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Nonlinear evolution equations generated by subdifferentials with nonlocal constraints

Tom 86 / 2009

Risei Kano, Yusuke Murase, Nobuyuki Kenmochi Banach Center Publications 86 (2009), 175-194 MSC: Primary 35K45; Secondary 35K50. DOI: 10.4064/bc86-0-11

Streszczenie

We consider an abstract formulation for a class of parabolic quasi-variational inequalities or quasi-linear PDEs, which are generated by subdifferentials of convex functions with various nonlocal constraints depending on the unknown functions. In this paper we specify a class of convex functions $\{\varphi^t(v;\cdot)\}$ on a real Hilbert space $H$, with parameters $0\le t \le T$ and $v$ in a set of functions from $[-\delta_0,T]$, $0<\delta_0< \infty,$ into $H$, in order to formulate an evolution equation of the form $$ u'(t)+\partial \varphi^t(u;u(t)) \ni f(t),\, 0< t < T,\hbox{ in }H.$$ Our objective is to discuss the existence question for the associated Cauchy problem.

Autorzy

  • Risei KanoDepartment of Mathematics
    Graduate School of Science and Technology
    Chiba University
    1-33 Yayoi-cho, Inage-ku
    Chiba, 263-8522 Japan
    e-mail
  • Yusuke MuraseDepartment of Mathematics
    Graduate School of Science and Technology
    Chiba University
    1-33 Yayoi-cho, Inage-ku
    Chiba, 263-8522 Japan
    e-mail
  • Nobuyuki KenmochiDepartment of Mathematics
    Faculty of Education, Chiba University
    1-33 Yayoi-cho, Inage-ku
    Chiba, 263-8522 Japan
    e-mail

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