Consider the parabolic equation: \begin{equation*} \tag{1} \displaystyle \sum_{i,j}^{1 \cdots m} a_{ij}(x,t) \frac{\partial^{2}z(x,t)}{\partial x_{i}\partial x_{j}} - \frac{\partial z(x,t)}{\partial t} = f(x,t,z,p),\end{equation*} assuming that $f(x,t,z,p)$, defined in $\overline{D} = \{ x \in \overline{\Omega}; t,z,p \in J = (-\infty,+\infty) \}$, is measurable and bounded on every bounded set of $\overline{D}$. Given an appropriate definition of solution, we prove that if $f (x,t,z,p)$ is monotone increasing in $z$, then all solutions of (1) satisfying the condition $z(x,t) |_{x \in \partial \Omega} = 0$ have the same asymptotic behaviour for $t \to + \infty$. Moreover, if there exists a solution bounded on $J$, this is the only bounded solution.