TY - JOUR

T1 - Doubly nonlinear evolution equations with non-monotone perturbations in reflexive banach spaces

AU - Akagi, Goro

N1 - Funding Information:
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. Mathematics Subject Classification (2000): Primary 34G25; Secondary 35K65 Keywords: Doubly nonlinear evolution equation, Subdifferential, Non-monotone perturbation, Reflexive Banach space, Fixed point theorem.

PY - 2011/3

Y1 - 2011/3

N2 - 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.

AB - 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.

KW - Doubly nonlinear evolution equation

KW - Fixed point theorem

KW - Non-monotone perturbation

KW - Reflexive banach space

KW - Subdifferential

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U2 - 10.1007/s00028-010-0079-6

DO - 10.1007/s00028-010-0079-6

M3 - Article

AN - SCOPUS:79959781841

SN - 1424-3199

VL - 11

SP - 1

EP - 41

JO - Journal of Evolution Equations

JF - Journal of Evolution Equations

IS - 1

ER -