Existence Theorems for Some Weak Abstract Variable Domain Hyperbolic Problems

Carroll, Robert ; State, Emile

Canadian journal of mathematics, Tome 23 (1971) no. 4, pp. 611-626

Résumé

In this paper we prove some existence theorems for some weak problems with variable domains arising from hyperbolic equations of the type 1.1 where A = {A(t)} is, for example, a family of elliptic differential operators in space variables x = (x1, ..., xn ). Thus let H be a separable Hilbert space and let V(t) ⊂ H be a family of Hilbert spaces dense in H with continuous injections i(t): V(t) → H (0 ≦ t ≦ T < ∞). Let V’ (t) be the antidual of V(t) (i.e. the space of continuous conjugate linear maps V(t) → C) and using standard identifications one writes V(t) ⊂ H ⊂ V‘(t).

Détail

DOI : 10.4153/CJM-1971-069-9

Bibliographie

[1] 1. Baiocchi, C., Soluzioni ordinarie e generalizzate del problema di Cauchy per equazioni differenziali astratte lineari del secondo ordine in spazi di Hilbert, Ricerche Mat. 16 (1967), 27–95. Google Scholar

[2] 2. Baiocchi, C., Teoremi di esistenza e regolarità per certe classi di equazioni differenziali astratte, Ann. Mat. Pura Appl. 72 (1966), 395–418. Google Scholar

[3] 3. Brezis, H., On some degenerate nonlinear parabolic problems, Proc. Symp. Pure Math., Amer. Math. Soc., Vol. 18, part 1 (1970), 28–38. Google Scholar

[4] 4. Brezis, H. and Lions, J., Sur certains problèmes unilatéraux hyperboliques, Comptes Rendus, Acad. Sci. Paris 264 (1967), 928–931. Google Scholar

[5] 5. Browder, F., Nonlinear maximal monotone operators in Banach space, Math. Annalen 175 (1968), 89–113. Google Scholar

[6] 6. Browder, F., Qn nonlinear wave equations, Math. Z. 80 (1962), 249–264. Google Scholar

[7] 7. Browder, F. and Strauss, W., Scattering for nonlinear wave equations, Pacific J. Math. 13 (1963), 23–43. Google Scholar

[8] 8. Caby, D., Application de la méthode de pénalisation à la résolution d'équations différentielles opérationelles à domaine variable, Ph.D. Thèse, Paris, 1969. Google Scholar

[9] 9. Carroll, R., Abstract methods in partial differential equations (Harper-Row, New York, 1969). Google Scholar

[10] 10. Carroll, R., On the nature of weak solutions and some abstract Cauchy problems, Bull. Amer. Math. Soc. 72 (1966), 1068–1072. Google Scholar

[11] 11. Carroll, R., On the propagator equation, Illinois J. Math. 11 (1967), 506–527. Google Scholar

[12] 12. Carroll, R., Solutions of possibly noncoercive evolution problems with regular data (summary), Proc. Conf. Evolution Eqs. and Functional Anal., Univ. of Kansas, 1970 (to appear). Google Scholar

[13] 13. Carroll, R., Some variable domain problems in abstract evolution equations, Proc. Conf. Eqs. of Evolution and Nonlinear Semigroups, Univ. of Kentucky, 1969, 11–24. Google Scholar

[14] 14. Carroll, R., Some remarks on kernels, recovery formulas, and evolution equations, Proc. Conf. Evolution Eqs. and Functional Anal., Univ. of Kansas, 1970 (to appear). Google Scholar

[15] 15. Carroll, R. and Cooper, J., Solutions des équations d?évolution abstraites à domaine variable, Comptes Rendus, Acad. Sci. Paris 270 (1970), 319–321. Google Scholar

[16] 16. Carroll, R. and Cooper, J., Remarks on some variable domain problems in abstract evolution equations, Math. Annalen 188 (1970), 143–164. Google Scholar

[17] 17. Carroll, R. and Mazumdar, T., Solutions of some possibly non-coercive evolution problems with regular data (to appear in J. Appl. Anal.). Google Scholar

[18] 18. Carroll, R. and Neuwirth, J., Some uniqueness theorems for differential equations with operator coefficients, Trans. Amer. Math. Soc. 110 (1964), 459–472. Google Scholar

[19] 19. Carroll, R. and State, E., Solutions faibles des équations d'évolution hyperboliques à domaine variable, Comptes Rendus, Acad. Sci. Paris 271 (1970), 931–933. Google Scholar

[20] 20. Cooper, J., Some aspects of abstract linear evolution equations in Hilbert space, Ph.D. Thesis, University of Illinois, 1967. Google Scholar

[21] 21. Cooper, J., Evolution equations in Banach space with variable domain (to appear). Google Scholar

