Multivalued solutions of multidimensional linear equations of heat conduction and rivertons

Background. The article considers the problem of calculating multivalued solutions of multidimensional linear parabolic equations. Solutions for this type of equations of heat conductivity in dimension $d>2$ were not previously known and represent an important new element of the general properties of this type of equations' solutions. Materials and methods. The main method used in this work is the riverton method, which is associated with solutions of multidimensional systems of first order quasilinear equations of a special form. The method is adapted to problems for the equations of heat conduction, diffusion, and other equations of parabolic type. Results. Special attention is paid to 2D and 3D equations of heat conduction, for which a complete procedure for deriving solutions is presented. For the case of a coordinate space of dimension greater than 3, a general scheme for constructing multivalued solutions is given. Conclusions. The developed approach demonstrates the presence of multivalued solutions for heat equations in the dimension of coordinate space 3 and higher, as well as for equations of hyperbolic and elliptic types. Consequently, diffusion processes can lead to the formation of discontinuous structures in the medium.