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### Open Access Journals

PROC

Convergence to convection-diffusion waves for solutions to dissipative nonlinear evolution equations

In this paper we consider the global existence and the asymptotic
behavior of solutions to the Cauchy problem for the following
nonlinear evolution equations with ellipticity and damping
$$ \left\{\begin{array}{l} \psi_t = -(1-\alpha) \psi -
\theta_x + \alpha \psi_{x x} + \psi\psi_x, (E)\\
\theta_t = -(1-\alpha)\theta + \nu \psi_x + 2\psi\theta_x +
\alpha \theta_{x x},
\end{array}
\right.
$$
with initial data converging to different constant states at
infinity
$$(\psi,\theta)(x,0)=(\psi_0(x),
\theta_0(x)) \rightarrow (\psi_{\pm}, \theta_{\pm})
\ \ {as} \ \ x \rightarrow \pm \infty,
(I)
$$
where $\alpha$ and $\nu$ are positive constants such that $\alpha
<1$, $\nu <4\alpha(1-\alpha)$. Under the assumption that
$|\psi_+ - \psi_- |+| \theta_+ - \theta_-|$ is sufficiently small, we show
that if the initial data is a small perturbation of the
convection-diffusion waves defined by (11) which
are obtained by the parabolic system (9), solutions to
Cauchy problem (E) and (I) tend asymptotically to those
convection-diffusion waves with exponential rates. We mainly
propose a better asymptotic profile than that in the previous work
by [13,3], and derive its decay rates by weighted energy
method instead of considering the linearized structure as in
[3].

DCDS

In this paper we introduce an obstacle thermistor system. The existence
of weak solutions to the steady-state systems and capacity solutions to the time dependent
systems are obtained by a penalized method under reasonable assumptions
for the initial and boundary data. At the same time, we prove that there exists
a uniform absorbing set for nonnegative initial data in $L_2(\Omega)$. Finally for smooth
initial data a global attractor to the system is obtained by a series of Campanato
space arguments.

DCDS-B

In this paper we consider a thermistor problem with a current source,
i.e., a nonlocal boundary condition. The electric potential is unknown on part of the
boundary, but the current through it is known. We apply a decomposition technique
and transform the equation satisfied by the potential into two elliptic problems with
usual boundary conditions. The unique solvability of the initial boundary value
problem is achieved.

DCDS-B

In this paper we study a box scheme (or finite volume element method) for a
non-local nonlinear parabolic variational inequality arising in the study
of thermistor problems. Under some assumptions on
the data and regularity of the solution, optimal error estimates in the
$H^1$-norm are attained.

DCDS

In this paper we discuss initial-boundary problems for second order
parabolic equations with rapidly oscillating coefficients in a
bounded convex domain. The asymptotic expansions of the solutions
for problems with multiple spatial and temporal scales are presented
in four different cases. Higher order corrector methods are
constructed and associated explicit convergence rates obtained.

CPAA

In this paper we consider a biosensor model in $R^3$, consisting of
a coupled parabolic differential equation with Robin boundary
condition and an ordinary differential equation. Theoretical
analysis is done to show the existence and uniqueness of a Holder
continuous solution based on a maximum principle, weak solution
arguments. The long-time convergence to a steady state is also
discussed as well as the system situation. Next, a finite volume
method is applied to the model to obtain an approximate solution.
Drawing in part on the analytical results given earlier, we
establish the existence, stability and error estimates for the
approximate solution, and derive $L^2$ spatial norm convergence
properties.
Finally, some illustrative numerical simulation results are presented.

DCDS-B

In this paper, we present the a posteriori error analysis for the
finite element approximation of American option valuation problems. We introduce an efficient and reliable error estimator both for the semi discrete and
fully discrete backward Euler scheme.

DCDS

In this paper we consider a class of integro-differential equations
of parabolic type arising in the study of a quasi-static thermoelastic contact
problem involving a critical parameter $\alpha$.
For $\alpha <1$, the problem is first transformed
into an equivalent standard parabolic equation with non-local
and non-linear boundary conditions. Then the existence, uniqueness and
continuous dependence of the solution upon the data are demonstrated via
solution representation techniques and the maximum principle. Finally
the asymptotic behavior of the solution as $ t \rightarrow \infty$ is examined, and we
show that the non-local term has no impact on the asymptotic behavior
for $ \alpha <1$. The paper concludes with some remarks on the case $\alpha >1$.

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