Are there any existence results for parabolic PDE of the type $$u_t - Au = f$$ in some Gelfand triple setting ($V \subset H \subset V^*$) with $A$ an operator that it is not quite monotone but close: $$(Au-Av, u-v) \geq -C|u-v|_{H}^p$$ and satisfies some coercivity and hemicontinuity conditions, etc?
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$\begingroup$ Are you thinking of a linear $A$? Should $V$ be a Hilbert space as in the usual notion of Gel'fand triple? $\endgroup$– Delio MugnoloFeb 18, 2014 at 8:34
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$\begingroup$ @DelioM. $A$ is nonlinear for what I am thinking. $V$ can be taken to be a Hilbert space if necessary.. I thought Banach was enough but if not let it be Hilbert. $\endgroup$– weasdFeb 18, 2014 at 9:13
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What you need is the theory of subdifferentials of convex lower semicontinuous functionals. You can find a lot in these nice lecture notes, chapter 8.
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$\begingroup$ Thank you for these. In the existence proof in chapter 8, the convergence $u_m' \rightharpoonup u'$ in $L^2(0,T;H)$ is used to identify $\chi = \epsilon'(u)$. Unfortunately I don't have this convergence of the time deriviatives at all. Wonder if you have any idea how to get round this? $\endgroup$– weasdFeb 20, 2014 at 12:04
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$\begingroup$ I am not sure to understand. That convergence is automatic if your functional satisfies the assumption of that theorem (coercivity etc.). And those assumptions are of course formulated thinking of a relatively general parabolic setting. How comes that that convergence fails? Do you have an oscillating behaviour as $m\to\infty$? $\endgroup$ Feb 21, 2014 at 9:59
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$\begingroup$ In my case I do not have all the assumptions of that theorem. I was hoping to modify the proof of the theorem for my setting. In my case I have only the monotonicity property I mentioned above and my operator $A$ is not gradient system like in that notes, and so I am unable to find a good bound on the $u_m'$.. $\endgroup$– weasdFeb 21, 2014 at 19:12