I am studying a paper of Liu Yacheng which named "On potential wells and applications to semilinear hyperbolic equations and parabolic equations" it considers a nonlinear parabolic equation \begin{equation} \left\{ \begin{array}{ll} u_{t}-\triangle u=f(u), & \hbox{$x\in\Omega,t>0$;} \\ u(x,0)=u_{0}(x), & \hbox{$x\in\Omega$;} \\ u(x,t)=0, & \hbox{$x\in\partial\Omega, t\geq 0$.} \end{array} \right. \end{equation} where $f(u)=u|u|^{p-1}$ and $\Omega\subset \mathbb{R}^n (n\geq 3)$ is a bounded smooth domain, the constant $p$ satisfies $2<p+1\leq \frac{2n}{n-2}$. In the paper of Liu, he defined a weak solution $u(x,t)$ of above nonlinear parabolic equation which satisfy $u\in L^{\infty}(0,T;H_{0}^{1}(\Omega)), u_{t}\in L^2(0,T;L^2(\Omega))$ for some $T>0$, and also defined a functional $$ I(u)=\|\nabla u\|_{L^2(\Omega)}^{2}-\int_{\Omega}|u|^{p+1}dx. $$ I want to ask that: "if $u\in L^{\infty}(0,T;H_{0}^{1}(\Omega)), u_{t}\in L^2(0,T;L^2(\Omega))$ is a weak solution of above nonlinear parabolic equation, can we deduce that the functional $I(u)$ is a continuous function with respect $t$ ?"
In the paper of L.E. Payne and D.H. Sattinger "Saddle points and instability of nonlinear hyperbolic equaitons", it gave a positive answer of above question but without a complete proof (see Page 294). From the Holder's inequality and Sobolev embedding, I can only deduce that the functional $I(u)$ is Lipschitz continuous in $H_{0}^{1}(\Omega)$, i.e. $$ |I(u)-I(v)|\leq C\|u-v\|_{H^{1}(\Omega)},~~\forall u,v\in H_{0}^{1}(\Omega) $$ for some positive $C>0$.
Can anyone help me? Thank you very much in advance!
