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

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: \({\partial_V \psi^t (u{^\prime}(t)) + \partial_V \varphi(u(t)) + B(t, u(t)) \ni f(t)}\) in V*, 0 < t < T, u(0) = u 0, where \({\partial_V \psi^t, \partial_V \varphi : V \to 2^{V^*}}\) denote the subdifferential operators of proper, lower semicontinuous and convex functions \({\psi^t, \varphi : V \to (-\infty, +\infty]}\), respectively, for each \({t \in [0,T]}\), and f : (0, T) → V* and \({u_0 \in V}\) 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., \({B \equiv 0}\)) 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.

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Correspondence to Goro Akagi.

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Dedicated to Professor Mitsuharu Ôtani on the occasion of his 60th birthday

This work is supported in part by the Shibaura Institute of Technology grant for Project Research (2006, 2007, 2008, 2009), and the Grant-in-Aid for Young Scientists (B) (No. 19740073), Ministry of Education, Culture, Sports, Science and Technology.

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Akagi, G. Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces. J. Evol. Equ. 11, 1–41 (2011). https://doi.org/10.1007/s00028-010-0079-6

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Mathematics Subject Classification (2000)

  • Primary 34G25
  • Secondary 35K65

Keywords

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