DCDS
Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions
Xi Tu Zhaoyang Yin
In this paper we mainly study the Cauchy problem for a generalized Camassa-Holm equation. First, by using the Littlewood-Paley decomposition and transport equations theory, we establish the local well-posedness for the Cauchy problem of the equation in Besov spaces. Then we give a blow-up criterion for the Cauchy problem of the equation. we present a blow-up result and the exact blow-up rate of strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion. Finally, we verify that the system possesses peakon solutions.
keywords: blow-up A generalized Camassa-Holm equation Besov spaces local well-posedness peakon solutions.
DCDS
Global dynamics in a two-species chemotaxis-competition system with two signals
Xinyu Tu Chunlai Mu Pan Zheng Ke Lin
In this paper, we consider a chemotaxis-competition system of parabolic-elliptic-parabolic-elliptic type
$\begin{eqnarray*}\label{1}\left\{\begin{array}{llll}u_t = Δ u-χ_{1}\nabla·(u\nabla v)+μ_{1}u(1-u-a_{1}w), &x∈ Ω, ~~~t>0, \\0 = Δ v-v+w, &x∈Ω, ~~~t>0, \\w_t = Δ w-χ_{2}\nabla·(w\nabla z)+μ_{2}w(1-a_{2}u-w), &x∈ Ω, ~~~ t>0, \\0 = Δ z-z+u, &x∈Ω, ~~~t>0, \\\end{array}\right.\end{eqnarray*}$
with homogeneous Neumann boundary conditions in an arbitrary smooth bounded domain
$Ω\subset R^n$
,
$n≥2$
, where
$χ_{i}$
,
$μ_{i}$
and
$a_{i}$
$(i = 1, 2)$
are positive constants. It is shown that for any positive parameters
$χ_{i}$
,
$μ_{i}$
,
$a_{i}$
$(i = 1, 2)$
and any suitably regular initial data
$(u_{0}, w_{0})$
, this system possesses a global bounded classical solution provided that
$\frac{χ_{i}}{μ_{i}}$
are small. Moreover, when
$a_{1}, a_{2}∈ (0, 1)$
and the parameters
$μ_{1}$
and
$μ_{2}$
are sufficiently large, it is proved that the global solution
$(u, v, w, z)$
of this system exponentially approaches to the steady state
$\left(\frac{1-a_{1}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{1}}{1-a_{1}a_{2}}\right)$
in the norm of
$L^{∞}(Ω)$
as
$t\to ∞$
; If
$a_{1}≥1>a_{2}>0$
and
$μ_{2}$
is sufficiently large, the solution of the system converges to the constant stationary solution
$\left(0, 1, 1, 0\right)$
as time tends to infinity, and the convergence rates can be calculated accurately.
keywords: Chemotaxis Lotka-Volterra-type competition boundedness Logistic source asymptotic stability

Year of publication

Related Authors

Related Keywords

[Back to Top]