Global existence and blowup of solutions for a parabolic equation with a gradient term (Q2701612)

The author investigates the semilinear parabolic equation NEWLINE\[NEWLINEu_t= \Delta u+ f(u)+ g(u)|\nabla u|^2NEWLINE\]NEWLINE in a bounded domain with homogeneous Dirichlet data and nonnegative initial datum \(\phi\). Under suitable assumptions he proves that for small initial datum the solution exists globally in time, while for large initial datum it blows up in finite time. Here small resp. large means strictly less resp. larger than a positive equilibrium solution.