Degenerate evolution equations in hilbert space

Avner Friedmano; Zeev Schuss

doi:10.1090/S0002-9947-1971-0283623-9

Degenerate evolution equations in hilbert space

Avner Friedmano, Zeev Schuss

Northwestern University

Research output: Contribution to journal › Article › peer-review

18 Scopus citations

Abstract

We consider the degenerate evolution equation Cx(t) du_ldt + c₂(t)A(t)u =f(t) in Hilbert space, where c₁, c₂, c₁+ c₂>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 -f A(t)u = fit) in (0, T), u(0) = u0. We prove the existence and uniqueness of a weak solution, and differentiability theorems. Applications to degenerate parabolic equations are given.

Original language	English
Pages (from-to)	401-427
Number of pages	27
Journal	Transactions of the American Mathematical Society
Volume	161
DOIs	https://doi.org/10.1090/S0002-9947-1971-0283623-9
State	Published - Nov 1971
Externally published	Yes

Keywords

Cauchy problem
Degenerate equation
Differentiability of solutions
Evolution equations
Existence
Parabolic equations
Uniqueness
Weak solution

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10.1090/S0002-9947-1971-0283623-9

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title = "Degenerate evolution equations in hilbert space",

abstract = "We consider the degenerate evolution equation Cx(t) duldt + c2(t)A(t)u =f(t) in Hilbert space, where c1, c2, c1+ c2>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 -f A(t)u = fit) in (0, T), u(0) = u0. We prove the existence and uniqueness of a weak solution, and differentiability theorems. Applications to degenerate parabolic equations are given.",

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N2 - We consider the degenerate evolution equation Cx(t) duldt + c2(t)A(t)u =f(t) in Hilbert space, where c1, c2, c1+ c2>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 -f A(t)u = fit) in (0, T), u(0) = u0. We prove the existence and uniqueness of a weak solution, and differentiability theorems. Applications to degenerate parabolic equations are given.

AB - We consider the degenerate evolution equation Cx(t) duldt + c2(t)A(t)u =f(t) in Hilbert space, where c1, c2, c1+ c2>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 -f A(t)u = fit) in (0, T), u(0) = u0. We prove the existence and uniqueness of a weak solution, and differentiability theorems. Applications to degenerate parabolic equations are given.

KW - Cauchy problem

KW - Degenerate equation

KW - Differentiability of solutions

KW - Evolution equations

KW - Existence

KW - Parabolic equations

KW - Uniqueness

KW - Weak solution

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JO - Transactions of the American Mathematical Society

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Degenerate evolution equations in hilbert space

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Keywords

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