DCDS-B
Global existence for a thin film equation with subcritical mass
Jian-Guo Liu Jinhuan Wang
In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin film equation in one-dimensional case
$h_t+\partial_x (h^n\,\partial_{xxx} h)+\partial_x (h^{n+2}\partial_{x} h)=0,$
where
$n≥q 1$
. There exists a critical mass
$M_c=\frac{2\sqrt{6}π}{3}$
found by Witelski et al.(2004 Euro. J. of Appl. Math. 15,223-256) for
$n=1$
. We obtain global existence of a non-negative entropy weak solution if initial mass is less than
$M_c$
. For
$n≥q 4$
, entropy weak solutions are positive and unique. For
$n=1$
, a finite time blow-up occurs for solutions with initial mass larger than
$M_c$
. For the Cauchy problem with
$n=1$
and initial mass less than
$M_c$
, we show that at least one of the following long-time behavior holds:the second moment goes to infinity as the time goes to infinity or
$ h(·, t_k)\rightharpoonup 0$
in
$L^1(\mathbb{R})$
for some subsequence
${t_k} \to \infty $
.
keywords: Long-wave instability free-surface evolution equilibrium the Sz. Nagy inequality long-time behavior
DCDS
Parabolic elliptic type Keller-Segel system on the whole space case
Jinhuan Wang Li Chen Liang Hong
This note is devoted to the discussion on the existence and blow up of the solutions to the parabolic elliptic type Patlak-Keller-Segel system on the whole space case. The problem in two dimension is closely related to the Logarithmic Hardy-Littlewood-Sobolev inequality, which directly introduced the critical mass $8\pi$. While in the higher dimension case, it is related to the Hardy-Littlewood-Sobolev inequality. Therefore, a porous media type nonlinear diffusion has been introduced in order to balance the aggregation. We will review the critical exponents which were introduced in the literature, namely, the exponent $m=2-2/n$ which comes from the scaling invariance of the mass, and the exponent $m=2n/(n+2)$ which comes from the conformal invariance of the entropy. Finally a new result on the model with a general potential, inspired from the Hardy-Littlewood-Sobolev inequality, will be given.
keywords: global existence nonlocal aggregation critical diffusion exponent Chemotaxis critical stationary solution mass concentration.

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