Generalized solutions to a semilinear wave equation. (English) Zbl 1078.35077
The authors study generalized solutions to semilinear wave equations with singular data and various types of nonlinearities in space dimension \(n \leq 9\). The generalized functions setting employed for this purpose is the space \({\mathcal G}_{L^2}\), a version of Colombeau’s algebra based on asymptotic (with respect to a regularization parameter \(\varepsilon\)) \(L^2\)-estimates. The basic strategy is to regularize the nonlinear term and analyze the solutions of the resulting family of Cauchy problems using Sobolev-type embedding and interpolation theorems. For dimensions \(7,8,9\), more precise results, based on work by [H. Pecher, Math. Z. 161, 9–40 (1978; Zbl 0384.35039); Math. Z. 150, 159–183 (1976; Zbl 0318.35054)] are required. In the case of cubic nonlinearities also solutions without regularizations are considered and compatibility results with the regularized equations are derived.
Reviewer: Michael Kunzinger (Wien)
MSC:
| 35L70 | Second-order nonlinear hyperbolic equations |
| 35D05 | Existence of generalized solutions of PDE (MSC2000) |
| 46F30 | Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.) |
References:
| [1] | Biagioni, H. A.; Oberguggenberger, M., Generalized solutions to the Korteweg-de Vries and the regularized long-wave equations, SIAM J. Math. Anal., 23, 923-940 (1993) · Zbl 0757.35068 |
| [2] | Colombeau, J. F., Elementary Introduction to New Generalized Functions (1985), North-Holland: North-Holland Amsterdam · Zbl 0627.46049 |
| [3] | Grosser, M.; Kunzinger, M.; Oberguggenberger, M.; Steinbauer, R., Geometric Theory of Generalized Functions with Applications to General Relativity (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0998.46015 |
| [4] | Kapitanski, L., Weak and yet weaker solutions to a semilinear wave equation, Commun. Part. Diff. Eq., 19, 1629-1676 (1994) · Zbl 0831.35109 |
| [5] | Kichenassamy, S., Nonlinear Wave Equations (1996), Marcel Dekker Inc.: Marcel Dekker Inc. New York · Zbl 0845.35001 |
| [6] | Lions, J. L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (1969), Dunod: Dunod Gauthier-Villars, Paris · Zbl 0189.40603 |
| [7] | Pecher, H., \(L^p\)-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen, I, Math. Z., 150, 159-183 (1976) · Zbl 0318.35054 |
| [8] | Pecher, H., Ein nichtlinearer Interpolationssatz und seine Anwendung auf nichtlineare Wellengleichungen, Math. Z., 161, 9-40 (1978) · Zbl 0384.35039 |
| [9] | Reed, M., Abstract Non-linear Wave Equations (1976), Springer: Springer New York · Zbl 0317.35002 |
| [10] | J. Shatah, M. Struwe, Geometric Wave Equations, CIMS Lecture Notes, vol. 2, Courant Inst. Math. Sci., New York, 1998.; J. Shatah, M. Struwe, Geometric Wave Equations, CIMS Lecture Notes, vol. 2, Courant Inst. Math. Sci., New York, 1998. · Zbl 0993.35001 |
| [11] | W. Strauss, Nonlinear Wave Equations, CBMS Lecture Notes, vol. 73, Am. Math. Soc., Providence, RI, 1989.; W. Strauss, Nonlinear Wave Equations, CBMS Lecture Notes, vol. 73, Am. Math. Soc., Providence, RI, 1989. |
| [12] | Struwe, M., Semilinear wave equations, Bull. Am. Math. Soc., 26, 53-85 (1992) · Zbl 0767.35045 |
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