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Existence Theorems for Some Weak Abstract Variable Domain Hyperbolic Problems

Published online by Cambridge University Press:  20 November 2018

Robert Carroll  and
Emile State
Robert Carroll
Affiliation:
University of Illinois, Urbana, Illinois
Emile State
Affiliation:
University of Western Ontario, London, Ontario
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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).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 4 , August 1971 , pp. 611 - 626
Copyright
Copyright © Canadian Mathematical Society 1971

References

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. 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. 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. Brezis, H. and Lions, J., Sur certains problèmes unilatéraux hyperboliques, Comptes Rendus, Acad. Sci. Paris 264 (1967), 928–931.Google Scholar
5. Browder, F., Nonlinear maximal monotone operators in Banach space, Math. Annalen 175 (1968), 89–113.Google Scholar
6. Browder, F., Qn nonlinear wave equations, Math. Z. 80 (1962), 249–264.Google Scholar
7. Browder, F. and Strauss, W., Scattering for nonlinear wave equations, Pacific J. Math. 13 (1963), 23–43.Google Scholar
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. Carroll, R., Abstract methods in partial differential equations (Harper-Row, New York, 1969).Google Scholar
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. Carroll, R., On the propagator equation, Illinois J. Math. 11 (1967), 506–527.Google Scholar
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. 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. 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. 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. Carroll, R. and Cooper, J., Remarks on some variable domain problems in abstract evolution equations, Math. Annalen 188 (1970), 143–164.Google Scholar
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. 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. 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. Cooper, J., Some aspects of abstract linear evolution equations in Hilbert space, Ph.D. Thesis, University of Illinois, 1967.Google Scholar
21. Cooper, J., Evolution equations in Banach space with variable domain (to appear).Google Scholar
22. Cooper, J., Un problème relatif aux équations d'évolution abstraites, Comptes Rendus, Acad. Sci. Paris 268 (1967), 422–424.Google Scholar
23. Cooper, J., Two point problems for abstract evolution equations (to appear).Google Scholar
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. Goldstein, J., Time dependent hyperbolic equations, J. Functional Analysis 4 (1969), 31–49.Google Scholar
26. Goldstein, J., Semigroups and second order differential equations, J. Functional Analysis 4 (1969), 50–70.Google Scholar
27. Jörgens, K., Das Anfangswertproblem im Grossen fur eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77 (1961), 295–308.Google Scholar
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. Krein, S., Linear differential equations in Banach spaces, Izd. Nauka, Glav. Red. Fis.-Mat. Lit., Moscow, 1967.Google Scholar
30. Lax, P. and Phillips, R., Scattering theory (Academic Press, New York, 1967).Google Scholar
31. Lee, K., A mixed problem for hyperbolic equations with time dependent domain, Jour. Math. Anal. Appl. 16 (1966), 455–471.Google Scholar
32. Lions, J., Equations différentielles-opérationnelles (Springer, Berlin, 1961).Google Scholar
33. Lions, J., Quelques méthodes de résolution des problèmes aux limites nonlinéaires (Dunod, Paris, 1969).Google Scholar
34. Lions, J., Une remarque sur les applications du théorème de Hille-Yosida, J. Math. Soc. Japan 9 (1957), 62–70.Google Scholar
35. Lions, J. and Magenes, E., Problèmes aux limites nonhomogènes et applications. Vols. 1, 2, 3 (Dunod, Paris, 1968 and 1970).Google Scholar
36. Lions, J. and Strauss, W., Some nonlinear evolution equations, Bull. Soc. Math. France 93 (1965), 43–96.Google Scholar
37. Mazumdar, T., Existence theorems for noncoercive variable domain evolution problems, Ph.D. Thesis, University of Illinois, 1971.Google Scholar
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. 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. 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. 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. 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. Segal, I., Nonlinear semigroups, Ann. of Math. 78 (1963), 339–364.Google Scholar
44. State, E., Weak hyperbolic problems with variable domains, Ph.D. Thesis, University of Illinois, 1971.Google Scholar
45. Strauss, W., Scattering for hyperbolic equations, Trans. Amer. Math. Soc. 108 (1963), 13–37.Google Scholar
46. Strauss, W., The initial value problem for certain nonlinear evolution equations, Amer. J. Math. 89 (1967), 249–259.Google Scholar
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. 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. 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. Yosida, K., An operator-theoretical integration of the wave equation, J. Math. Soc. Japan 8 (1956), 79–92.Google Scholar