Boundedness vs. blow-up in a degenerate diffusion equation with gradient nonlinearity Christian StinnerMichael Winkler 35B3335B3535K5535K65degenerate diffusionblow-upgradient nonlinearity This work deals with positive solutions of the Dirichlet problem for the degenerate equation \[ u_t = u^p \Delta u + u^q + u^r |\nabla u|^2, \quad p > 0,\ q > 1,\ r > -1, \] in a smoothly bounded domain $\Omega \subset \mathbb{R}^n$. It addresses the question in which cases the gradient nonlinearity enforces blow-up as compared to the well-understood equation $u_t = u^p \Delta u + u^q$. Particularly, it is shown that each of the parameter regimes where either \begin{enumerate}[\textbullet] \item all solutions are global and bounded, or \item both global bounded and blow-up solutions exist, or \item all solutions blow up in finite time \end{enumerate} is bounded by one of the critical hyperplanes $q = p + 1$ and $r = 2p - q$ in the $(p,q,r)$-space. Moreover, unlike for most parameter constellations in the equation without gradient term, the size of $\Omega$ is seen to play an important role in respect of blow-up. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3081 10.1512/iumj.2007.56.3081 en Indiana Univ. Math. J. 56 (2007) 2233 - 2264 state-of-the-art mathematics http://iumj.org/access/