Complicated Asymptotic Behavior of Solutions for the Cauchy Problem of Doubly Nonlinear Diffusion Equation

In this paper,we analyze the large time behavior of nonnegative solutions to the doubly nonlinear diffusion equation u_(t)−div(|∇u^(m)|^(p−2)∇u^(m))=0 in R^(N)with p>1,m>0 and m(p−1)−1>0.By using the finite propagation property and the L^(1)−L^(∞)smoothing effect,we find that the complicated asymptotic behavior of the rescaled solutions t^(μ/2)u(t^(β)⋅,t)for 0<μ<2 N/(N[m(p−1)−1]+p)andβ>(2−μ[m(p−1)−1])/(2 p)can take place.

A Class of New Special Solution of Nonlinear Diffusion Equation

One of the most interesting problems of nonlinear differential equations is the construction of partial solutions. A new method is presented in this paper to seek special solutions of nonlinear diffusion equations. This method is based on seeking suitable function to satisfy Bernolli equation. Many new special solutions are obtained.

EXPONENTIAL TIME DIFFERENCING-PADE FINITE ELEMENT METHOD FOR NONLINEAR CONVECTION-DIFFUSION-REACTION EQUATIONS WITH TIME CONSTANT DELAY

In this paper,ETD3-Padéand ETD4-PadéGalerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions.An ETD-based RK is used for time integration of the corresponding equation.To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator,the Padéapproach is used for such an exponential operator approximation,which in turn leads to the corresponding ETD-Padéschemes.An unconditional L^(2) numerical stability is proved for the proposed numerical schemes,under a global Lipshitz continuity assumption.In addition,optimal rate error estimates are provided,which gives the convergence order of O(k^(3)+h^(r))(ETD3-Padé)or O(k^(4)+h^(r))(ETD4-Padé)in the L^(2)norm,respectively.Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.

Solvability for Nonlinear Reaction Diffusion Equation with Boundary Perturbation

In this paper, the nonlinear reaction diffusion equation with boundary perturbation is considered. Using discussions on solvability, the perturbed solution of original problem is obtained, and the uniform validity of the solution is proved.

EVANS FUNCTIONS AND INSTABILITY OF A STANDING PULSE SOLUTION OF A NONLINEAR SYSTEM OF REACTION DIFFUSION EQUATIONS

In this paper, we consider a nonlinear system of reaction diffusion equa- tions arising from mathematical neuroscience and two nonlinear scalar reaction diffusion equations under some assumptions on their coefficients. The main purpose is to couple together linearized stability criterion （the equivalence of the nonlinear stability, the linear stability and the spectral sta- bility of the standing pulse solutions） and Evans functions to accomplish the existence and instability of standing pulse solutions of the nonlinear system of reaction diffusion equations and the nonlinear scalar reaction diffusion equa- tions. The Evans functions for the standing pulse solutions are constructed explicitly.

Streamlines in the Two-Dimensional Spreading of a Thin Fluid Film: Blowing and Suction Velocity Proportional to the Height

The two-dimensional spreading under gravity of a thin fluid film with suction (fluid leak-off) or blowing (fluid injection) at the base is considered. The thin fluid film approximation is imposed. The height of the thin film satisfies a nonlinear diffusion equation with a source/sink term. The Lie point symmetries of the nonlinear diffusion equation are derived and exist, which provided the fluid velocity at the base, vn satisfies a first order linear partial differential equation. The general form has algebraic time dependence while a special case has exponential time dependence. The solution in which vn is proportional to the height of the thin film is studied. The width of the base always increases with time even for suction while the height decreases with time for sufficiently weak blowing. The streamlines of the fluid flow inside the thin film are plotted by first solving a cubic equation. For sufficiently weak blowing there is a dividing streamline, emanating from the stagnation point on the centre line which separates the fluid flow into two regions, a lower region consisting of rising fluid and dominated by fluid injection at the base and an upper region consisting of descending fluid and dominated by spreading due to gravity. For sufficiently strong blowing the lower region expands to completely fill the whole thin film.

