ON THE METHOD OF STATIONARY STATES FOR QUASILINEAR PARABOLIC EQUATIONS

Abstract

A method is presented for investigating the space-time structure of unbounded nonnegative solutions of quasilinear parabolic equations of the form , where is a nonlinear elliptic operator. Three examples are considered in detail: the Cauchy problem for the equation where 0$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img4.gif/> and 1$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img5.gif/> are constants; the boundary value problem in 0\,\}$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img6.gif/> 0,\quad x\in\Omega;$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img7.gif/> 0,\quad x_3=0;\qquad \alpha=\textrm{const}>0;$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img8.gif/> and the Cauchy problem for the system It is assumed that at the point the solution grows without bound as . The derivation of an estimate of the solution near , is based on an analysis of an appropriate family of stationary solutions : , , 0$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img17.gif/> a parameter. It is shown that the behavior of a solution as depends to large extent on the structure of the envelope 0}U_\lambda(x)$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img19.gif/>. In particular, if , then grows without bound as at points arbitrarily far from . If for , then determines a lower bound for in a neighborhood of , .Bibliography: 28 titles.