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Implicit Degenerate Evolution Equations and Applications

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  • The initial-value problem is studied for evolution equations in Hilbert space of the general form d/dt A(u) + B(u) ϶ f, where and are maximal monotone operators. Existence of a solution is proved when A is a subgradient and either is strongly monotone or B is coercive; existence is established also in the case where A is strongly monotone and B is subgradient. Uniqueness is proved when one of A or B is continuous self-adjoint and the sum is strictly monotone; examples of nonuniqueness are given. Applications are indicated for various classes of degenerate nonlinear partial differential equations or systems of mixed elliptic-parabolic-pseudoparabolic types and problems with nonlocal nonlinearity.
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  • Di Benedetto, E., & Showalter, R. E. (1981). Implicit degenerate evolution equations and applications. SIAM Journal on Mathematical Analysis, 12(5), 731-751. doi:10.1137/0512062
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  • 12
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  • 5
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  • The published article can be found at the SIAM Journal on Mathematical Analysis (Society for Industrial and Applied Mathematics).
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  • This research wassponsored in part by the United States Army under contracts DAAG29-75-C-0024 and DAAG29-80-C-0041. This material is based upon work supported by the National Science Foundation under grantsMCS78-09525 A01 and MCS75-07870 A01.
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