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Existence Theorems for Some Non-Linear Equations of Evolution

Published online by Cambridge University Press:  20 November 2018

John C. Clements
John C. Clements*
Affiliation:
University of Toronto, Toronto, Ontario University of Washington, Seattle, Washington
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In recent years considerable attention has been focused on non-linear hyperbolic differential equations with the object of establishing the existence of global solutions. It is our aim here to establish the existence of weak solutions of boundary value problems for non-linear equations of the form

(1-1)

where d is a real constant called the damping coefficient, u(t) is a vector-valued function defined on a subinterval of the real line into a space of complex-valued functions u(x) defined on a bounded domain Ω in the real Euclidean space EN of N dimensions, ut(t) ≡ du(t)/dt, and A(t) is the family of partial differential operators of order 2m (m = 1, 2, …) on Ω given in generalized divergence form by

(1-2)

with

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 4 , 01 August 1970 , pp. 726 - 745
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Browder, F. E., On non-linear wave equations, Math. Z. 80 (1962), 249–264.Google Scholar
2. Browder, F. E., Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc 69 (1963), 862–874.Google Scholar
3. Browder, F. E., Non-linear parabolic boundary value problems of arbitrary order, Bull. Amer. Math. Soc. 69 (1963), 858–861.Google Scholar
4. Browder, F. E., Existence and uniqueness theorems for solutions of non-linear boundary value problems, Proc. Sympos. Appl. Math., Vol. 17, pp. 24–49 (Amer. Math. Soc, Providence, R.I., 1965).Google Scholar
5. Browder, F. E., Existence of periodic solutions for non-linear equations of evolution, Proc. Nat. Acad. Sci. 58 (1963), 1100–1103.Google Scholar
6. Cesari, L., Existence in the large of periodic solutions of hyperbolic partial differential equations, Arch. Rational Mech. Anal. 20 (1965), 170–190.Google Scholar
7. Cesari, L., Smoothness properties of periodic solutions in the large of nonlinear hyperbolic differential systems, Funkcial. Ekvac. 9 (1966), 325–338.Google Scholar
8. Diaz, J. B. and Ludford, G. S. S., On the singular Cauchy problem for a generalization of the Euler-Poisson-Darboux equation in two space variables, Ann. Mat. Pura Appl. (4) 38 (1955), 33–50.Google Scholar
9. Ficken, F. A. and Fleishman, B. A., Initial value and time-periodic solutions for a non-linear wave equation, Comm. Pure Appl. Math. 10 (1957), 331–356.Google Scholar
10. Hale, J. K., Periodic solutions of a class of hyperbolic equations, Arch. Rational Mech. Anal. 28 (1967), 380–398.Google Scholar
11. Hille, E. and Phillips, R. S., Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., Vol. 31, rev. éd. (Amer. Math. Soc, Providence, R.I., 1957).Google Scholar
12. K., Jörgens, Das Anfangswertproblem im Grossen fur eine Klasse nichtlinearer Wellengleichungen, Math. Z. 77 (1961), 295–308.Google Scholar
13. Keller, J. B., Electrodynamics. I. The equilibrium of a charged gas in a container, J. Rational Mech. Anal. 5 (1956), 715–724.Google Scholar
14. Keller, J. B., On solutions of nonlinear wave equations, Comm. Pure Appl. Math. 10 (1957), 523–530.Google Scholar
15. Lindsay, R. B., Mechanical radiation (McGraw-Hill, New York, 1960).Google Scholar
16. J.-L., Lions, Equations différentielles opérationnelles et problèmes aux limites, Die Grundlehrender mathematischen Wissenschaften, Bd. III (Springer-Verlag, Berlin, 1961).Google Scholar
17. J.-L., Lions and Strauss, W. A., Some non-linear evolution equations, Bull. Soc Math. France 98 (1965), 43–96.Google Scholar
18. Minty, G. J., Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341–346.Google Scholar
19. Minty, G. J., Qn a “monotonicity” method for the solution of non-linear equations in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 1038–1041.Google Scholar
20. Nirenberg, L., Estimates and existence of solutions of elliptic equations, Comm. Pure Appl. Math. 9 (1956), 509–530.Google Scholar
21. Prodi, G., Soluzioni periodiche di equazioni a derivate parziali di tipo iperbolico non lineari, Ann. Mat. Pura Appl. (4) 42 (1956), 25–49.Google Scholar
22. Rabinowitz, P. H., Periodic solutions of non-linear hyperbolic partial differential equations, Comm. Pure Appl. Math. 20 (1967), 145–205.Google Scholar
23. Rabinowitz, P. H., Periodic solutions of non-linear hyperbolic partial differential equations. II, Comm. Pure Appl. Math. 22 (1969), 15–39.Google Scholar
24. 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
25. Wilcox, C. H., Initial-boundary value problems for linear hyperbolic partial differential equations of the second order, Arch. Rational Mech. Anal. 10 (1962), 361–400.Google Scholar