Dynamics of the supports of energy solutions of mixed problems for quasi-linear parabolic equations of arbitrary order

A. E. Shishkov ; A. G. Shchelkov

Izvestiya. Mathematics , Tome 62 (1998) no. 3, pp. 601-626

Résumé

We study the geometry of the supports of solutions of the Cauchy–Dirichlet problem for a wide class of quasi-linear degenerate parabolic equations of any order, whose model representative is the equation of non-stationary filtration with non-linear absorption: $$ \dfrac{\partial}{\partial t}\bigl(|u|^{q-1}u\bigr)-\sum_{i=1}^n\,\dfrac{\partial}{\partial x_i}\biggl(|D_x u|^{p-1}\dfrac{\partial u}{\partial x_i}\biggr)+b_0|u|^{\lambda-1}u=0,\qquad b_0>0,\quad n\geqslant 1. $$ In the cases when $0\lambda$ and $0\lambda$, which correspond to “fast” and “slow” diffusion, we find conditions on the behaviour of the initial function $u_0(x)\in L_{q+1}(\Omega)$ in a neighbourhood of the boundary of its support that ensure the effect of finite and infinite inertia of the support of an arbitrary energy solution; these conditions are, in a certain sense, exact. We establish a condition for the reverse motion of the front of the support boundary.