The limit problems for linear and nonlinear polyparabolic equations and asymptotic behaviour of solutions of parabolic systems (Q2781445)

The author brings together the newest results obtained at present in the study of polyparabolic equations, systems of parabolic equations and the asymptotic behaviour \((t\to\infty)\) of their solutions. The enumeration of the paper's chapters will show the area of the domain considered by the author better and shorter that any other presentation.NEWLINENEWLINENEWLINE1. The first linear polyparabolic problem for the curvilinear time-spatial trapezium. 2. The first linear\dots for the time-spatial half-unbounded domain. 3. The nonlinear polyparabolic initial-boundary value problem with boundary conditions of Lauricella type for time-spatial trapezium. 4. The time-periodic solution to a polyparabolic equation for the three-dimensional time-spatial cylinder with Riquier's boundary conditions. 5. Solution to the nonlinear polyparabolic problem for the three-dimensional time-spatial cylinder with Riquier's conditions. 6. The time-periodic solution to a polyparabolic factorized equation. 7. The nonlinear factorized polyparabolic initial-boundary value problem for the strip. 8. The asymptotic behaviour of solutions to the first limit problem for parabolic systems of differential equations of higher order. The paper ends with three short appendices: 1. on the physical models for partial differential equations of higher orders, 2. the physical model for a factorized polyparabolic equation in a spherical shell, 3. examples of differential equations with dissipative structures.NEWLINENEWLINENEWLINENearly all chapters have the same plan: a short introduction in the treated problem, the motivation of the considered problem after which he solves it and analyses the properties of the solutions. The solutions are expressed in terms of polyparabolic thermal potentials with unknown densities and of potentials corresponding to a source function. For the determination of the unknown densities, he constructs an equivalent system of Volterra integral equations or integro-differential nonlinear system of equations, corresponding to the analysed problem. In the case analysed in the sixth chapter he uses a suitable Green's function.NEWLINENEWLINENEWLINENearly all the presented results in the paper are generalizations of these previously obtained and known by him a fertile searcher with 22 papers published after 1990 and 9 in preparation. From this point of view the paper is an original contribution to the problem development and a comprehensive synthesis of the known, till now, results in the domain.NEWLINENEWLINENEWLINEAn observation for the author. Two times in the paper appears the assertion: ``The first paper considering the bicaloric initial-boundary problem \(\dots\) was published by \textit{M. Nicolescu} in 1954'' [Acad. Republ. Popul. Romîne, Studii Cerc. Mat. 5, 243-332 (1954; Zbl 0059.09102)]. The quoted paper is a synthesis, extended on hundred pages, of results obtained by him and other authors till 1954, where he analyses and comments them. In 1952, about he has given a course to the advanced students in mathematics on bicaloric equation, whose analytical programme constitutes, probably, the quoted, by the author, paper.