带有爆破现象的反应扩散方程的移动网格方法

研究了带有指数非线性项的反应扩散方程的数值解.针对方程在有限时间内会变得非常奇异,提出了移动网格方法和维数分析方法来解该方程.数值结果验证了当移动网格方程具有等分布占优这个性质的时候,移动网格方法求解方程非常有效.另外,数值结果同样显示了等分布占优不是一个必要条件.     

Moving mesh method for a reaction-diffusion equation with traveling heat source on unbounded domain

对无界区域上带移动热源的反应扩散方程提出了局部吸收边界条件.移动网格方法对导出的有界区域问题进行了求解.数值例子显示了当热源移动的速度比较慢的时候,方程会在有限时间内发生爆破现象.而当热源移动的速度足够快时,爆破现象不会发生.数值例子验证了新方法的有效性和精确性.     

Semilinear problems in unbounded domains

As formulated by Silva [E.A. de B.e. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal. 16 (1991) 455-477] and Schechter [M. Schechter, A generalization of the saddle point method with applications, Ann. Polon. Math. 57 (3) (1992) 269-281; M. Schechter, New saddle point theorems, in: Generalized Functions and Their Applications, Varanasi, 1991, Plenum, New York, 1993, pp. 213-219], the sandwich theorem has become a very useful tool in finding critical points of functionals leading to solutions of partial differential equations. In the present paper, this theorem is strengthened to apply to more general situations. We present some applications.     

The blow-up rate for semilinear parabolic problems on general domains

We derive results on blow-up rates for parabolic equations and systems from Fujita-type theorems. We complement a previous study by allowing (possibly unbounded) domains with boundary. Received May 2000     

Nonlinear eigenvalue problems for quasilinear operators on unbounded domains

We prove several existence results for eigenvalue problems involving the p-Laplacian and a nonlinear boundary condition on unbounded domains. We treat the non-degenerate subcritical case and the solutions are found in an appropriate weighted Sobolev space. Received May 2000     

The weighted Ritz‐Galerkin method for elliptic boundary value problems on unbounded domains

Recently Babus?ka‐Oh introduced the method of auxiliary mapping (MAM) which efficiently handles elliptic boundary value problems containing singularities. In this paper, a special weighted residue method, the Weighted Ritz‐Galerkin Method (WRGM), is investigated by introducing special weight functions. Together with this method, MAM is modified to yield highly accurate finite element solutions to general elliptic boundary value problems on the exterior of bounded domains at low cost. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 301–326, 2003.     

Gradient estimates for Dirichlet parabolic problems in unbounded domains

We consider autonomous parabolic Dirichlet problems in a regular unbounded open set Ω⊂R^N involving second-order operator A with (possibly) unbounded coefficients. We determine new conditions on the coefficients of A yielding global gradient estimates for the bounded classical solution.     

Nonlinear eigenvalue problems for quasilinear equations in unbounded domains

We prove the existence of a solution of the nonlinear equation in IR^N and in exterior domains, respectively. We concentrate to the case when p ≥ N and the nonlinearity f(x, · ) is “superlinear” and “subcritical”.     

On the moving boundary formulation for parabolic problems on unbounded domains

A parabolic problem defined on unbounded domains and admitting invariance property to Lie group of scalings can be transformed into an equivalent boundary value problem governed by an ODE defined on an infinite interval. A free boundary formulation and a convergence theorem for this kind of transformed problems are available in [Fazio, SIAM J. Numer. Anal., 33 (1996), pp. 1473-1484]. Depending on its scaling invariance properties, the free boundary problem is then solved numerically using either an iterative or a non-iterative method. Finally, the solution of the parabolic problem is retrieved applying the inverse map of similarity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elliptic and parabolic problems in unbounded domains

We consider elliptic and parabolic problems in unbounded domains. We give general existence and regularity results in Besov spaces and semi‐explicit representation formulas via operator‐valued fundamental solutions which turn out to be a powerful tool to derive a series of qualitative results about the solutions. We give a sample of possible applications including asymptotic behavior in the large, singular perturbations, exact boundary conditions on artificial boundaries and validity of maximum principles. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the blow-up of positive solutions to elliptic equations in plane unbounded domains

Mathematical Notes - Using model equations of the form Δu + u σ = 0 as an example, in both linear (σ = 1) and nonlinear (σ > 1) cases, we present some direct methods of...     

Indefinite quasilinear elliptic problems with subcritical and supercritical nonlinearities on unbounded domains

By using the fibering method, we study the existence of non-negative solutions for a class of indefinite quasilinear elliptic problems on unbounded domains with noncompact boundary, in the presence of competing subcritical and supercritical lower order nonlinearities.     

A new method for two dimensional hyperbolic problems in semi‐unbounded irregular domains

Anewmethod called the full rational differential quadrature method is presented to deal with two dimensional linear and nonlinear hyperbolic problems in semi‐unbounded irregular domains. The spacial and temporal discretizations are both implemented by the rational differential quadrature method (RDQM). The RDQM, proven to be A‐stable (Chen and Tanaka, Comput Mech 28 (2002) 331–338) in the temporal discretization, is much more efficient than the finite difference schemes widely used in earlier works. In addition, the irregular boundary conditions are treated by the direct expansion method(DEM). Numerical experiments show that the present method is of high efficiency and easy to implement. © 2011Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A computational procedure for planar seepage problems in unbounded domains

We consider the boundary-value problem modeling planar seepage of groundwater from a single source contour to a regular infinite system of perfect drainage ditches. A fast converging iterative method is developed for such problems, based on the P-transformation method and the method of accelerated successive over relaxation.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 86–92, 1987.     

Multiplicity results for a class of quasilinear eigenvalue problems on unbounded domains

Using a recent result of Ricceri [10] we prove a multiplicity result for a class of quasilinear eigenvalue problems with nonlinear boundary conditions on an unbounded domain. Our paper completes previous results obtained by Carstea and Rădulescu [4], Chabrowski [1], [2], Kandilakis and Lyberopoulos [6] and Pflüger [7]. Received: 17 April 2007     

Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains

We consider nonlinear elliptic eigenvalue problems on unbounded domains G?$\text{R}$ⁿ. Using an extended Ljusternik-Schnirelman theory we prove the existence of infinitely many eigenfunctions on every sphere in L²(G). Moreover, we establish that the infimum λ1 of the spectrum of the linearized problem L is always a bifurcation point. In addition, there is an infinity of branches emanating at λ1 from the trivial line of solutions if λ1 belongs to the essential spectrum of L.