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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Degenerate evolution equations in Hilbert space

Authors: Avner Friedman and Zeev Schuss
Journal: Trans. Amer. Math. Soc. 161 (1971), 401-427
MSC: Primary 47.60; Secondary 35.00
MathSciNet review: 0283623
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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.

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Keywords: Evolution equations, degenerate equation, Cauchy problem, weak solution, existence, uniqueness, differentiability of solutions, parabolic equations
Article copyright: © Copyright 1971 American Mathematical Society