On equations of mathematical physics containing multi-homogeneous functions of derivatives

We introduce the concept of a multi-homogeneous function for which the homogeneity property holds for some subsets of its set of arguments. Some properties of such functions have been formulated. A class of mathematical physics equations containing a multi-homogeneous function of the first order derivatives with respect to spatial variables and a linear differential operator in time is considered. Using the method of separation of variables, we obtain solutions of equations of this kind in the form of finite sums in which each term depends on the time and spatial variables belonging to one of the above homogeneity subvectors $X_k$. It is shown that if all the constants of separation of variables are equal to zero, then the solution depends on arbitrary functions of some linear combinations of spatial variables $z_k$ forming subvector $X_k$. For the cases of non-zero values of constants of separation of variables, we obtained solutions characterized by linear dependence on the values of $z_k$, solutions with the power and exponential dependence on these variables, and solutions containing arbitrary functions of variables forming subvectors $X_k$. The obtained results are illustrated by an example of an equation of second order in time with a multi-homogeneous function of four variables.