On the uniform stabilization of solutions of the second mixed problem for a parabolic equation

A. K. Gushchin

Sbornik. Mathematics, Tome 47 (1984) no. 2, pp. 439-498

Résumé

A number of properties of the Green function of the second mixed problem for a parabolic equation of second order on $(t>0)\times\Omega$ ($\Omega$ an arbitrary domain in $\mathbf R_n$) are established. By means of these results a criterion is proved for the uniform stabilization of a solution: the existence of a uniform limit of the spherical mean of the initial function (extended by zero outside $\Omega$) is necessary and sufficient for the uniform stabilization of a solution of the problem considered, with a bounded initial function under a certain condition on the unbounded domain $\Omega$. The basic properties of the Green function are obtained on the basis of an estimate of the solution of the problem with a compactly supported initial function in terms of the norm of the initial function in $L_1(\Omega)$. Bibliography: 53 titles.