Ω,(45)ε∂V∂t−k≤t≤0=h2x−ε∂∂tW+W1+U−k≤t≤0,x∈Ω.We introduce a stretched variable, see [1, 2]: τ=t/ε and let(46)V∼∑i,j=0∞vijτ,xεiμj.Substituting equations (14), (22), (34), and (46) into equations (42)–(45), expanding nonlinear terms in ε and μ, and equating the coefficients of like powers of εiμj, respectively, for i,j=0,1,…, we have(47)v00ττ+cv00=F0,x,W00+W100+u00+v00−F0,x,W00+W100+u00,(48)v00x∈∂Ω=0,(49)v00−b≤τ≤0=h1x−W000,x+W1000,x+u000,x,x∈Ω,(50)∂v00∂τ−b≤τ≤0=h2x−H¯00−b≤τ≤0,(51)vijττ+cvij=G¯ij,i,j=0,1,…,i+j≠0,(52)vijx∈∂Ω=−Wij+W1ij+uijx∈∂Ω,i,j=0,1,…,i+j≠0,(53)vij−b≤τ≤0=H¯ij−b≤τ≤0,x∈Ω,i,j=0,1,…,i+j≠0,(54)∂vij∂τ−b≤τ≤0=H¯ij−b≤τ≤0,x∈Ω,i,j=0,1,…,i+j≠0,where G¯ij,H¯iji=0,1,…,i+j≠0 are determined functions, and their constructions are omitted too.From the problem equations (47)–(50), we can have v00. From v00 and equations (51)–(54), we can obtain solutions vij,i,j=0,1,…,i+j≠0 successively.From the hypotheses, it is easy to see thatvij,i,j=0,1,… possesses boundary layer behavior:(55)vij=Oexp−δ˜ijtε,i,j=0,1,…,where δ˜ij>0,i,j=1,2,… are constants.Then, from equation (46), we have the initial corrective term V.From equation (41), we obtain the formal asymptotic expansion of solution w to the nonlinear singular perturbation time-delay evolution equation initial boundary value problem (8)–(10) with two parameters:(56)w∼W00+∑i.j=0,i+j≠0∞Wijεiμj+∑i.j=0∞w1ij+uij+vijεiμj,0<ε,μ≪1,


## 6. Main Result

Now, we prove that this expansion (56) is uniformly valid in Ω and we have the following theorem.Theorem 1.
Under the hypothesesH1−H4, there exists a solution wt,x to the nonlinear singular perturbation time-delay evolution equation initial boundary value problem (8)–(10) with two parameters which holds the uniformly valid asymptotic expansion (56) for ε and μ in t,x∈−k,T0×Ω¯.Proof.
First, we get the remainder termRt,x of the initial boundary value problem with two parameters (8)–(10). Let(57)wt,x=w⇀t,x+Rt,x,where(58)w¯t,x=W00+∑i,j=0,i+j≠0mWijεiμj+∑i.j=0mw1ij+uij+vijεiμj.
Using equations (14), (27), (39), (55), and (57), we obtain(59)HR≡ε2∂2Rt,x∂t2−μLRt,x+cRt−τ,x−Ft,x,u¯+Rt,x−Ft,x,u¯t,x=Oλm+1,x∈Ω,λ=maxε,μ,R=Oλm+1,x∈∂Ω,λ=maxε,μ,R−k≤t≤0=Oλm+1,x∈Ω,λ=maxε,μ,ε∂R∂t−k≤t≤0=Oλm+1,x∈Ω,λ=maxε,μ.
We can have the linearized differential operatorL¯ as follows:(60)L¯p=ε2∂2pt,x∂t2−μ2Lpt,x+cpt−τ,x,and therefore,(61)Ψp≡Hpt,x−L¯pt,x−Ft,x,u¯t,x−Ft,x,u¯t,x+pt,x+Fu¯t,x,u¯t,x+pt,xpt,x.
For fixedε,μ, the normed linear space N is chosen as(62)N=pp∈C2−k,T0×Ω,p∂Ω=g,p−k≤t≤0=h1,pt−k≤t≤0=h2,with norm(63)p=maxt∈−k,T0,x∈Ωp,and the Banach space B as(64)B=qq∈C−k,T0×Ω,with norm(65)q=maxt∈−k,T0,x∈Ωq.
From the hypotheses, we may show that the condition(66)L−1g≤l−1g,∀g∈B,by the fixed point theorem [1, 2] is fulfilled where l−1 is independent of ε and μ, i.e., L−1 is continuous. The Lipschitz condition of the fixed point theorem becomes(67)Ψp2−Ψp1<C1maxt∈−k,T0,x∈Ωp1+p2p2−p1+C2maxt∈−k,T0,x∈Ωp12⋅p2−p1<C3rp2−p1,where C1,C2, and C3 are constants; this inequality is valid for all p1,p2 in a ball KNr with r≤1. Finally, we obtain the result that the remainder term exists; moreover,(68)maxt∈−k,T0,x∈ΩRt,x=Oλm+1,λ=maxε,μ.
From equation (56), we have(69)wt,x≡W00+∑i,j=0,i+j≠0mWijεiμj+∑i,j=0mw1ij+uij+vijεiμj+Oλm+1,λ=maxε,μ.
The proof of the theorem is completed.


## 7. Conclusions

Nonlinear singularly perturbed problem for time-delay evolution equation with two parameters is an attractive investigated subject. In this paper, we use the variables of the multiple scales method, homogeneous equilibrium method, and approximation expansion method from the singularly perturbed theories, and then we discuss the structure of the solution to the time-delay problem with two small parameters; after setting up a local coordinate system, we obtained the formal asymptotic expansion solution for the nonlinear singular perturbation time-delay evolution equation initial boundary value problem with two parameters; using the singularly perturbed theory and the fixed point theorem from functional analysis, the uniform validity of asymptotic expansion solution to the problem is proved. In addition, the proposed method possesses the advantages of being very convenient to use.



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