Blow-up with logarithmic nonlinearities
The authors study nonnegative solutions of the nonlinear diffusion equation \[ u_t=u_{xx}-\lambda(u+1)\log^p(u+1) \] in the half-line \((0,\infty)\) under the nonlinear boundary condition \[ -u_x(0,t)=\lambda(u+1)\log^q(u+1)(0,t) \] and a suitable initial condition. They determine the ranges of positive parameters \(p\), \(q\) and \(\lambda\) for which the solution is global in time and ranges for which it blows up in finite time. For blow-up solutions the blow-up rate and the blow-up set are found and the asymptotic behaviour close to the blow-up time is described. The appearance of extinction in finite time is also studied.
scientific article; zbMATH DE number 5190783