Differential and Integral Equations

Extremal solutions of quasilinear parabolic subdifferential inclusions

Siegfried Carl and Dumitru Motreanu

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In this paper we consider initial boundary value problems for quasilinear parabolic differential inclusions governed by general operators of Leray-Lions type and continuously perturbed subdifferentials. Our main goal is to show the existence of extremal solutions within a sector formed by appropriately defined upper and lower solutions. One of the main difficulties that arises in the treatment of the parabolic problems considered here is due to the fact that the underlying solution space does not possess lattice structure. Furthermore, the variational techniques that can be used for the corresponding stationary problems are not applicable here. The main tools used in the proof of our result are abstract results on nonlinear evolution equations, regularization, comparison and truncation techniques as well as special test function techniques.

Article information

Differential Integral Equations, Volume 16, Number 2 (2003), 241-255.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K85: Linear parabolic unilateral problems and linear parabolic variational inequalities [See also 35R35, 49J40]
Secondary: 34G25: Evolution inclusions 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35K60: Nonlinear initial value problems for linear parabolic equations


Carl, Siegfried; Motreanu, Dumitru. Extremal solutions of quasilinear parabolic subdifferential inclusions. Differential Integral Equations 16 (2003), no. 2, 241--255. https://projecteuclid.org/euclid.die/1356060687

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