Solution Theory to Semilinear Parabolic Stochastic Partial Differential Equations with Polynomially Bounded Coefficients

In the present paper we deal with stochastic semilinear partial differential equations Lu=γ(u)+σ(u)Ξ˙ of parabolic type with (t, x)-depending coefficients which may admit a polynomial growth with respect to the space variable. Under suitable assumptions on the coefficients of the parabolic operator L, on the initial data and on the stochastic noise Ξ (more precisely, on the spectral measure M associated with Ξ) we prove existence of a unique (mild) function-valued solution for the associated Cauchy problem.