Behavior as \(t\to+\infty\) of positive solutions of the first boundary value problem for semilinear parabolic equations (Q1178163)

The behavior for \(t\to\infty\) of positive solutions of the following first boundary value problem is considered \[ \text{(IC)}\qquad u|_ S=0,\quad u|_{t=0}=0 \] \[ \text{(IN1)}\qquad Lu+uf(u)-u_ t\geq 0\qquad\text{or }\qquad\text{(IN2)}\quad Lu+uf(u)+u_ t\leq 0, \] where \(L\) is a second-order elliptic operator. It is proven that solutions of the problem (IN1), (IC) tend to zero for \(t\to\infty\) and that solutions of the problem (IC), (IN2) tend to infinity as \(t\to\infty\).