On blow-up regimes in one nonlinear parabolic equation

A nonlinear heat equation with a special source on a straight line is considered. The family of exact solutions to this equation that have the form p(t) + q(t)cosx/√2, where functions p(t) and q(t) satisfy a certain dynamic system, is constructed. The system is comprehensively analyzed, and the behavior of p(t) and q(t) depending on initial data is revealed. It is found that some of the unbounded solutions from the aforementioned family are close, in a certain sense, to an analytical solution to the heat equation with power nonlinearities. The Cauchy problem for the equations considered is studied as well. It is proved that, depending on the initial solution function, solutions may develop in a blow-up regime or decay.