Let $ E \subset H \subset E^{*}$ , where $ E$ is a real Banach space, $ E^{*}$ its dual and $ H$ is a real Hilbert space. Embendings are continuous and dense. Let $ \langle v_1,v_2\rangle$ be a dual pair for $ v_1 \in E^{*}, v_2 \in E$ .
An operator (nonlinear, in general) $ A \colon E \to E^{*}$ is called strongly $ \alpha$ –monotone ($ \alpha > 1$ ) if there is a constant $ M>0$ such that $ $ \langle Au-Av,u-v\rangle \geq M \|u-v\|^{\alpha}_{H}.$ $
Consider the corresponding evolution equation $ $ \frac{d}{dt}u+Au=0$ $
Are there any natural examples of equations with such operators? I know a good example for monotone condition (i.e. the above inequality with $ M=0$ ) is the Laplace operator and non-linear term $ |u|^{p-2}u$ in the nonlinear Schrödinger equation. But there are many papers I’ve seen (mainly on almost periodic functions), where the strong monotonicity condition is used, but no useful example of such operators is given.
The same question stands for finite dimensional spaces, i. e. for the case $ E=H=E^{*} = \mathbb{R}^n$ and $ \langle\cdot,\cdot\rangle$ is just scalar product. Are there any known natural examples?
