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Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Banach spaces. (English) Zbl 1115.34059
The authors prove the existence of solutions of the Cauchy problem for the doubly nonlinear evolution equation $\frac{d}{dt}v(t)+\partial_{v}\phi^{t}(u(t))\ni f(t), \;v(t)\in\partial_{H}\psi(u(t)), \;0<t<T,$ where $$\partial_{H}\psi$$ (respectively, $$\partial_{v}\phi^{t}$$) denotes the subdifferential operator of a proper lower semicontinuous functional $$\psi$$ (respectively, $$\phi^{t}$$ explicitly depending on $$t$$) from a Hilbert space $$H$$ (respectively, reflexive Banach space $$V$$) into $$(-\infty,+\infty]$$ and $$f$$ is a given function. The proofs are based upon approximation problems in $$H$$ and a couple of energy inequalities. The initial boundary value problem of a non-autonomous degenerate elliptic-parabolic problem is treated as an application to partial differential equations.

##### MSC:
 34G25 Evolution inclusions 35F25 Initial value problems for nonlinear first-order PDEs 35K65 Degenerate parabolic equations
##### Keywords:
elliptic-parabolic problem
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##### References:
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