LINEARIZATION OF NOVEL NONLINEAR DIFFUSION-EQUATIONS WITH THE HILBERT KERNEL AND THEIR EXACT-SOLUTIONS

Two novel nonlinear diffusion equations with the Hilbert kernel are proposed. The equations can be linearized by introducing appropriate dependent variable transformations. The initial value problems for the proposed equations are then solved exactly through the linearization and explicit nonperiodic and periodic solutions are constructed. The properties of solutions are investigated in detail. It is found that the blow up of solutions occurs at a finite time for both the nonperiodic and periodic cases due to the breakdown of certain analytic conditions imposed on the dependent variables.