Large time behaviour for the heat equation with absorption and convection (Q1914782)

The large time behaviour of solutions of the Cauchy problem for the equation \[ u_t+ \Biggl({u^m\over m}\Biggr)= u_{xx}- u^p\quad \text{in }\mathbb{R}\times \mathbb{R}^+\tag{1} \] is considered. The authors study the competition between the diffusion, convection and absorption terms. Moreover, they discuss the solution of the Cauchy problem for the equation \[ u_t+\mu \Biggl({u^m\over m}\Biggr)= \varepsilon u_{xx}-\nu u^p\quad\text{in } \mathbb{R}\times \mathbb{R}^+\tag{2} \] and prove some auxiliary estimates for solutions of (2).