Thời gian: 9h30-10h15, ngày 10/12/2025

Địa điểm: P. 508, A6, Viện Toán học

Tóm tắt:  In this talk, we would like to study 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 establish 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 show that the $L^p$-estimates of solutions, $1<p\leq \infty$ are decreasing in time. Moreover, we demonstrate 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 investigate the asymptotic profile of $u$ when $t\to\infty$. Precisely, for any $q\in[1,\infty)$ we have
\[
\big\|u(t)-W(t)\big\|_{L^q(\mathbb{R}^N)} \leq Ct^{-\frac{q-1+2^{1-N}}{qm}} ,\quad t>0, \]
where $W(x,t)$ is the vortex patch solution.%$$W(x,t)=\frac{\|u_0\|_{L^\infty(\mathbb{R}^N)}}{\big(1+m\|u_0\|^m_{L^\infty(\mathbb{R}^N)} t\big)^{1/m}} \mathbf{1}_{\big\{|x|\leq R(t)\big\}} ,\quad R(t)=R_0\big(1+m\|u_0\|^m_{L^\infty(\mathbb{R}^N)} t\big)^{\frac{1}{Nm}} \,,$$

Hence, we extend 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 solutionsof the above problem. In particular, we obtain some comparison results in a suitable sense for the symmetrization solutions.