On A Class Of Doubly Nonlinear Evolution Equations

Colli, Pierluigi; Visintin, Augusto

doi:10.1080/03605309908820706

Cited by 165 publications

(174 citation statements)

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“…Note that existence results for this class of equation have been obtained in [3,7,8] for more restrictive vector field a. A work is in progress in this direction.…”

Section: Conclusion and Possible Extensionsmentioning

confidence: 99%

Numerical analysis of nonlinear elliptic-parabolic equations

Maitre

2002

ESAIM: M2AN

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Abstract. This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). We introduce a new and simple algorithm based on Halpern's iteration for nonexpansive operators (Bauschke, 1996;Halpern, 1967;Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995;Kačur, 1999).Mathematics Subject Classification. 65M12, 35K65, 35K55, 65N22.

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“…Note that existence results for this class of equation have been obtained in [3,7,8] for more restrictive vector field a. A work is in progress in this direction.…”

Section: Conclusion and Possible Extensionsmentioning

confidence: 99%

Numerical analysis of nonlinear elliptic-parabolic equations

Maitre

2002

ESAIM: M2AN

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“…Using an implicit time scheme such as (1.5) to solve doubly nonlinear evolutions in reflexive Banach spaces has been carried out with great success; see [2,4,11,12,16,27,31]. In our view, the main insight that makes this approach work is a certain compactness feature of weak solutions that we now explore.…”

Section: Weak Solutionsmentioning

confidence: 99%

A doubly nonlinear evolution for the optimal Poincaré inequality

Hynd

Lindgren

2016

Calc. Var.

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We study the large time behavior of solutions of the PDE |v t | p−2 v t = ∆ p v. A special property of this equation is that the Rayleigh quotient Ω |Dv(x, t)| p dx/ Ω |v(x, t)| p dx is nonincreasing in time along solutions. As t tends to infinity, this ratio converges to the optimal constant in Poincaré's inequality. Moreover, appropriately scaled solutions converge to a function for which equality holds in this inequality. An interesting limiting equation also arises when p tends to infinity, which provides a new approach to approximating ground states of the infinity Laplacian.

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“…We already know that the weak equation (6.2) is equivalent to the PDE the mutually orthogonal spaces 8) and the functionals…”

Section: Single-scale Formulation (Homogenization)mentioning

confidence: 99%

“…The homogenization of several other quasilinear equations may also be studied, including doubly-nonlinear systems of the form 6) with α and γ as above. Existence of a solution for an associated boundary-and initial-value problem was proved in [8].…”

Section: Introductionmentioning

confidence: 99%

Variational approach to homogenization of doubly-nonlinear flow in a periodic structure

Nandakumaran

Visintin

2015

Nonlinear Analysis: Theory, Methods & Applications

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This work deals with the homogenization of an initial-and boundary-value problem for the doubly-nonlinear systemHere ε is a positive parameter, and the prescribed mappings α and γ are maximal monotone with respect to the first variable and periodic with respect to the second one. The inclusions (0.2) and (0.3) are here formulated as null-minimization principles, via the theory of Fitzpatrick [MR 1009594]. As ε → 0, a two-scale formulation is derived via Nguetseng's notion of two-scale convergence, and a (single-scale) homogenized problem is then retrieved. (2000): 35B27, 35K60, 49J40, 78M40. AMS Classification

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On A Class Of Doubly Nonlinear Evolution Equations

Cited by 165 publications

References 10 publications

Numerical analysis of nonlinear elliptic-parabolic equations

Numerical analysis of nonlinear elliptic-parabolic equations

A doubly nonlinear evolution for the optimal Poincaré inequality

Variational approach to homogenization of doubly-nonlinear flow in a periodic structure

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