Uniqueness and qualitative behavior of solutions to a porous medium equation with Newtonian potential pressure of nonlinear density function

In this paper, we studied the nonnegative solutions of the following problem: $ \left\{ \begin{array} {ll} u_t = \operatorname{div}\left( u \nabla (-\Delta)^{-1} u^{m} \right)\; \; &\text{in } \mathbb{R}^N\times (0,T) \,, \\ u(x,0) = u_0(x) &\text{in} \; \; \mathbb{R}^N \,, \\ \end{array} \right. $ with $ m\geq 1 $. We established the well-posedness theory for densities $ u_0(x) $ in $ \mathcal{C}^\gamma( \mathbb{R}^N) $, $ \gamma\in(0,1) $; or in $ H^s( \mathbb{R}^N) $, $ s>\frac{N}{2} $ with compact support, respectively.Concerning the qualitative behavior of solutions, we showed that the $ L^p $-estimates of solutions, $ 1<p\leq \infty $ are decreasing in time. Moreover, we demonstrated that the solutions satisfy the following universal bound $ u(x,t)\leq (m t)^{-\frac{1}{m}},\quad \text{for } (x,t)\in \mathbb{R}^N\times (0,\infty). $ In addition, we investigated the asymptotic profile of $ u $ when $ t\to\infty $. Precisely, for any $ q\in[1,\infty) $, we have$ \big\|u(t)-W(t)\big\|_{L^q( \mathbb{R}^N)} \leq C t^{-\frac{q-1+2^{1-N}}{qm}} ,\quad t>0, $where $ W(x,t) $ is the vortex patch solution. Hence, we extended the known results of the case $ q = m = 1 $ in the literature.We end the paper with a section devoted to the study of symmetrization solutions of the above problem. In particular, we obtained some comparison results in a suitable sense for the symmetrization solutions.