infinite dimensional hamiltonian system

An infinite dimensional Hamiltonian system takes the form

(IHS)\\;\\;\\left\\{\\begin{array}[]{rll}\\partial_{t}u-\\Delta_{x}u&=H_{v}(t,x,u,v)&%\\\\-\\partial_{t}v-\\Delta_{x}v&=H_{u}(t,x,u,v)&\\forall(t,x)\\in\\mathbb{R}\\times%\\Omega\\end{array}\\right.

where $\\Omega\\subset{\\mathbb{R}}^{N}$ is a smoothly bounded domain, $H\\in\\mathcal{C}^{1}(\\mathbb{R}\\times\\overline{\\Omega}\\times\\mathbb{R}^{2},%\\mathbb{R})$ .

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