Geodesic
Limits of solutions to the semilinear wave equation with small parameter
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2006), pp. 65-84.

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We study the existence of the limits of solution to singularly perturbed initial boundary value problem of hyperbolic – parabolic type with boundary Dirichlet condition for the semilinear wave equation. We prove the convergence of solutions and also the convergence of gradients of solutions to perturbed problem to the corresponding solutions to the unperturbed problem as the small parameter tends to zero. We show that the derivatives of solution relative to time-variable possess the boundary layer function of the exponential type in the neighborhood of t=0. 
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Andrei Perjan. Limits of solutions to the semilinear wave equation with small parameter. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2006), pp. 65-84. https://geodesic-test.mathdoc.fr/item/BASM_2006_1_a7/

[1] Hsiao C. G., Weinacht R. J., “Singular perturbations for semilinear hyperbolic equation”, SIAM J. Math. Anal., 14 (1983), 1168–1179 | DOI | MR | Zbl

[2] Benaouda A., Madaune-Tort M., “Singular perturbations for nonlinear hyperbolic-parabolic problems”, SIAM J. Math. Anal., 18 (1987), 137–148 | DOI | MR | Zbl

[3] Milani A. J., “Long time existence and singular perturbation results for quasi-linear hyperbolic equations with small parameter and dissipation term, II”, Nonlinear Analysis. Theory, Methods and Applications, 11:12 (1987), 1371–1381 | DOI | MR

[4] Esham B. F., Weinacht R. J., “Hyperbolic-parabolic singular perturbations for quasilinear problems”, SIAM J. Math. Anal., 20 (1989), 1344–1365 | DOI | MR | Zbl

[5] Mora X., Sola-Morales J., “The Singular Limit Dynamics of Semilinear Damped Wave Equations”, J. Differ. Eq., 78 (1989), 262–307 | DOI | MR | Zbl

[6] Najman B., “Time Singular Limit of Semilinear Wave Equation with Damping”, J. Math. Anal. Appl., 174 (1993), 95–117 | DOI | MR | Zbl

[7] Lions J. L., Necotorye metody reshenia nelineinych craevych zadach, Mir, Moskva, 1972 (in Russian) | MR

[8] Barbu V., Semigroups of nonlinear contractions in Banach spaces, Ed. Acad. Rom., Bucharest, 1974 (in Romanian) | MR | Zbl

[9] Moroşanu Gh., Nonlinear Evolution Equations and Applications, Ed. Acad. Rom., Bucharest, 1988 | Zbl

[10] Lavrenitiev M. M., Reznitskaia K. G., Yahno B. G., The inverse one-dimentional problems from mathematecal physics, Nauka, Novosibirsk, 1982 (in Russian)

[11] Perjan A., “Linear singular perturbations of hyperbolic-parabolic type”, Buletinul Academiei de Ştiinţe a Republicii Moldova, Matematica, 2003, no. 2(42), 95–112 | MR | Zbl

[12] Evans L. C., Partial Differential Equations, Graduate Studies in Mathematics, 19, AMS, 1998 | Zbl