Asymptotic behaviour of supports of solutions of

D. A. Sapronov ; A. E. Shishkov

Sbornik. Mathematics, Tome 197 (2006) no. 5, pp. 753-790

Résumé

We study the phenomenon of the finiteness of the rate of propagation of the supports of generalized energy solutions of mixed problems for a broad class of doubly degenerate parabolic equations of high order; a model example here is the equation $$ (|u|^{q-1}u)_t+(-1)^m \sum_{|\alpha|=m} D_x^\alpha(|D_x^\alpha u|^{p-1} D_x^\alpha u)+(|u|^{\lambda-1}u)_{x_1}=0, $$ $m \geqslant 1$, $p>0$, $q>0$, $\lambda>0$. Bounds (that are sharp in a certain sense) for the early evolution of the supports of solutions (in particular, of the ‘right’ and the ‘left’ fronts of the solutions), which depend on local properties of the initial function and the parameters of the equation, are established. The behaviour of the supports for large times is also studied. Bibliography: 31 titles.