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

Abstract

We consider the solutions of degenerate parabolic equations and inequalities of the formLu-ut= ¦u¦q sgnu and sgnu(Lu−ut)−¦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.