Doubly nonlinear evolution equations with non-monotone perturbations in reflexive banach spaces

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)

Abstract

Let V and V* be a real reflexive Banach space and its dual space, respectively. This paper is devoted to the abstract Cauchy problem for doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations of the form: ∂V ψt (u'(t))+∂V (u(t))+ B(t, u(t)) e{cyrillic} f (t)in V*, 0 < t < T, u(0) = u0, where ∂V ψt, ∂V : V → 2V * denote the sub differential operators of proper, lower semicontinuous and convex functions ψt, : V → (-∞,+∞], respectively, for each t j∈ [0, T ], and f: (0, T) → V* and u0 j∈ V are given data. Moreover, let B be a (possibly) multi-valued operator from (0, T)×V into V*. We present sufficient conditions for the local (in time) existence of strong solutions to the Cauchy problem as well as for the global existence. Our framework can cover evolution equations whose solutions might blow up in finite time and whose unperturbed equations (i.e., B ≡ 0) might not be uniquely solved in a doubly nonlinear setting. Our proof relies on a couple of approximations for the equation and a fixed point argument with a multi-valued mapping. Moreover, the preceding abstract theory is applied to doubly nonlinear parabolic equations.

Original languageEnglish
Pages (from-to)1-41
Number of pages41
JournalJournal of Evolution Equations
Volume11
Issue number1
DOIs
Publication statusPublished - 2011 Mar
Externally publishedYes

Keywords

  • Doubly nonlinear evolution equation
  • Fixed point theorem
  • Non-monotone perturbation
  • Reflexive banach space
  • Subdifferential

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

Fingerprint

Dive into the research topics of 'Doubly nonlinear evolution equations with non-monotone perturbations in reflexive banach spaces'. Together they form a unique fingerprint.

Cite this