Degenerate evolution equations in Hilbert space
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- by Avner Friedman and Zeev Schuss PDF
- Trans. Amer. Math. Soc. 161 (1971), 401-427 Request permission
Abstract:
We consider the degenerate evolution equation ${c_1}(t)du/dt + {c_2}(t)A(t)u = f(t)$ in Hilbert space, where ${c_1} \geqq 0,{c_2} \geqq 0,{c_1} + {c_2} > 0;A(t)$ is an unbounded linear operator satisfying the usual conditions which ensure that there is a unique solution for the Cauchy problem $du/dt + A(t)u = f(t){\rm {in}}(0,T],u(0) = {u_0}$. We prove the existence and uniqueness of a weak solution, and differentiability theorems. Applications to degenerate parabolic equations are given.References
- Gaetano Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8) 5 (1956), 1–30 (Italian). MR 89348
- Gaetano Fichera, On a unified theory of boundary value problems for elliptic-parabolic equations of second order, Boundary problems in differential equations, Univ. Wisconsin Press, Madison, Wis., 1960, pp. 97–120. MR 0111931
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
- V. P. Gluško and S. G. Kreĭn, Degenerate linear differential equations in a Banach space, Dokl. Akad. Nauk SSSR 181 (1968), 784–787 (Russian). MR 0232067
- J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of second order, Comm. Pure Appl. Math. 20 (1967), 797–872. MR 234118, DOI 10.1002/cpa.3160200410 O. A. Oleĭnik, On linear equations of the second order with a nonnegative characteristic form, Mat. Sb. 69 (111) (1966), 111-140. (Russian) MR 33 #1603.
- Claude Bardos and Haïm Brezis, Sur une classe de problèmes d’évolution non linéaires, J. Differential Equations 6 (1969), 345–394 (French). MR 242020, DOI 10.1016/0022-0396(69)90023-0
- Haïm Brezis, On some degenerate nonlinear parabolic equations, Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 28–38. MR 0273468
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 161 (1971), 401-427
- MSC: Primary 47.60; Secondary 35.00
- DOI: https://doi.org/10.1090/S0002-9947-1971-0283623-9
- MathSciNet review: 0283623