Singularities of solutions to the non-Newtonian polytropic filtration

In this paper, we study the non-Newtonian polytropic filtration equation $u_t-\mathrm{div}\left({\left|\nabla u^m\right|}^{p-2}\nabla u^m\right)=0$ with a positive initial data on a smooth bounded domain $\Omega \subset R^n$ for $n\ge 3$, where $0<m<1$, $2<p<1+\frac{1}{m}$, and in particular $p<\frac{n\left(m+1\right)}{1+mn}$. To investigate the regularity of solutions to the Dirichlet problem for this equation when the initial data exhibit a singularity of the form $u_0\left(x\right)\sim A{\left|x\right|}^{-\gamma }$ for $x\in \Omega \setminus \left\{0\right\}$ with $A>0$ and $\gamma >0$, we introduce a linear diffusion term in the regularization process. This addition ensures that the equation remains uniformly parabolic, thereby satisfying both the maximum principle and the comparison principle. The desired results are obtained provided that the coefficient of this regularization term converges to zero in the norm of the appropriate function space. This paper shows that the behavior of the solution depends critically on the value of the exponent $\gamma$ in the initial data, leading to the following distinct cases: finite-time boundedness, infinite-time boundedness, singular stabilization, and infinite-time blow-up.