Large time behavior of solutions to Newtonian filtration equation with nonlinear boundary sources

This paper deals with the exterior problem of the Newtonian filtration equation with nonlinear boundary sources. The large time behavior of solutions including the critical Fujita exponent are determined or estimated. An interesting phenomenon is illustrated that there exists a threshold value for the coefficient of the lower order term, which depends on the spacial dimension. Exactly speaking, the critical global exponent is strictly less than the critical Fujita exponent when the coefficient is under this threshold, while these two exponents are identically equal when the coefficient is over this threshold. Supported by the NNSF of China and the China Postdoctoral Science Foundation.     

Large time behavior for a viscous Hamilton-Jacobi equation with Neumann boundary condition

We prove the existence and the uniqueness of strong solutions for the viscous Hamilton-Jacobi equation: with Neumann boundary condition, and initial data μ₀, a continuous function. The domain Ω is a bounded and convex open set with smooth boundary, a∈R,a≠0 and p>0. Then, we study the large time behavior of the solution and we show that for p∈(0,1), the extinction in finite time of the gradient of the solution occurs, while for p?1 the solution converges uniformly to a constant, as t→∞.     

Large time behavior of solutions to the nonlinear pseudo-parabolic equation

In this paper, we investigate the initial value problem for the nonlinear pseudo-parabolic equation. Global existence and optimal decay estimate of solution are established, provided that the initial value is suitably small. Moreover, when n?2$n?2$ and the nonlinear term f(u)$f(u)$ disappears, we prove that the global solutions can be approximated by the linear solution as time tends to infinity. When n=1$n=1$ and the nonlinear term f(u)$f(u)$ disappears, we show that as time tends to infinity, the global solution approaches the nonlinear diffusion wave described by the self-similar solution of the viscous Burgers equation.     

Existence of nonlinear boundary layer to the Boltzmann equation with reverse reflection boundary condition

Many physical models have boundaries. When the Boltzmann equation is used to study a physical problem with boundary, there usually exists a layer of width of the order of the Knudsen number along the boundary. Hence, the research on the boundary layer problem is important both in mathematics and physics. Based on the previous work, in this paper, we consider the existence of boundary layer solution to the Boltzmann equation for hard sphere model with positive Mach number. The boundary condition is imposed on incoming particles of reverse reflection type, and the solution is assumed to approach to a global Maxwellian in the far field. Similar to the problem with Dirichlet boundary condition studied in [S. Ukai, T. Yang, S.H. Yu, Nonlinear boundary layers of the Boltzmann equation: I. Existence, Comm. Math. Phys. 3 (2003) 373-393], the existence of a solution is shown to depend on the Mach number of the far field Maxwellian. Moreover, there is an implicit solvability condition on the boundary data. According to the solvability condition, the co-dimension of the boundary data related to the number of the positive characteristic speeds is obtained.     

Large time behavior of solutions to parabolic equations with Neumann boundary conditions

In this paper we are interested in the large time behavior as t→+∞ of the viscosity solutions of parabolic equations with nonlinear Neumann type boundary conditions in connection with ergodic boundary problems which have been recently studied by Barles and the author in [G. Barles, F. Da Lio, On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linèaire 22 (5) (2005) 521-541].     

Large time behavior of interface solutions to the heat equation with Fisher-Wright white noise

Summary The one-dimensional heat equation driven by Fisher-Wright white noise is studied. From initial conditions with compact support, solutions retain this compact support and die out in finite time. There exist interface solutions which change from the value 1 to the value 0 in a finite region. The motion of the interface location is shown to approach that of a Brownian motion under rescaling. Solutions with a finite number of interfaces are approximated by a system of annihilating Brownian motions.     

The nonlinear boundary layer to the Boltzmann equation for cutoff soft potential with physical boundary condition

We consider the nonlinear boundary layer to the Boltzmann equation for cutoff soft potential with physical boundary condition, i.e., the Dirichlet boundary condition with weak diffuse effect. Under the assumption that the distribution function of gas particles tends to a global Maxwellian in the far field, we will show the boundary layer exist if the boundary data satisfy the solvability condition. Moreover, the codimensions of the boundary data which satisfies the solvability condition change with the Mach number of the far field Maxwellian like Chen et al. (2004) [5], Ukai et al. (2003) [6] and Wang et al. (2007) [7].     

