Existence and nonexistence of weak solution to Neumann problem for degenerate qusilinear parabolic equations in domains with noncompact boundary. Slow diffusion case. (Q2901612)

This article deals with the following problem NEWLINE\[NEWLINE u_t= div ( u^{m-1}| {Du}| ^{\lambda- 1} Du) - u^p, \qquad in\,\,\, \Omega\times (0,T), NEWLINE\]NEWLINE NEWLINE\[NEWLINE u^{m-1}| {Du}| ^{\lambda- 1} {{\partial u}\over{\partial \overrightarrow{n}}} =0, \qquad in \,\,\, \partial \Omega \times (0, T ), NEWLINE\]NEWLINE NEWLINE\[NEWLINE u (x,0) =\mu ,\qquad x \in \Omega, NEWLINE\]NEWLINE \(\lambda>0\), \(m+\lambda-2>0\), \(p>1\), \(\mu \) is nonnegative bounded Radon measure. The author prove that weak solution of this problem exists if \(p<m+\lambda-1+{{\lambda +1}\over{N}}\) and doesn't exists if \(p>m+\lambda-1+{{\lambda +1}\over{N}}\) under condition that domain \(\Omega\in R^N\) belongs to some class \(B(g)\). In geometric sense the belonging of a domain to \(B(g)\) means that this domain doesn't narrow in infinity.