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

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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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Authors and Affiliations

  1. Department of Machinery and Control Systems, School of Systems Engineering and Science, Shibaura Instituteof Technology, 307 Fukasaku, Minuma-ku, Saitama-shi, Saitama, 330-8570, Japan

    Goro Akagi

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  1. Goro Akagi
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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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