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

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.-en, 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.


Introduction
ere 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. ey proved an exponential stability result under a suitable condition between the internal damping and the boundary layers. e 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 u tt − Δu t − di v(|∇U| α− 2 ∇u) − di v(|∇u t | β− 2 ∇u t ) + a|u t | m− 2 + μ 1 u t (x, t) + μ 2 u t (x, t − τ) � b|u| p− 2 u, associated with initial and Dirichlet boundary conditions at one part and acoustic boundary conditions at another part, respectively. ey 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: x ∈ R n , 0 < t < τ, (1) 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 initialvalue 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.
us, the conduction reference state may not be regarded quasisteady. ey accordingly considered the initial-value problem and obtained a Volterra-type integral equation governing the evolution of surface-temperature perturbations.
For instance, Feng and Mo [32] in 2015 considered the nonlinear elliptic boundary value problem with two parameters: where L is the uniform elliptic operator which can be expressed as follows: 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: where L denotes the uniform elliptic type: 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. e same authors considered the singular perturbation differential-integral initial boundary value problem of the form where 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. en, 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. e 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.
where L signifies a uniformly elliptic operator: ε and μ are small parameters and k � bε is a time-delay parameter, b > 0 and c are constants, x ≡ (x 1, x 2 , . . . , x n ) ∈ Ω, Ω is a bounded convex region in R n , zΩ denotes boundary of Ω for class C 1+α , where α ∈ (0, 1) is Hölder exponent, T 0 is a large enough positive constant, and F is a disturbed term.
We have the hypotheses that [H 2 ]a ij and b i with regard to x are Hölder continuous functions and g and h i are sufficiently smooth functions in correspondence ranges.
[H 3 ]F is a sufficiently smooth function in correspondence ranges except e rest of this paper is organized as follows. In Section 2, 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.

Outer Solution
Now, we construct the outer solution to the problem (8)- (10). First, we develop w(t − k, x, y) in small parameter k � bε: e degradation of problem (8)-(10) is From the hypotheses, there is a solution W 00 (t, x)(x ≠ x 0 ) to equation (13).
where (14), we have the outer solution W(t, x) to the original problem. But it does not continue at (t, x 0 ) 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.

Spike Layer Corrective Term
Set up a local coordinate system (ρ, ϕ) near x 0 ∈ Ω. Define the coordinate of every point Q in the neighborhood of x 0 in the following way: the coordinate ρ( ≤ ρ 0 ) is the distance from the point Q to x 0 , where ρ 0 is small enough. ϕ � (ϕ 1 , ϕ 2 , . . . , ϕ n− 1 ) is a nonsingular coordinate.
In the neighborhood of x 0 : (0 ≤ ρ ≤ ρ 0 ) ⊂ Ω, we have Shock and Vibration 3 where We introduce the variables of multiple scales [1,2] on where h(ρ, ϕ) is a function to be determined. For convenience, we still substitute ρ, ϕ for ρ, ϕ below, respectively. From equation (17), we have while K 0 � a nn h 2 ρ (z 2 /zσ 2 ) and K 1 , K 2 are determined operators and their constructions are omitted.

Shock and Vibration
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 where K 0 � a nn h 2 ρ (z 2 /zσ 2 ) and K 1 , K 2 are determined operators and their constructions are omitted too.
We set the solution w to original boundary value problem (8)- (10), where where U is a boundary layer corrective function.
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:

Main Result
Now, we prove that this expansion (56) is uniformly valid in Ω and we have the following theorem. Proof. First, we get the remainder term R(t, x) of the initial boundary value problem with two parameters (8)- (10). Let where Using equations (14), (27), (39), (55), and (57), we obtain We can have the linearized differential operator L as follows: and therefore, For fixed ε, μ, the normed linear space N is chosen as with norm and the Banach space B as with norm From the hypotheses, we may show that the condition by the fixed point theorem [1,2] is fulfilled where l − 1 is independent of ε and μ, i.e., L − 1 is continuous. e Lipschitz condition of the fixed point theorem becomes where C 1 , C 2 , and C 3 are constants; this inequality is valid for all p 1 , p 2 in a ball K N (r) with ‖r‖ ≤ e proof of the theorem is completed.

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.