Suppose I have an $L^2$ gradient system $\dot{x}=-\nabla V(x)$, where $V:H\rightarrow\mathbb{R}$ is an analytic potential on an infinite dimensional Hilbert space $H$. Suppose also that I have Lojasiewicz estimates and I know that my flow converges $x(t)\rightarrow x_\infty$ for all initial conditions. I know that my potential only has one stable equilibrium, call it $x^\ast$ that satisfies a second variation inequality, i.e. the Hessian is uniformly positive definite (all other equilibria are "unstable"). Then is it possible to say that my flow necessarily has to converge to $x^*$ for all initial conditions? In other words, can I say that $x^\ast = x_\infty$?
More generally, if an equilibrium satisfies the second variation inequality, is it asymptotically stable?
I know this is true in finite dimensional gradient flows, but I can't find any references that state such a result in infinite dimensions.