CPAA

This paper is concerned with the initial-boundary value problem for
the generalized Benjamin-Bona-Mahony-Burgers equation in the half
space $R_+$
\begin{eqnarray}
u_t-u_{txx}-u_{xx}+f(u)_{x}=0,\quad t>0, x\in R_+,\\
u(0,x)=u_0(x)\to u_+, \quad as \ \ x\to +\infty,\\
u(t,0)=u_b.
\end{eqnarray}
Here $u(t,x)$ is an unknown function of $t>0$ and $x\in R_+$,
$u_+\not=u_b$ are two given constant states and the nonlinear
function $f(u)$ is assumed to be a non-convex function which has one or finitely many inflection points. In this paper, we consider $u_b

NHM

Critical threshold phenomena in a one dimensional quasi-linear
hyperbolic model of blood flow with viscous damping are
investigated. We prove global in time regularity and finite time
singularity formation of solutions simultaneously by showing the
critical threshold phenomena associated with the blood flow model.
New results are obtained showing that the class of data that leads
to global smooth solutions includes the data with negative initial
Riemann invariant slopes and that the magnitude of the negative
slope is not necessarily small, but it is determined by the
magnitude of the viscous damping. For the data that leads to shock
formation, we show that shock formation is delayed due to viscous
damping.

DCDS-B

We study a traffic flow model with inhomogeneous road conditions
such as obstacles. The model is a system of nonlinear hyperbolic equations
with both relaxation and sources. The flux and the source terms depend on
the space variable. Waves for such a system propagate in a more complicated
way than those do for models with homogeneous road conditions.

The $L^1$ well-posedness theory for the model is established. In particular,
we derive the continuous dependence of the solution on its initial data in $L^1$
topology. Moreover, the $L^1$-convergence to the unique zero relaxation limit is
proved. Finally, the asymptotic states of a general solution whose initial data
tend to constant states as $|x| \rightarrow +\infty$ are constructed.

NHM

We review our previous results on partial differential equation(PDE) models of traffic flow.
These models include the first order PDE models,
a nonlocal PDE traffic flow model with Arrhenius look-ahead dynamics,
and the second order PDE models, a discrete model which captures the essential features of
traffic jams and chaotic behavior. We study the well-posedness of such PDE problems, finite time blow-up,
front propagation, pattern formation and asymptotic behavior of solutions including the
stability of the traveling fronts. Traveling wave solutions are wave front solutions propagating with a constant speed
and propagating against traffic.

DCDS-B

This paper aims at an initial-boundary value problem on bounded domains for a one-dimensional quasilinear hyperbolic model of blood flow with viscous damping. It is shown that, for given smooth initial data close to a constant equilibrium state, there exists a unique global smooth solution to the model. Time asymptotically, it is shown that the solution converges to the constant equilibrium state exponentially fast as time goes to infinity due to viscous damping and boundary effects.

NHM

We consider a nonlocal traffic flow model with Arrhenius look-ahead dynamics.
We provide a complete local theory and give the blowup alternative of
solutions to the conservation law with a nonlocal flux.
We show that the finite time blowup of solutions must
occur at the level of the first order derivative of the solution.
Furthermore,
we prove that finite time singularities do occur for several types of physical
initial data
by analyzing the solutions on different characteristic lines. These results are new and are consistent with the blowups observed
in previous numerical simulations on the nonlocal traffic flow model [6].

DCDS

We prove the global-in-time existence of intermediate weak solutions of the equations of chemotaxis system in a bounded domain of $\mathbb{R}^2$ or $\mathbb{R}^3$ with initial chemical concentration small in $H^1$. No smallness assumption is imposed on the initial cell density which is in $L^2$. We first show that when the initial chemical concentration $c_0$ is small only in $H^1$ and $(n_0-n_\infty,c_0)$ is smooth, the classical solution exists for all time. Then we construct weak solutions as limits of smooth solutions corresponding to mollified initial data. Finally we determine the asymptotic behavior of the global solutions.

DCDS

We study critical threshold phenomena in a dynamic continuum traffic flow model known
as the Payne and Whitham (PW) model.
This model is a quasi-linear hyperbolic relaxation system, and when equilibrium velocity
is specifically associated with pressure, the equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system.
For a scenario of physical interest we identify a lower threshold for
finite time singularity in solutions and an upper threshold for the global existence of the smooth solution.
The set of initial data leading to global smooth solutions is large, in particular allowing
initial velocity of negative slope.

NHM

This paper is concerned with an initial-boundary value problem on bounded domains for a one
dimensional quasilinear hyperbolic model of blood flow with viscous damping.
It is shown that $L^\infty$ entropy weak solutions exist globally in time
when the initial data are large, rough and contains vacuum states.
Furthermore, based on entropy principle and the theory of divergence measure field,
it is shown that any $L^\infty$ entropy weak solution converges to a constant equilibrium state
exponentially fast as time goes to infinity.
The physiological relevance of the theoretical results obtained in this paper is demonstrated.

CPAA

We establish the existence and stability of smooth large-amplitude traveling waves to nonlinear conservation laws modeling image processing with general flux functions. We innovatively construct a weight function in the weighted energy estimates to overcome the difficulties caused by the absence of the convexity of fluxes in our model. Moreover, we prove that if the integral of the initial perturbation decays algebraically or exponentially in space, the solution converges to the traveling waves with rates in time, respectively. Furthermore, we are able to construct another new weight function to deal with the degeneracy of fluxes in establishing the stability.