zbMATH — the first resource for mathematics

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.

34G25 Evolution inclusions
35F25 Initial value problems for nonlinear first-order PDEs
35K65 Degenerate parabolic equations
Full Text: DOI
[1] Akagi, G.; Ôtani, M., Time-dependent constraint problems arising from macroscopic critical-state models for type-II superconductivity and their approximations, Adv. math. sci. appl., 14, 683-712, (2004) · Zbl 1088.35085
[2] Alt, H.W.; Luckhaus, S., Quasilinear elliptic – parabolic differential equations, Math. Z., 183, 311-341, (1983) · Zbl 0497.35049
[3] Barbu, V., Nonlinear semigroups and differential equations in Banach spaces, (1976), Noordhoff Leiden
[4] Barbu, V., Existence for nonlinear Volterra equations in Hilbert spaces, SIAM J. math. anal., 10, 552-569, (1979) · Zbl 0462.45021
[5] Brézis, H., Operateurs maximaux monotones et semi-groupes de contractions dans LES espaces de Hilbert, Math. studies, vol. 5, (1973), North-Holland Amsterdam/New York · Zbl 0252.47055
[6] Brézis, H., Monotonicity methods in Hilbert spaces and some applications to non-linear partial differential equations, (), 101-156
[7] DiBenedetto, E.; Showalter, R.E., Implicit degenerate evolution equations and applications, SIAM J. math. anal., 12, 731-751, (1981) · Zbl 0477.47037
[8] Gajewski, H.; Skrypnik, I.V., To the uniqueness problem for nonlinear parabolic equations. partial differential equations and applications, Discrete contin. dyn. syst., 10, 315-336, (2004) · Zbl 1060.35053
[9] Kenmochi, N., Some nonlinear parabolic variational inequalities, Israel J. math., 22, 304-331, (1975) · Zbl 0327.49004
[10] Kenmochi, N., Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. fac. educ. chiba univ., 30, 1-87, (1981)
[11] Kenmochi, N.; Pawlow, I., A class of nonlinear elliptic – parabolic equations with time-dependent constraints, Nonlinear anal., 10, 1181-1202, (1986) · Zbl 0635.35043
[12] Maitre, E.; Witomski, P., A pseudo-monotonicity adapted to doubly nonlinear elliptic – parabolic equations, Nonlinear anal., 50, 223-250, (2002) · Zbl 1001.35091
[13] Shirakawa, K., Large time behavior for doubly nonlinear systems generated by subdifferentials, Adv. math. sci. appl., 10, 417-442, (2000) · Zbl 0960.34050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.