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I read "SADDLE POINTS AND INSTABILITY OF NONLINEAR HYPERBOLIC EQUATIONS" Payne, Sattinger. I have hyperbolic equation:

\begin{equation} \label{eq:intro_main_eq} \begin{cases} \tag{A} w_{tt}=\Delta w + f(w), x\in \Omega \\ w\bigr{|}_{\partial \Omega}=0 \\ u(x,0) = u_0(x), u_t(x,0) = v_0(x) \end{cases} \end{equation} $ w\in W_0^{1,2}, \Omega\in\mathbb{R}^n $.

In fourth part there is energy functional, that decreasing: $ E(t)=\frac{1}{2} (|u_t|_{L_2}^2+|\nabla u|_{L_2}^2)-\int_D F(u) $, i.e. $\forall t_1, t_2: t_1<t_2<T \Rightarrow E(t_1)\ge E(t_2)$; where $F(u)=\int_0^u f(s)ds$

How can I prove, that this functional is decreasing?

PS To be precise, I can't understand why $\int_{\partial\Omega} u_t \frac{\partial u}{\partial n} $ should be less zero or equal. Or what is my mistake?

Thank you!

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  • $\begingroup$ If $u$ vanishes on the boundary at all times, then so does $u_t$... $\endgroup$ – Anthony Carapetis Feb 20 '17 at 12:00
  • $\begingroup$ And energy is constant zero? Is that right? $\endgroup$ – Student Feb 20 '17 at 12:43
  • $\begingroup$ Could someone help me? $\endgroup$ – Student Feb 20 '17 at 16:51
  • $\begingroup$ Ok, energy functional is constant and not decreasing $\endgroup$ – Student Feb 22 '17 at 10:21

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