Large time behavior of free boundary for a degenerate parabolic equation with absorption

This work studies the large time behavior of free boundary and continuous dependence on nonlinearity for the Cauchy problem of a degenerate parabolic partial differential equation with absorption. Our objective is to give an explicit expression of speed of propagation of the solution and to show that the solution depends on the nonlinearity of the equation continuously.     

Large time behavior of solutions of a dissipative semilnear heat equation

We study the current of the Pauli operator in a strong constant magnetic field. We prove that in the semi-classical limit the persistent current and the current from the interaction of the spin with the magnetic field cancel, in the case where the magnetic field is very strong. Furthermore we calculate the next term in the asymptotics and estimate the error. Finally, we discuss the connection between this work and the semi-classical estimate of the energy in strong magnetic fields proved by Lieb, Solovej and Yngvason [8] Lieb, E., Solovej, J. P. and Yngvason, J. 1994. Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions. Commun. Math. Phys., 161: 77–124. [Crossref], [Web of Science ®] , [Google Scholar].

     

Global solutions of the heat equation with a nonlinear boundary condition

We consider the heat equation with a nonlinear boundary condition, $$(P) \left\{\begin{array}{ll} \partial_t u = \Delta u, & x \in \Omega, \quad t > 0, \\ \partial_\nu u=u^p, & x \in \partial \Omega,\quad t > 0,\\ u (x,0) = \phi (x),& x\in\Omega, \end{array}\right.$$ where ${\Omega = \{x = (x^{\prime},x_N) \in {\bf R}^{N} : x_N > 0\}, N \ge 2, \partial_t = \partial{/}\partial t , \partial_\nu = -\partial{/}\partial x_{N}}$ , p > 1 + 1/N, and (N ? 2)p < N. In this paper we give a complete classification of the large time behaviors of the nonnegative global solutions of (P).     

Global renormalized solutions and Navier–Stokes limit of the Boltzmann equation with incoming boundary condition for long range interaction

For the long range interaction, we prove the global existence of renormalized solutions to the Boltzmann equation with incoming boundary condition. Furthermore, as Knudsen number ? goes to zero, the limit to the incompressible Navier–Stokes limit with homogeneous Dirichlet boundary condition is justified when the boundary data of the scaled Boltzmann equation is close to the Maxwellian with order $O({?}^3)$ in the sense of boundary relative entropy.     

Large time behavior of solutions for the nonlinear wave equation with viscosity in odd dimensions

We study the dissipation of solutions of the Cauchy problem for the nonlinear dissipative wave equation in odd multi-spatial dimensions. Pointwise estimates of the time-asymptotic shape of the solutions are obtained and shown to exhibit the generalized Huygens principle. Our approach is based on the detailed analysis of the Green function of the linearized system. This is used to study the coupling of nonlinear diffusion waves.     

Large time behavior of solutions to a degenerate parabolic equation not in divergence form

In this paper, we investigate positive solutions of the degenerate parabolic equation not in divergence form: ut=upΔu+auq−bur, subject to the null Dirichlet boundary condition. We at first discuss the existence and nonexistence of global solutions to the problem, and then study the large time behavior for the global solutions. When the positive source dominates the model, we prove that the global solutions uniformly tend to the positive steady state of the problem as t→∞. In particular, we establish the uniform asymptotic profiles for the decay solutions when the problem is governed by the nonlinear diffusion or absorption.     

Large time asymptotics of solutions to the generalized Benjamin-Ono equation

We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Benjamin-Ono (BO) equation: , where is the Hilbert transform, , when the initial data are small enough. If the power of the nonlinearity is greater than , then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. In the case critical for the asymptotic behavior i.e. for the modified Benjamin-Ono equation, we prove that the solutions have the same time decay as in the corresponding linear BO equation. Also we find the asymptotics for large time of the solutions of the Cauchy problem for the BO equation in the critical and noncritical cases. For the critical case, we prove the existence of modified scattering states if the initial function is sufficiently small in certain weighted Sobolev spaces. These modified scattering states differ from the free scattering states by a rapidly oscillating factor.