Quasilinear evolution equations in Banach spaces

Murphy, Michael G.

doi:10.1090/S0002-9947-1980-0567096-X

Quasilinear evolution equations in Banach spaces
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Trans. Amer. Math. Soc. 259 (1980), 547-557 Request permission

Abstract:

This paper is concerned with the quasi-linear evolution equation $u’(t) + A(t, u(t))u(t) = 0$ in $[0, T], u(0) = {x_0}$ in a Banach space setting. The spirit of this inquiry follows that of T. Kato and his fundamental results concerning linear evolution equations. We assume that we have a family of semigroup generators that satisfies continuity and stability conditions. A family of approximate solutions to the quasi-linear problem is constructed that converges to a “limit solution.” The limit solution must be the strong solution if one exists. It is enough that a related linear problem has a solution in order that the limit solution be the unique solution of the quasi-linear problem. We show that the limit solution depends on the initial value in a strong way. An application and the existence aspect are also addressed.

References

Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York, Inc., New York, 1967. MR 0230022, DOI 10.1007/978-3-642-46066-1
Paul R. Chernoff and Jerome A. Goldstein, Admissible subspaces and the denseness of the intersection of the domains of semigroup generators, J. Functional Analysis 9 (1972), 460–468. MR 0298465, DOI 10.1016/0022-1236(72)90021-3
Michael G. Crandall and L. C. Evans, On the relation of the operator $\partial /\partial s+\partial /\partial \tau$ to evolution governed by accretive operators, Israel J. Math. 21 (1975), no. 4, 261–278. MR 390853, DOI 10.1007/BF02757989
J. R. Dorroh, A class of nonlinear evolution equations in a Banach space, Trans. Amer. Math. Soc. 147 (1970), 65–74. MR 253085, DOI 10.1090/S0002-9947-1970-0253085-5
J. R. Dorroh, A simplified proof of a theorem of Kato on linear evolution equations, J. Math. Soc. Japan 27 (1975), no. 3, 474–478. MR 380504, DOI 10.2969/jmsj/02730474
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Tosio Kato, Integration of the equation of evolution in a Banach space, J. Math. Soc. Japan 5 (1953), 208–234. MR 58861, DOI 10.2969/jmsj/00520208
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Tosio Kato, Linear evolution equations of “hyperbolic” type, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970), 241–258. MR 279626
Tosio Kato, Linear evolution equations of “hyperbolic” type. II, J. Math. Soc. Japan 25 (1973), 648–666. MR 326483, DOI 10.2969/jmsj/02540648
Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975, pp. 25–70. MR 0407477
Tosio Kato and Hiroki Tanabe, On the abstract evolution equation, Osaka Math. J. 14 (1962), 107–133. MR 140954
Tosio Kato and Hiroki Tanabe, On the analyticity of solution of evolution equations, Osaka Math. J. 4 (1967), 1–4. MR 222430
Yoshikazu Kobayashi, Difference approximation of Cauchy problems for quasi-dissipative operators and generation of nonlinear semigroups, J. Math. Soc. Japan 27 (1975), no. 4, 640–665. MR 399974, DOI 10.2969/jmsj/02740640
Yukio K\B{o}mura, Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan 19 (1967), 493–507. MR 216342, DOI 10.2969/jmsj/01940493
G. E. Ladas and V. Lakshmikantham, Differential equations in abstract spaces, Mathematics in Science and Engineering, Vol. 85, Academic Press, New York-London, 1972. MR 0460832

Equations of parabolic type in a Banach space

49

Hiroki Tanabe, A class of the equations of evolution in a Banach space, Osaka Math. J. 11 (1959), 121–145. MR 114137
Hiroki Tanabe, Remarks on the equations of evolution in a Banach space, Osaka Math. J. 12 (1960), 145–166. MR 125454
Hiroki Tanabe, On the equations of evolution in a Banach space, Osaka Math. J. 12 (1960), 363–376. MR 125455

Functional analysis