Instantaneous shrinking and localization of functions in \(Y_\lambda(m,p,q,N)\) and their applications.

*(English)*Zbl 0992.35008Summary: The aim of this paper is to discuss the instantaneous shrinking and localization of the support of functions in \(Y_\lambda (m,p,q,N)\) and their applications to some nonlinear parabolic equations including the porous medium equation \(u_t=\Delta u^m-u^q\), \(m>0\), \(q>0\) and the \(p\)-Laplace equation \(u_t=\text{div} (|\nabla u|^{p-2} \nabla u)-u^q\), \(p>1\), \(q>0\). In particular, as an application of the results, a necessary and sufficient condition for the solutions of the above \(p\)-Laplace equation with nonnegative finite Borel measures as initial conditions to have the instantaneous shrinking property of the support is obtained. This is an answer to an open problem posed by R. Kersner and A. Shishkov.

##### MSC:

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

35K65 | Degenerate parabolic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

35K15 | Initial value problems for second-order parabolic equations |

35K55 | Nonlinear parabolic equations |