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.
Highlights
In order to solve the problem in the noncompactness of some operators, they introduced some weighted spaces
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
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
Summary
We construct the outer solution to the problem (8)–(10). Substitute equation (14) into equation (8), develop the nonlinear term F in ε and μ, and equate coefficients of the powers εiμj(i, j 0, 1, . I + j ≠ 0) into equation (14), we have the outer solution W(t, x) to the original problem. 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
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