Blow-up and global existence profile of a class of fully nonlinear degenerate parabolic equations
The Cauchy problem is investigated: \[ \begin{cases} u_t=u^q|\Delta u|^{m-1}\Delta u+u^p,\quad x\in \mathbb R^n, t>0,\\ u(x, 0)=u_0(x),\quad x\in \mathbb R^n, \end{cases} \tag{1} \] where \(q\geq 1\), \(m\geq 1\), \(p>0\), and \(u_0\) is nonnegative and appropriately smooth. The authors give conditions for the initial data \(u_0\) and for the parameters \(p\), \(q\) and \(m\) for local and global existence of generalized solutions of (1), for existence and uniqueness of the limit solutions of the Cauchy problem (1), for blow-up in finite time of the nontrivial solutions of (1), and for global existence of the limit solution of the considered problem (1).
scientific article; zbMATH DE number 5987918
scientific article