Numerical Resolution Near t=0 of Nonlinear Evolution Equations in the Presence of Corner Singularities in Space Dimension 1

The incompatibilities between the initial and boundary data will cause singularities at the time-space corners,which in turn adversely affect the accuracy of the numerical schemes used to compute the solutions.We study the corner singularity issue for nonlinear evolution equations in 1D,and propose two remedy procedures that effectively recover much of the accuracy of the numerical scheme in use.Applications of the remedy procedures to the 1D viscous Burgers equation,and to the 1D nonlinear reaction-diffusion equation are presented.The remedy procedures are applicable to other nonlinear diffusion equations as well.

Ma’s Variation of Parameters Method for Fisher’s Equations

In this paper,we apply Ma’s variation of parameters method(VPM)for solving Fisher’s equations.The suggested algorithm proved to be very efficient and finds the solution without any discretization,linearization,perturbation or restrictive assumptions.Numerical results reveal the complete reliability of the proposed VPM.

Hydrodynamic Limit for Exclusion Processes

The exclusion process,sometimes called Kawasaki dynamics or lattice gas model,describes a system of particles moving on a discrete square lattice with an interaction governed by the exclusion rule under which at most one particle can occupy each site.We mostly discuss the symmetric and reversible case.The weakly asymmetric case recently attracts attention related to KPZ equation;cf.Bertini and Giacomin(Commun Math Phys 183:571–607,1995)for a simple exclusion case and Gonçalves and Jara(Arch Ration Mech Anal 212:597–644,2014)for an exclusion process with speed change,see also Gonçalves et al.(Ann Probab 43:286–338,2015),Gubinelli and Perkowski(J Am Math Soc 31:427–471,2018).In Sect.1,as a warm-up,we consider a simple exclusion process and discuss its hydrodynamic limit and the corresponding fluctuation limit in a proper space–time scaling.From this model,one can derive a linear heat equation and a stochastic partial differential equation(SPDE)in the limit,respectively.Section 2 is devoted to the entropy method originally invented by Guo et al.(Commun Math Phys 118:31–59,1988).We consider the exclusion process with speed change,in which the jump rate of a particle depends on the configuration nearby the particle.This gives a non-trivial interaction among particles.We study only the case that the jump rate satisfies the so-called gradient condition.The hydrodynamic limit,which leads to a nonlinear diffusion equation,follows from the local ergodicity or the local equilibrium of the system,and this is shown by establishing one-block and twoblock estimates.We also discuss the fluctuation limit which follows by showing the so-called Boltzmann–Gibbs principle.Section 3 explains the relative entropy method originally due to Yau(Lett Math Phys 22:63–80,1991).This is a variant of GPV method and gives another proof for the hydrodynamic limit.The difference between these two methods is as follows.Let N^(d)be the volume of the domain on which the system is defined(typically,d-dimensional discrete box with side length N)and denote the(relative)entropy by H.Then,H relative to a global equilibrium behaves as H=O(N^(d))(or entropy per volume is O(1))as N→∞.GPV method rather relies on the fact that the entropy production I,which is the time derivative of H,behaves as O(N^(d−2))so that I per volume is o(1),and this characterizes the limit measures.On the other hand,Yau’s method shows H=o(Nd)for H relative to local equilibria so that the entropy per volume is o(1)and this proves the hydrodynamic limit.In Sect.4,we considerKawasaki dynamics perturbed by relatively largeGlauber effect,which allows creation and annihilation of particles.This leads to the reaction–diffusion equation in the hydrodynamic limit.We discuss especially the equation with reaction term of bistable type and the problem related to the fast reaction limit or the sharp interface limit leading to the motion by mean curvature.We apply the estimate on the relative entropy due to Jara and Menezes(Non-equilibrium fluctuations of interacting particle systems,2017;Symmetric exclusion as a random environment:invariance principle,2018),which is actually obtained as a combination of GPV and Yau’s estimates.This makes possible to study the hydrodynamic limit for microscopic systems with another diverging factors apart from that caused by the space–time scaling.

ONE DIMENSIONAL FILTRATION PROBLEM IN PARTIALLY SATURATED LAYERED POROUS MEDIA

We study in this paper the first boundary value problem of one dimensional degenerate quasilinear elliptic-parabolic equation with discontinuous coefficients (layered media). The uniquenessof the weak solutions is proved under natural conditions.