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New Class of Doubly Nonlinear Evolution Equations Governed by Time-Dependent Subdifferentials

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Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs

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Abstract

We discuss a new class of doubly nonlinear evolution equations governed by time-dependent subdifferentials in uniformly convex Banach spaces, and establish an abstract existence result of solutions. Also, we show non-uniqueness of solution, giving some examples. Moreover, we treat a quasi-variational doubly nonlinear evolution equation by applying this result extensively, and give some applications to nonlinear PDEs with gradient constraint for time-derivatives.

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Acknowledgements

This work is dedicated to Professor Gianni Gilardi on the occasion of his 70th birthday and it is supported by Grant-in-Aid for Scientific Research (C), No. 26400179 and 16K05224, JSPS. The authors express their gratitude to an anonymous referee for reviewing the original manuscript and for many valuable comments that helped clarify and refine this paper.

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Correspondence to Noriaki Yamazaki .

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Kenmochi, N., Shirakawa, K., Yamazaki, N. (2017). New Class of Doubly Nonlinear Evolution Equations Governed by Time-Dependent Subdifferentials. In: Colli, P., Favini, A., Rocca, E., Schimperna, G., Sprekels, J. (eds) Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs. Springer INdAM Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-64489-9_11

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