Nonlinear Diffusion with Absorption

We consider the nonlinear diffusion-absorption problem in L¹() u - div a(, Du) + j (, u) f on , u = 0 on where is an arbitrary open set in ^N. Conditions for the existence of a strong solution are presented and the existence of a natural generalized solution in the general case is established. We study the dependence of the generalized solution on the absorption term j and characterize generalized solutions which are essentially bounded.     

具有吸收和非线性边界流的非线性扩散方程的爆破估计

本文研究一类具有非线性吸收和非线性边界流的非线性扩散方程，建立了解的爆破速率估计，所得结果依赖于模型中三种非线性机制之间的相互作用。     

Attractors for a Nonclassical Diffusion Equation

Abstract For a nonclassical diffusion equation,the asymptotic behavior is investigated,and the existence ofa global attractor is proved.     

热方程的非古典势对称群与不变解

主要研究了热方程与波方程的非古典势对称群生成元及相应的群不变解．研究表明对于守恒形式的偏微分方程,可通过其伴随系统求得的非古典势对称群生成元来构造其显式解．这些显式解不能由方程本身的Lie对称群生成元或Lie-B?cklund对称群生成元构造得到．     

Global Attractors for a Nonclassical Diffusion Equation

We prove the existence of global attractors in H0^1 （Ω） for a nonclassical diffusion equation. Two types of nonlinearity f are considered： one is the critical exponent, and the other is the polynomial growth of arbitrary order.     

The Cauchy Problem for a Nonlinear Diffusion Equation with Absorption and Convection

In this paper we construct the large-time solution of the equation which models the effectof nonlinear absorption on thin film flow of a viscous fluidon a sloping bed. For initial data with finite support we showhow the asymptotic structure of the Cauchy problem depends onthe absorption index . An important addition motivation forthe study is to provide a basis for extending the results tothe more general equation which has applications in plasma physics.     

具有非线性梯度吸收项的快速扩散方程的有限熄灭

本文研究快速扩散方程ut-Δum +｜ u｜p =0的柯西问题 ,其中m ,p∈ ( 0 ,1) .对于 0

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Nonclassical Symmetries and the Singular Manifold Method: Theory and Six Examples

In this paper we discuss a new approach to the relationship between integrability and symmetries of a nonlinear partial differential equation. The method is based heavily on ideas using both the Painlevé property and the singular manifold analysis, which is of outstanding importance in understanding the concept of integrability of a given partial differential equation. In our examples we show that the solutions of the singular manifold possess Lie point symmetries that correspond precisely to the so-called nonclassical symmetries. We also point out the connection between the singular manifold method and the direct method of Clarkson and Kruskal. Here the singular manifold is a function of its reduced variable. Although the Painlevé property plays an essential role in our approach, our method also holds for equations exhibiting only the conditional Painlevé property. We offer six full examples of how our method works for the six equations, which we believe cover all possible cases.     

一类非经典反应扩散方程全局吸引子的正则性

考虑了一类非经典反应扩散方程全局吸引子的正则性。利用渐近先验估计证明了系统在H01(Ω)中的全局吸引子A1在D(A)中有界,并进一步获得A1即为系统在D(A)中的全局吸引子A2。     

Random Attractor for the Nonclassical Diffusion Equation with Fading Memory

In this paper,we consider the stochastic nonclassical diffusion equationwith fading memory on a bounded domain. By decomposition of the solution operator, we give the necessary condition of asymptotic smoothness of the solution to the initial boundary value problem, and then we prove the existence of a random attractor in the space $M_1=D(A^{\frac{1}{2}}) × L^2_μ(R^+, D(A^{\frac{1}{2}}))$, where A=-Δ with Dirichlet boundary condition.     

一类非经典扩散方程在强拓扑空间中的指数吸引子

讨论了非经典反应扩散方程ut-△ut-△u=f(u)+g(x)当非线性项满足临界指数增长时,该方程在强拓扑空间H2(Ω)∩H10(Ω)中的指数吸引子的存在性.特别的,通过证明指数吸引子的存在性,可知文献[7,12,14]中的强拓扑空间中的全局吸引子有有限的分形维数.     

Nonclassical Solutions of Fully Nonlinear Elliptic Equations

We prove the existence of non-smooth solutions to fully non-linear elliptic equations. Received: June 2006, Accepted: January 2007     

一类带非线性边界条件的非经典扩散方程解的爆破

In this article,under suitable conditions on the internal term f(·)and the bound-ary nonlinear term g(·),we prove blow-up results for a class of nonclassical diffusion equations with nonlinear boundary condition.     

Conditional Lie–Bäcklund Symmetries and Invariant Subspaces to Nonlinear Diffusion Equations with Convection and Source

The conditional Lie–Bäcklund symmetry method is used to study the invariant subspace of the nonlinear diffusion equations with convection and source terms. We obtain a complete list of canonical forms for such equations which admit higher order conditional Lie–Bäcklund symmetries and multidimensional invariant subspaces. The functionally generalized separable solutions to the resulting equations are constructed due to the corresponding symmetry reductions. For most of the cases, they are reduced to solving finite‐dimensional dynamical systems.     

Nonlinear Diffusion Processes

We study elliptic systems of strongly nonlinear first-order differential equations in complex form on the plane. For such systems we develop the theory of Hilbert boundary value problems which is very much similar to the well-known theory for a holomorphic vector. Systems of nonlinear elliptic equations describe problems of interaction of several nonlinear stationary processes in the diffusive and convective mass and heat transport by hydrodynamic fluid flows.     

Blow-up of Solutions of Nonclassical Nonlocal Nonlinear Model Equations

Computational Mathematics and Mathematical Physics - For a nonlinear nonlocal operator differential equation of the first order, an abstract Cauchy problem is considered that is a generalization of...     

Existence of Random Attractors for 2D-Stochastic Nonclassical Diffusion Equations on Unbounded Domains

In this article, we prove the existence of a random attractor for stochastic nonclassical diffusion equations on unbounded domains, and the asymptotic compactness of the random dynamical system is established by a tail-estimates method.     

Self-similar Asymptotics for Linear and Nonlinear Diffusion Equations

The long-time asymptotic solutions of initial value problems for the heat equation and the nonlinear porous medium equation are self-similar spreading solutions. The symmetries of the governing equations yield three-parameter families of these solutions given in terms of their mass, center of mass, and variance. Unlike the mass and center of mass, the variance, or “time-shift,” of a solution is not a conserved quantity for the nonlinear problem. We derive an optimal linear estimate of the long-time variance. Newman's Lyapunov functional is used to produce a maximum entropy time-shift estimate. Results are applied to nonlinear merging and time-dependent, inhomogeneously forced diffusion problems.