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Some common asymptotic properties of semilinear parabolic, hyperbolic and elliptic equations

Peter Poláčik (2002)

Mathematica Bohemica

We consider three types of semilinear second order PDEs on a cylindrical domain Ω × ( 0 , ∞ ) , where Ω is a bounded domain in ℝ N , N ≥ 2 . Among these, two are evolution problems of parabolic and hyperbolic types, in which the unbounded direction of Ω × ( 0 , ∞ ) is reserved for time t , the third type is an elliptic equation with a singled out unbounded variable t . We discuss the asymptotic behavior, as t → ∞ , of solutions which are defined and bounded on Ω × ( 0 , ∞ ) .

Some fast finite-difference solvers for two-dimensional evolutionary equations on special domains

Ta Van Dinh (1982)

Aplikace matematiky

The author proves the existence of the asymptotic error expansion to the Peaceman-Rachford finite-difference scheme for the first boundary value problem of the two-dimensional evolationary equation on the so-called uniform and nearly uniform domains. This expansion leads, by Richardson extrapolation, to a simple process for accelerating the convergence of the method. A numerical example is given.

Some generic properties of nonlinear second order diffusional type problem

Vladimír Ďurikovič, Mária Ďurikovičová (1999)

Archivum Mathematicum

We are interested of the Newton type mixed problem for the general second order semilinear evolution equation. Applying Nikolskij’s decomposition theorem and general Fredholm operator theory results, the present paper yields sufficient conditions for generic properties, surjectivity and bifurcation sets of the given problem.

Some solutions for a class of singular equations

Abdullah Altin, Ayşegül Erençin (2004)

Czechoslovak Mathematical Journal

In this paper we obtain all solutions which depend only on r for a class of partial differential equations of higher order with singular coefficients.

Stability analysis of phase boundary motion by surface diffusion with triple junction

Harald Garcke, Kazuo Ito, Yoshihito Kohsaka (2009)

Banach Center Publications

The linearized stability of stationary solutions for the surface diffusion flow with a triple junction is studied. We derive the second variation of the energy functional under the constraint that the enclosed areas are preserved and show a linearized stability criterion with the help of the H - 1 -gradient flow structure of the evolution problem and the analysis of eigenvalues of a corresponding differential operator.

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