Blowup solutions in a problem for the nonlinear heat equation on a small interval (Q1037051)

The article investigates extremely fast processes with blowup in nonlinear media. The dynamics of the solutions of the one-dimensional quasi-linear heat equation with a volume heat source and nonlinear thermal conductivity has been analyzed in the parameter region where self-similar solutions of the equations evolve in the blowup regime when the source intensity is greater than the diffusion. In this case, the heat is localized, and the advanced combustion process proceeds in the form of simple or complex spatial structures with contracting half-width. The evolution of different initial distributions and the reaching of the self-similar regime by these distributions have been studied as well as the dependence of the size of the localization region on the shape of the initial compactly supported distribution. The possibility of the cyclic evolution of solutions against the background of overall growth with blowup has been shown. The article particularly focuses on the case when the observation interval is much lesser than the characteristic size of the heat localization region for given initial distributions. In such a case, all initial perturbations achieve the self-similar regime, although the corresponding scenario possesses certain specific features. An example of the formation of complex spatial structures that evolve with blowup on a small interval is demonstrated.