Geodesic
A point stabilization criterion for second order parabolic equations with almost periodic coefficients
Sbornik. Mathematics, Tome 38 (1981) no. 2, pp. 279-292 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {V. V. Zhikov},
     title = {A~point stabilization criterion for second order parabolic equations with almost periodic coefficients},
     journal = {Sbornik. Mathematics},
     pages = {279--292},
     year = {1981},
     volume = {38},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_38_2_a8/}
}
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V. V. Zhikov. A point stabilization criterion for second order parabolic equations with almost periodic coefficients. Sbornik. Mathematics, Tome 38 (1981) no. 2, pp. 279-292. http://geodesic.mathdoc.fr/item/SM_1981_38_2_a8/

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