Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena
We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions : either the space derivative blows up in finite time (with itself remaining bounded), or is global and converges in norm to the unique steady state. The main difficulty is to prove boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out the method of...