On qualitative properties of sign-constant solutions of some quasilinear parabolic problems

We study the Cauchy problem for quasilinear parabolic inequalities containing squares of the first derivatives of an unknown function (the so-called nonlinearities of the KPZ type). The coefficients of the leading nonlinear terms of the inequalities considered either can be continuous functions (the regular case) or can admit power singularities (the singular case) of degree ho greater than $1$. For the regular case, we prove the damping of global nonnegative solutions to the problem studied. By damping, we mean the boundedness of the support of a solution for each positive $t$, uniform (with respect to $t$) convergence to zero as $|x|\to\infty$, and vanishing (for any $x$) starting with a certain sufficiently large $t$. For the singular case, we proved that the problem considered has no global positive solutions.