Existence and nonexistence of global positive solutions for the evolution \(p\)-Laplacian equations in exterior domains (Q881621)

The paper deals with the existence and nonexistence of global positive solutions for two evolution \(p\)-Laplacian equations in exterior domains with inhomogeneous boundary conditions. It is shown that \(q_c=n(p-1)/(n-p)\) is its critical exponent provided \(2n/(n+1)<p<n.\) Furthermore, it is proved that, if \(\max \{1,p-1\}<q\leq q_c,\) every positive solution of the equations blows up in finite time; whereas for \(q>q_c,\) the equations admit the global positive solutions for some boundary value \(f(x)\) and some initial data \(u_0(x).\) It is also shown that every positive solution of the equations blows up in finite time provided \(n\leq p.\)