| Abstract: |
| We consider the following double quasi-variational evolution equations governed by time-dependent subdifferentials in a uniformly convex Banach space $V^*$:
$${\rm (QP)} \ \ \partial_* \psi^t(u;u`(t)) + \partial_* \varphi^t(u; u(t)) +g(t,u(t))\ni f(t)
\ \mbox{ in } V^* \ \mbox{ for a.a.} \ t\in(0,T),$$ where $V$ is a (real) uniformly convex Banach space with the uniformly convex dual space $V^*$, $0< T< \infty $, $u`=du/dt$ in $V$, $g(t, \cdot)$ is a single-valued Lipschitz operator in $V^*$, and $f$ is a given $V^*$-valued function. The time-dependent function $ \psi^t (v;z)$ is proper, lower semi-continuous (l.s.c.), and convex in $z\in V$. Also, $\varphi^t(v;z)$ is a time-dependent, non-negative, continuous convex function in $z\in V$. Note that $(t,v) \in [0,T]\times C([0,T];H)$ is a parameter that determines the convex functions $ \psi^t (v;\cdot )$ and $\varphi^t(v;\cdot)$ on $V$. The dependence of function $v$ upon $\psi^t(v;\cdot)$ and $\varphi^t(v;\cdot)$ is, in general, allowed to be non-local. In addition, the subdifferentials $ \partial_* \psi^t (v;z) $ of $ \psi^t (v;z) $ with respect to $z\in V$ is a multivalued operator in $V^*$, and $ \partial_* \varphi^t (v;z) $ of $ \varphi^t(v;z) $ with respect to $z\in V$ is a single-valued linear operator in $V^*$. \
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In this talk, we establish the existence theory of abstract double quasi-variational evolution equations (QP). We also give some applications to nonlinear PDEs with gradient constraint for time-derivatives. \
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This is a joint work with Nobuyuki Kenmochi (ICM, University of Warsaw, Warsaw, Poland) and Ken Shirakawa (Chiba University, Chiba, Japan). |
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