The Verigin problem: review of results; filtration in a porous soil; formulation of the problem; self-similar solutions; Stefan's problem as a limit case of Verigin's problem; one-dimensional problem - main statements and formulation of results; proofs of theorems 5.1, 5.2 - Verigin's problem with the given mass flux on the known boundaries and the Cauchy-Verigin problem; proof of theorem 5.3; Verigin's problem with the given pressure on the known boundaries. equivalence transformations of evolution equations: main ideas; a historical survey; reciprocity transformations of second-order equations; hidden symmetry of evolution equations; linearization by means of Lagrangian coordinates; Lagrange-invariant equations; equations with spherical and cylindrical symmetries; equivalence transformations for higher-order equations and systems of equations; a remarkable equation of nonlinear heat conduction; the on-phase Stefan problem - explicit solutions with functional arbitrariness. One dimensional parabolic equations: introduction; Lagrangian coordinates in one-dimensional evolution equations; analysis of the problem in Langragian terminology; uniform estimates; the inverse transformation; some starting properties of the interface; estimates for the time derivatives and the higher-order derivatives; the interface regularity; regularity of interfaces for a generalized porous medium equation; axially symmetrical solutions of the porous medium equation - long-time asymptotic behaviour; a non-classical problem for a degenerate parabolic equation - uniqueness of unbounded solutions; the Stefan problem with degeneracy at the free boundary - example of exact solution. Parabolic equations in several space dimensions: review of results; Langragian coordinates in the one-phase Stefan problem; correctness of the linear model; similarity solutions of the Stefan problem; solvability of the nonlinear problem; canonical Lagrangian coordinates; Boussinesq's equation in filtration theory; local regularity of interfaces; bibliography; notation.
