Dead cores and instantaneous compactification of the supports of energy solutions of quasilinear parabolic equations of arbitrary order

A general sufficient condition for the formation of “dead cores” of generalized solutions of a wide class of quasilinear parabolic equations of non-linear diffusion-absorption type is obtained. On that basis a sufficient and close to necessary condition for the instantaneous compactification of the support of an arbitrary local energy solution of the corresponding Cauchy problem is derived, which is expressed in terms of the behaviour at infinity of some integral norm (with respect to balls of fixed radius) of the initial function. A precise upper bound for the compactification radius is obtained.