Analytical Analysis of One Nonlinear Diffusion Problem With a Source Specified by Nonlinear Boundary Conditions in Multidimensional Space

This paper studies the properties of one nonlinear diffusion problem with a source specified by nonlinear boundary conditions. The main results are global existence of the solution; regularity and positivity of solutions; asymptotic behaviour of solutions as time goes to infinity; comparison principles; and maximum principles for solutions. The proofs are based on the energy method, comparison methods, and asymptotic techniques. The conditions for the global existence of the solution in time and the unsolvability of the solution of the diffusion problem in a homogeneous medium are founded based on comparison principle and self-similar analysis. We obtain the critical exponent of the Fujita type and the critical global existence exponent.