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.

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 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: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: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.

The same authors considered the singular perturbation differential-integral initial boundary value problem of the formwhere

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.where signifies a uniformly elliptic operator:

and are small parameters and is a time-delay parameter, and are constants, is a bounded convex region in denotes boundary of for class , where is Hölder exponent, is a large enough positive constant, and is a disturbed term.

We have the hypotheses that  as .  and with regard to are Hölder continuous functions and and are sufficiently smooth functions in correspondence ranges.  is a sufficiently smooth function in correspondence ranges except , , where is a constant.  For , there exists a solution and .

The 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.

2. Outer Solution

Now, we construct the outer solution to the problem (8)–(10). First, we develop in small parameter :

The degradation of problem (8)–(10) is

From the hypotheses, there is a solution to equation (13).

We set as the outer solution to problem (8)–(10), and we have

Substitute equation (14) into equation (8), develop the nonlinear term in and , and equate coefficients of the powers , respectively.where

From equation (15), we can obtain . Substituting and into equation (14), we have the outer solution to the original problem. But it does not continue at 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 system near . Define the coordinate of every point in the neighborhood of in the following way: the coordinate is the distance from the point to , where is small enough. is a nonsingular coordinate.

In the neighborhood of : , we havewhere

We introduce the variables of multiple scales [1, 2] on :where is a function to be determined. For convenience, we still substitute for below, respectively. From equation (17), we havewhile and are determined operators and their constructions are omitted.

Let , and the solution to the original problem (8)–(10) iswhere is the spike layer corrective term, and

Substituting equations (17)–(22) into equation (8), expanding nonlinear terms in and , and equating the coefficients of , respectively, we obtainwhere are determined functions and their constructions are omitted.

From problems (23)-(24), we can have , From and equations (25)-(26), we can obtain solutions successively.

From the hypotheses, it is easy to see that possesses spike layer behavior:where are constants.

Let , where is a sufficiently smooth function in and satisfies

For convenience, we still substitute for below. Then from equation (22), we have the spike layer corrective term near .

4. Boundary Layer Corrective Term

Now, we set up a local coordinate system in the neighborhood near as Ref. [14], where . In the neighborhood of where

We introduce the variables of multiple scales, see the Ref [1, 2], on :where is a function to be determined. For convenience, we still substitute for below, respectively. From (29), we havewhere and are determined operators and their constructions are omitted too.

We set the solution to original boundary value problem (8)–(10), wherewhere is a boundary layer corrective function.

Set . Thus, we have . And let

Substituting equation (34) into equations (8) and (9) and expanding nonlinear terms in , we equate the coefficients of the same powers for . And we obtainwhere are determined functions successively, and their constructions are omitted too.

From equations (35) and (36), we can have solution . And from equations (37) and (38), we can obtain successively. Substituting into equation (34), we obtain .

From the hypotheses, it is easy to see that possesses boundary layer behavior:where are constants.

Let , where is a sufficiently smooth function in and satisfies

For convenience, we still substitute for below. Then, from equation (34), we have the boundary corrective term near .

5. Initial Layer Corrective Term

The solution to the original problem (8)–(10) iswhere is an initial layer corrective term. Substituting (41) into equations (8)–(10), we have