A sufficient condition for the existence of a “dead zone” for solutions of degenerate semilinear parabolic equations and inequalities

We consider the solutions of degenerate parabolic equations and inequalities of the formLu-u _t = |u| ^q sgnu and sgnu(Lu−u _t )−|u| ^q ≥0, 0<q<1, with the elliptic operatorL in divergent or nondivergent form. We establish a dependence of the maximum modulus of the solution on the domain and on the equation (inequality) such that this dependence guarantees the existence of a “dead zone” of the solution. In this case, the character of degeneracy is unessential. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 824–831, December, 1996.