De Giorgi's conjecture for minimizers of the hyperbolic variational functional

Let $w_0,w_1\in C_0^{\infty}(\mathbb{R}^n)$ and let $k>1$ be an integer. For every positive real number $\varepsilon$, let $w_{\varepsilon}=w_{\varepsilon}(t,x)$ minimize

$$F_{\varepsilon}(u):=\int_0^{\infty}\int_{\mathbb{R}^n}e^{-t/\varepsilon}\left(|u”(t,x)|^2+\tfrac{1}{\varepsilon^2}|\nabla u(t,x)|^2+\tfrac{1}{\varepsilon^2}|u(t,x)|^{2k}\right)\\,dx\\,dt$$

in the class of all $u$ satisfying

$$u(0,x)=w_0(x),\qquad u'(0,x)=w_1(x).$$

De Giorgi's conjecture. There exists a limit

$$\lim_{\varepsilon\downarrow0}w_{\varepsilon}(t,x)=w(t,x),$$

where $w$ satisfies

$$w”=\Delta w-kw^{2k-1}.$$

The conjecture concerns the variational approximation of weak solutions to the defocusing nonlinear wave equation and was stated by E. De Giorgi; the supplied source does not indicate whether it is open or resolved.