Người báo cáo: TS. Đào Nguyên Anh (Đại học Kinh tế Tp. Hồ Chí Minh)
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
abla (-Delta)^{-1} u^{m} right)~~&text{in }mathbb{R}^Ntimes (0,T),,
\
u(x,0)=u_0(x) &text{in}
~~mathbb{R}^N,,
\
end{array}
right.
]
with $mgeq 1$.
We establish the well-posedness theory for densities $u_0(x)$ in $mathcal{C}^gamma(RR^N)$, $gammain(0,1)$; or in $H^s(RR^N)$, $s>frac{N}{2}$with compact support respectively.
Concerning the qualitative behavior of solutions, we show that the $L^p$-estimates of solutions, $1pleq infty$ are decreasing in time. Moreover, we demonstrate that the solutions satisfy the follwing universal bound
[
u(x,t)leq (m t)^{-frac{1}{m}},quadtext{for} (x,t)inRR^Ntimes (0,infty).
]
In addition, we investigate the asymptotic profile of $u$ when $ttoinfty$. Precisely, for any $qin[1,infty)$ we have
[
big|u(t)-W(t)big|_{L^q(RR^N)} leq Ct^{-frac{q-1+2^{1-N}}{qm}} ,quadt>0, ]
where$W(x,t)$ is the vortex patch solution.%$$W(x,t)=frac{|u_0|_{L^infty(RR^N)}}{big(1+m|u_0|^m_{L^infty(RR^N)} tbig)^{1/m}} mathbf{1}_{big{|x|leq R(t)big}} ,quad R(t)=R_0big(1+m|u_0|^m_{L^infty(RR^N)} tbig)^{frac{1}{Nm}} ,,$$
%and$R_0$ merely depends on $u_0, N,m$. %Note that $W(x,t)$ is the vortex patch solution.
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. |
