A point stabilization criterion for second order parabolic equations with almost periodic coefficients

We consider the Cauchy problem for the parabolic equation
$$ \frac{\partial u}{\partial t}-\frac\partial{\partial x_i}\biggl(a_{ij}(x,t)\frac\partial{\partial x_j}u\biggr)=0,\qquad u\big|_{t=0}(x)\in\mathscr L^\infty(\mathbf R^n), $$
with coefficients $a_{ij}(x_1,x_2,\dots,x_n,t)$ almost periodic on $\mathbf R^{n+1}$. We establish a necessary and sufficient condition on the initial function $u_0(x)$ under which the solution $u(t,x)$ is stabilized, i.e. $u(t, x)\to\lambda$ as $t\to\infty$. This condition consists in the existence of the mean value
$$ \lambda=\lim_{T\to\infty}T^{-n}\gamma^{-1}\int_{(\widehat A^{-1}x,x)\leqslant T^2}u_0(x)\,dx, $$
where $\widehat A = \{\widehat a_{ij}\}$ is the matrix of the coefficients of the “averaged” equation and $\gamma$ is the volume of the ellipsoid $(\widehat A^{-1}x,x)\leqslant1$.
Bibliography: 16 titles.