Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces
type: Item
instance of: scholarly article
title: Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces
zbMATH Open document ID: 1242.35156
author: Goro Akagi
published in: Journal of Evolution Equations
publication date: 2012-01-13T00:00:00Z
review text: The paper is concerned with the Cauchy problem for a doubly nonlinear evolution equation \[ \begin{cases}\partial_V\psi^t(u'(t))+\partial_V\varphi(u(t))+B(t,u(t))\ni f(t)\,\,\,\text{in}\,\,\,V^*,\,\,\,0<t<T,\\ u(0)=u_0,\end{cases} \] where \(V\) and \(V^*\) is a reflexive Banach space and its dual space, \(V\hookrightarrow H\equiv H^*\hookrightarrow V^*\) with continuous and densily defined canonical injections, \(H\) is a Hilbert space identified with its dual space, \(\partial _V\psi^t\) and \(\partial_V\varphi:V\to 2^{V^*}\) are the subdifferential operators of proper, lower semicontinuous and convex functions \(\psi^t\) and \(\varphi\), respectively, from \(V\) into \((-\infty,+\infty]\), \(u'(t)=du(t)/dt\), \(f:(0,T)\to V^*\) and \(u_0\in V\) are given, and \(B:(0,T)\times V\to V^*\) is a possibly multi-valued mapping such that \(B(t,\cdot)\) might be non-monotone in \(V\times V^*\) for each fixed \(t\). By using a couple of approximations for the equation and a fixed-point argument with a multi-valued mapping, the author gives sufficient conditions for the local (in time) existence of strong solutions to the above problem and for the global existence. The presented framework can cover evolution equations whose solutions might blow up in finite time and whose unperturbed equations (i.e. \(B\equiv 0\)) might not be uniquely solved in a doubly nonlinear setting. Some applications of the resulting theorems to doubly nonlinear parabolic equations are also presented.
reviewed by: Rodica Luca
Mathematics Subject Classification ID: 35K90
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 5997662
zbMATH Keywords: subdifferential
zbMATH Keywords: fixed-point theorem
MaRDI profile type: Publication
full work available at URL: s00028-010-0079-6
OpenAlex ID: W2098050438
cites work: Doubly nonlinear equations with unbounded operators
cites work: Doubly nonlinear periodic problems with unbounded operators
cites work: An abstract doubly nonlinear equation with a measure as initial value
cites work: Q3128678
cites work: Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Banach spaces
cites work: Evolution inclusions governed by subdifferentials in reflexive Banach spaces
cites work: Evolution inclusions governed by the difference of two subdifferentials in reflexive Banach spaces
cites work: Q5694396
cites work: Q3209098
cites work: Phase change problems with temperature dependent constraints for the volume fraction velocities
cites work: A new class of doubly nonlinear evolution equations
cites work: Averaged norms
cites work: Existence theorems for a class of two points boundary problems
cites work: Q4094029
cites work: Existence for Nonlinear Volterra Equations in Hilbert Spaces
cites work: A nonlinear system for phase change with dissipation
cites work: Q5665783
cites work: Problèmes unilateraux
cites work: Perturbations of nonlinear maximal monotone sets in banach space
cites work: On some doubly nonlinear evolution equations in Banach spaces
cites work: On A Class Of Doubly Nonlinear Evolution Equations
cites work: Implicit Degenerate Evolution Equations and Applications
cites work: Q5573683
cites work: A generalization of Tychonoff's fixed point theorem
cites work: Elliptic partial differential equations of second order
cites work: Q4421936
cites work: Sur la résolution d'une équation et d'une inéquation parabolique non linéaires
cites work: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance
cites work: Some nonlinear parabolic variational inequalities
cites work: Q3811038
cites work: Subdifferential operator approach to the Cahn-Hilliard equation with constraint
cites work: A class of nonlinear elliptic—parabolic equations with time-dependent constraints
cites work: A pseudo-monotonicity adapted to doubly nonlinear elliptic-parabolic equations
cites work: On rate-independent hysteresis models
cites work: EXISTENCE AND UNIQUENESS RESULTS FOR A CLASS OF RATE-INDEPENDENT HYSTERESIS PROBLEMS
cites work: Q4167197
cites work: Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems
cites work: Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, periodic problems
cites work: Q4172398
cites work: Sur la résolution de certaines équations paraboliques non linéaires
cites work: Nonlinear partial differential equations with applications
cites work: Well-posedness and long-time behavior for a class of doubly nonlinear equations
cites work: Global attractor for a class of doubly nonlinear abstract evolution equations
cites work: Q3762958
cites work: Q4514465
cites work: Compact sets in the space \(L^ p(0,T;B)\)
cites work: On solutions of some doubly nonlinear degenerate parabolic equations with absorption
cites work: Q4112274
cites work: Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems
DOI: 10.1007/S00028-010-0079-6
description: scientific article; zbMATH DE number 5997662