Pointwise decay for solutions of the 2D Neumann exterior problem for the wave equation
Secchi, Paolo
Bollettino dell'Unione Matematica Italiana, Tome 7-A (2004), p. 189-206 / Harvested from Biblioteca Digitale Italiana di Matematica
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We consider the exterior problem in the plane for the wave equation with a Neumann boundary condition. We are interested to the asymptotic behavior for large times for the solution, and in particular to the dependence on the norms of the initial data in the estimate for the pointwise decay rate. In the paper we prove such an estimate, by a combination of the estimate of the local energy decay and decay estimates for the free space solution.

In questo articolo si considera il problema esterno nel piano per le equazioni delle onde con una condizione di Neumann al bordo. Lo studio riguarda il comportamento per tempi grandi della soluzione, con particolare attenzione per la dipendenza dalla norma dei dati iniziali nella stima del tasso di decadimento puntuale. Nell'articolo si prova una tale stima, mediante una combinazione della stima di decadimento dell'energia locale e stime per la soluzione in tutto il piano.

Publié le : 2004-02-01
@article{BUMI_2004_8_7B_1_189_0,
     author = {Paolo Secchi},
     title = {Pointwise decay for solutions of the 2D Neumann exterior problem for the wave equation},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {7-A},
     year = {2004},
     pages = {189-206},
     zbl = {1178.35231},
     mrnumber = {2044266},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2004_8_7B_1_189_0}
}

Secchi, Paolo. Pointwise decay for solutions of the 2D Neumann exterior problem for the wave equation. Bollettino dell'Unione Matematica Italiana, Tome 7-A (2004) pp. 189-206. http://gdmltest.u-ga.fr/item/BUMI_2004_8_7B_1_189_0/

[1] Dan, W.-Shibata, Y., On a local energy decay of solutions of a dissipative wave equation, Funkcial. Ekvac., 38 (1995), 545-568. | MR 1374437 | Zbl 0848.35070

[2] John, F., Nonlinear wave equations, Formation of singularities, Univ. Lectures Series vol. 2, AMS 1990. | MR 1066694 | Zbl 0716.35043

[3] Klainerman, S., Global existence for nonlinear wave equations, Comm. Pure Appl. Math., 33 (1980), 43-101. | MR 544044 | Zbl 0405.35056

[4] Klainerman, S., Remarks on the global Sobolev inequalities in the Minkowski space Rn⁢1+1, Comm. Pure Appl. Math., 37 (1984), 443-455. | MR 745325 | Zbl 0599.35104

[5] Kleinman, R.-Vainberg, B., Full-low frequency asymptotic expansion for secondorder elliptic equations in two dimensions, Math. Methods Appl. Sci., 17 (1994), 989-1004. | MR 1293680 | Zbl 0812.35028

[6] Li, Ta-Tsien-Chen, Yunmei, Global classical solutions for nonlinear evolution equations, Longman, Harlow, 1992. | MR 1172318 | Zbl 0787.35002

[7] Morawetz, C. S., Decay for solutions of the exterior problem for the wave equation, Comm. Pure Appl. Math., 28 (1975), 229-264. | MR 372432 | Zbl 0304.35064

[8] Morawetz, C. S., Notes on time decay and scattering for some hyperbolic problems, Reg. Conf. Series Appl. Math., SIAM1975. | MR 492919 | Zbl 0303.35002

[9] Racke, R., Lectures on Nonlinear Evolution Equations, Initial Value Problems, Vieweg Verlag, 1992. | MR 1158463 | Zbl 0811.35002

[10] Ralston, J. V., Solutions of the wave equation with localized energy, Comm. Pure Appl. Math., 22 (1969), 807-824. | MR 254433 | Zbl 0209.40402

[11] Vainberg, B., On the short wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour as t→∞ of solutions of non-stationary problems, Russian Math. Surveys, 30 (1975), 1-58. | MR 415085 | Zbl 0318.35006