Research of compatibility of the redefined system for the multidimensional nonlinear heat equation
We study the multidimensional parabolic second-order equation with the implicit degeneration and the finite velocity of propagation of perturbations. This equation is given in the form of an overdetermined system of the differential equations with partial derivatives (the number of the equations exceeds the number of the required functions). It is known that an overdetermined system of the differential equations may not be compatible as well as may not have any solutions. Therefore, in order to determine the existence of the solutions and the degree of their arbitrariness the analysis of this overdetermined system is carried out. As a result of the research, the sufficient and the necessary and sufficient compatibility conditions for the overdetermined system of the differential equations with partial derivatives are received. On the basis of these results with the use of the equation of Liouville and the theorem of the potential operators, the exact non-negative solutions of the multidimensional nonlinear heat equation with the finite velocity of propagation of perturbations are constructed. In addition, the new exact non-negative solutions of the nonlinear evolution of Hamilton–Jacobi equations are obtained; the solutions of the nonlinear heat equation and the solutions of Riemann wave equation are also found. Some solutions are not invariant from the point of view of the groups of the pointed transformations and Lie–Bäcklund’s groups. Finally, the transformations of Bäcklund linking the solutions of the multidimensional nonlinear heat equation with the related nonlinear evolution equations are obtained.
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