A nonlinear stochastic heat equation with variable thermal conductivity and multiplicative noise

We consider a stochastic heat equation with variable thermal conductivity, on infinite domain, with both deterministic and stochastic source and with stochastic initial data. The stochastic source appears in the form of multiplicative generalized stochastic process. The generalized stochastic process appearing in the equation could be a generalized smoothed white noise process or any other generalized stochastic process of a certain growth. In our solving procedure we use regularized derivatives and the theory of generalized uniformly continuous semigroups of operators. We establish and prove the result concerning the existence and uniqueness of solution within certain generalized function space. We justify our procedure by proving that, under certain conditions, the operators of non-regularized and the corresponding regularized problems are associated.