The authors consider the problem \(u_ t=\Delta u-au^ p\), \(x\in\Omega\), \(t>0\), \({\partial u\over \partial n}=u^ q\), \(x\in\partial\Omega\), \(t>0\), \(u(x,0)=u_ 0(x)\geq 0\), \(x\in\overline\Omega\) with \(p,q>0\), \(a>0\), \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\), \(u_ 0\not\equiv 0\). Global existence, uniform boundedness of solutions, properties of stationary solutions, convergence to stationary solutions and blow up are investigated, depending on conditions on the parameters \(p,q,a\) and the initial function \(u_ 0\). For \(n=1\) a complete answer is given, for \(n>1\) the answer is far from being complete. In an earlier work the second author considered the case \(a=0\) [e.g.: Commentat. Math. Univ. Carol. 30, No. 3, 479-484 (1989; Zbl 0702.35141)].