Evolution equations generated by subdifferentials in the dual space of $(H^1(\Omega))$
Alain Damlamian, Nobuyuki Kenmochi · Discrete and Continuous Dynamical Systems · 1999
This paper is concerned with the subdifferential operator approachto nonlinear (possibly degenerate and singular) parabolic PDE's of the form $u_t-\Delta \beta(u) i f$ formulated in the dual space of $H^1(\Omega)$, where $\beta$ is a maximal monotonegraph in $\mathbf R\times \mathbf R$. In the set-up considered so far [8], some coerciveness condition hasbeen required for $\beta$, corresponding at least to the fact that it is onto $\mathbf R$. In thepresent paper, we show that the subdifferential operator approach is possible for anymaximal monotone graph $\beta$ without any growth condition.