[22] 22. Cooper, J., Un problème relatif aux équations d'évolution abstraites, Comptes Rendus, Acad. Sci. Paris 268 (1967), 422–424. Google Scholar

[23] 23. Cooper, J., Two point problems for abstract evolution equations (to appear). Google Scholar

[24] 24. Cooper, J., Evolution equations in Banach spaces with variable domain (summary), Proc. Conf. Evolution Eqs. and Functional Anal., Univ. of Kansas, 1970 (to appear). Google Scholar

[25] 25. Goldstein, J., Time dependent hyperbolic equations, J. Functional Analysis 4 (1969), 31–49. Google Scholar

[26] 26. Goldstein, J., Semigroups and second order differential equations, J. Functional Analysis 4 (1969), 50–70. Google Scholar

[27] 27. Jörgens, K., Das Anfangswertproblem im Grossen fur eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77 (1961), 295–308. Google Scholar

[28] 28. Kato, T., Some linear evolution equations of “hyperbolic” type, Proc. Conf. Eqs. of Evolution and Nonlinear Semigroups, Univ. of Kentucky, 1969, 25–27. Google Scholar

[29] 29. Krein, S., Linear differential equations in Banach spaces, Izd. Nauka, Glav. Red. Fis.-Mat. Lit., Moscow, 1967. Google Scholar

[30] 30. Lax, P. and Phillips, R., Scattering theory (Academic Press, New York, 1967). Google Scholar

[31] 31. Lee, K., A mixed problem for hyperbolic equations with time dependent domain, Jour. Math. Anal. Appl. 16 (1966), 455–471. Google Scholar

[32] 32. Lions, J., Equations différentielles-opérationnelles (Springer, Berlin, 1961). Google Scholar

[33] 33. Lions, J., Quelques méthodes de résolution des problèmes aux limites nonlinéaires (Dunod, Paris, 1969). Google Scholar

[34] 34. Lions, J., Une remarque sur les applications du théorème de Hille-Yosida, J. Math. Soc. Japan 9 (1957), 62–70. Google Scholar

[35] 35. Lions, J. and Magenes, E., Problèmes aux limites nonhomogènes et applications. Vols. 1, 2, 3 (Dunod, Paris, 1968 and 1970). Google Scholar

[36] 36. Lions, J. and Strauss, W., Some nonlinear evolution equations, Bull. Soc. Math. France 93 (1965), 43–96. Google Scholar

[37] 37. Mazumdar, T., Existence theorems for noncoercive variable domain evolution problems, Ph.D. Thesis, University of Illinois, 1971. Google Scholar

[38] 38. Medeiros, L., The initial value problem for nonlinear wave equations in Hilbert space, Trans. Amer. Math. Soc. 136 (1969), 305–327. Google Scholar

[39] 39. Raskin, V. and Sobolevskij, P., The Cauchy problem for second order differential operators in Banach spaces, Sibirsk. Mat. Z. 8 (1967), 70–90. Google Scholar

[40] 40. Rogak, E., A mixed problem for the wave equation in a time dependent domain, Arch. Rational Mech. Anal. 22 (1966), 24–36. Google Scholar

[41] 41. Sather, J., The initial-boundary value problem for a nonlinear hyperbolic equation in relativistic quantum mechanics, J. Math. Mech. 16 (1966), 27–50. Google Scholar

[42] 42. Sather, J., The existence of a global classical solution of the initial-boundary value problem for ▭u + u3 = f, Arch. Rational Mech. Anal. 22 (1966), 292–307. Google Scholar

[43] 43. Segal, I., Nonlinear semigroups, Ann. of Math. 78 (1963), 339–364. Google Scholar

[44] 44. State, E., Weak hyperbolic problems with variable domains, Ph.D. Thesis, University of Illinois, 1971. Google Scholar

[45] 45. Strauss, W., Scattering for hyperbolic equations, Trans. Amer. Math. Soc. 108 (1963), 13–37. Google Scholar

[46] 46. Strauss, W., The initial value problem for certain nonlinear evolution equations, Amer. J. Math. 89 (1967), 249–259. Google Scholar

[47] 47. Strauss, W., The energy method in nonlinear partial differential equations, Notas de Mat. 47, Inst. Mat. Pura Aplic, Rio de Janeiro, 1969. Google Scholar

[48] 48. Torelli, G., Un complemento ad un teorema di J. L. Lions sulle equazioni differenziali astratte del secondo ordine, Rend. Sem. Mat. Univ. Padova, 34 (1964), 224–241. Google Scholar

[49] 49. Wilcox, C., Initial-boundary value problems for linear hyperbolic partial differential equations of the second order, Arch. Rational Mech. Anal. 10 (1962), 361–400. Google Scholar

[50] 50. Yosida, K., An operator-theoretical integration of the wave equation, J. Math. Soc. Japan 8 (1956), 79–92. Google Scholar