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Degenerate evolution equations in Hilbert space

1971; American Mathematical Society; Volume: 161; Linguagem: Inglês

10.1090/s0002-9947-1971-0283623-9

1088-6850

Avner Friedman, Z. Schuss,

Differential Equations and Boundary Problems

We consider the degenerate evolution equation c 1 ( t ) d u / d t + c 2 ( t ) A ( t ) u = f ( t ) {c_1}(t)du/dt + {c_2}(t)A(t)u = f(t) in Hilbert space, where c 1 ≧ 0 , c 2 ≧ 0 , c 1 + c 2 > 0 ; A ( t ) {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 d u / d t + A ( t ) u = f ( t ) i n ( 0 , T ] , u ( 0 ) = u 0 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.