Nonlinear balance and asymptotic behavior of supercritical   reaction-diffusion equations with nonlinear boundary conditions
  We construct solutions of nonlinear reaction-diffusion equations with nonlinear boundary conditions in spaces where the problem is supercritical and show the nonlinear balance required between the nonlinear terms in order to obtain a dissipative system. Assuming this balance, the dynamics of the solutions is the same that take place in subcritical spaces. 
An\'ibal Rodr\'iguez-Bernal and Alejandro Vidal-L\'opez
math.AP