Advances in Differential Equations

On mild and weak solutions of elliptic-parabolic problems

Philippe Benilan and Petra Wittbold

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We consider an elliptic-parabolic equation in divergence form $b(v)_t = \text{div}\alpha(v, Dv)+ f$ with Dirichlet boundary conditions and initial condition. Under rather general assumptions, we prove existence of mild solutions satisfying an $L^1$-comparison principle; under some additional conditions, these solutions are shown to be weak solutions. Moreover, under the general assumptions, uniqueness of integral solutions is established; under certain conditions, we show that weak solutions are integral solutions. The notions of mild and integral solutions are derived from nonlinear semigroup theory; by this approach, we extend and make precise former results on existence and uniqueness of weak solutions.

Article information

Adv. Differential Equations, Volume 1, Number 6 (1996), 1053-1073.

First available in Project Euclid: 25 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35M10: Equations of mixed type
Secondary: 35D05 35K60: Nonlinear initial value problems for linear parabolic equations


Benilan, Philippe; Wittbold, Petra. On mild and weak solutions of elliptic-parabolic problems. Adv. Differential Equations 1 (1996), no. 6, 1053--1073.

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