## Abstract

We prove the existence of solutions of the Cauchy problem for the doubly nonlinear evolution equation: d v (t) / d t + ∂_{V} φ^{t} (u (t)) ∋ f (t), v (t) ∈ ∂_{H} ψ (u (t)), 0 < t < T, where ∂_{H} ψ (respectively, ∂_{V} φ^{t}) denotes the subdifferential operator of a proper lower semicontinuous functional ψ (respectively, φ^{t} explicitly depending on t) from a Hilbert space H (respectively, reflexive Banach space V) into (- ∞, + ∞] and f is given. To do so, we suppose that V {right arrow, hooked} H ≡ H^{*} {right arrow, hooked} V^{*} compactly and densely, and we also assume smoothness in t, boundedness and coercivity of φ^{t} in an appropriate sense, but use neither strong monotonicity nor boundedness of ∂_{H} ψ. The method of our proof relies on approximation problems in H and a couple of energy inequalities. We also treat the initial-boundary value problem of a non-autonomous degenerate elliptic-parabolic problem.

Original language | English |
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Pages (from-to) | 32-56 |

Number of pages | 25 |

Journal | Journal of Differential Equations |

Volume | 231 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 Dec 1 |

Externally published | Yes |

## Keywords

- Doubly nonlinear evolution equation
- Elliptic-parabolic problem
- Reflexive Banach space
- Subdifferential

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics