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The theorems stated in § 1 are proved. Analogous theorems concerning the solutions of equation (1.1) satisfying the condition ${\frac{\partial }{\partial \nu }z(x,t)|}_{x\in \partial \mathrm{\Omega }}=\psi (x,t,z)$ with $\psi (x,t,z)$ defined in $S=\{x\in \partial \mathrm{\Omega };t,z\in J\}$, measurable, increasing in $z$, are also proved.

Consider the parabolic equation: $${\displaystyle 1\sum _{i,j}^{1\cdots m}{a}_{ij}(x,t)\frac{{\partial }^{2}z(x,t)}{\partial x_i\partial x_j}-\frac{\partial z(x,t)}{\partial t}=f(x,t,z,p),}$$ assuming that$f(x,t,z,p)$, defined in $\overline{D}=\{x\in \overline{\mathrm{\Omega }};t,z,p\in J=(-\mathrm{\infty },+\mathrm{\infty })\}$, 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 ${z(x,t)|}_{x\in \partial \mathrm{\Omega }}=0$ have the same asymptotic behaviour for $t\to +\mathrm{\infty }$. Moreover, if there exists a solution bounded on $J$, this is the only bounded solution.