Global dynamics of a non-local delayed differential equation in the half plane
Tao Wang
Communications on Pure & Applied Analysis 2014, 13(6): 2475-2492 doi: 10.3934/cpaa.2014.13.2475
In this paper, we first derive an equation for a single species population with two age stages and a fixed maturation period living in the half plane such as ocean and big lakes. By adopting the compact open topology, we establish some a priori estimate for nontrivial solutions after describing asymptotic properties of the nonlocal delayed effect, which enables us to show the permanence of the equation. Then we can employ standard dynamical system theoretical arguments to establish the global dynamics of the equation under appropriate conditions. Applying the main results to the model with Ricker's birth function and Mackey-Glass's hematopoiesis function, we obtain threshold results for the global dynamics of these two models.
keywords: compact open topology global dynamics. non-local delayed differential equation a priori estimate The half plane permanence
One dimensional $p$-th power Newtonian fluid with temperature-dependent thermal conductivity
Tao Wang
Communications on Pure & Applied Analysis 2016, 15(2): 477-494 doi: 10.3934/cpaa.2016.15.477
We study the initial and initial-boundary value problems for the $p$-th power Newtonian fluid in one space dimension with general large initial data. The existence and uniqueness of globally smooth non-vacuum solutions are established when the thermal conductivity is some non-negative power of the temperature. Our analysis is based on some detailed estimates on the bounds of both density and temperature.
keywords: temperature-dependent thermal conductivity $p$-th power Newtonian fluid large initial data global solutions.
One-dimensional compressible Navier-Stokes equations with large density oscillation
Tao Wang Huijiang Zhao Qingyang Zou
Kinetic & Related Models 2013, 6(3): 649-670 doi: 10.3934/krm.2013.6.649
This paper is concerned with nonlinear stability of viscous shock profiles for the one-dimensional isentropic compressible Navier-Stokes equations. For the case when the diffusion wave introduced in [6, 7] is excluded, such a problem has been studied in [5, 11] and local stability of weak viscous shock profiles is well-established, but for the corresponding result with large initial perturbation, fewer results have been obtained. Our main purpose is to deduce the corresponding nonlinear stability result with large initial perturbation by exploiting the elementary energy method. As a first step toward this goal, we show in this paper that for certain class of ``large" initial perturbation which can allow the initial density to have large oscillation, similar stability result still holds. Our analysis is based on the continuation argument and the technique developed by Kanel' in [4].
keywords: large density oscillation. One-dimensional compressible Navier-Stokes equations viscous shock profiles nonlinear stability
Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation
Linjie Xiong Tao Wang Lusheng Wang
Kinetic & Related Models 2014, 7(1): 169-194 doi: 10.3934/krm.2014.7.169
The Cauchy problem to the Fokker-Planck-Boltzmann equation under Grad's angular cut-off assumption is investigated. When the initial data is a small perturbation of an equilibrium state, global existence and optimal temporal decay estimates of classical solutions are established. Our analysis is based on the coercivity of the Fokker-Planck operator and an elementary weighted energy method.
keywords: coercivity Fokker-Planck-Boltzmann weight function time decay.

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