The authors consider the semilinear equations of the following form: \[ \frac {\partial u}{\partial t}=Lu+\lambda V(u)\tag{1} \] \[ u(0)=f,\tag{2} \] where \(L\) is a Markov generator, \(V\) certain admissible nonlinear function and \(\lambda\) a coupling constant. The authors prove existence, smoothness and ergodicity results for (1)--(2) on infinite dimensional spaces assuming that the logarithmic Sobolev inequality is satisfied. As a result they construct a class of nonlinear Markov semigroups which are hypercontractive.