Boundedness of Global Solutions for a Heat Equation with Exponential Gradient Source

We consider a one-dimensional semilinear parabolic equation with exponential gradient source and provide a complete classification of large time behavior of the classical solutions: either the space derivative of the solution blows up in finite time with the solution itself remaining bounded or the solution is global and converges in C-1 norm to the unique steady state. The main difficulty is to prove C-1 boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov's functional by carrying out the method of Zelenyak.