Initial evolution of supports of solutions of quasilinear parabolic equations with degenerate absorption potential

The propagation of supports of solutions of second-order quasilinear parabolic equations is studied; the equations are of the type of nonstationary diffusion, having semilinear absorption with an absorption potential which degenerates on the initial plane. We find sufficient conditions, which are sharp in a certain sense, on the relationship between the boundary regime and the type of degeneration of the potential to ensure the strong localization of solutions. We also establish a weak localization of solutions for an arbitrary potential which degenerates only on the initial plane.
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