On Uniqueness of Boundary Blow-up Solutions of a Class of Nonlinear Elliptic Equations

We study boundary blow-up solutions of semilinear elliptic equations $Lu=u_+^p$ with $p>1$, or $Lu=e^{au}$ with $a>0$, where $L$ is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.Comment: To appear in Comm. Partial Differential Equations; 10 page

RESQME and Stand-Alone Simulation on a Workstation

RC 16037 (#71232) The Research Queueing Package Modeling Environment (RESQME) provides a graphical environment for constructing and solving extended queueing network models ofmanufacturing systems, for plotting graphs of results and for viewdng animations of models. The modeling environment can be run entirely on a workstation or optionally can execute large simulations on a host system using cooperative processing. In this paper we give a brief introduction to RESQME and to the RESQ modeling elements. We demonstrate how to use the package by constructing a simple model of part of a manufacturing line and solve this model to produce charts of performance measures and an animation which shows how the jobs flow through the system. By having the simulation available for use on the workstation and cooperatively on the host, RESQME provides a unique tool for understanding the performance of manufacturing systems. A user can do most of the model debugging locally on the workstation and make short püot runs to get a feeling for the amount of resources necessary to make more realistic experiments on the • host. Then long runs which investigate large parts ofthe parameter space can be done cooperatively on the host. Whether the model is solved on the workstation or on the hosMhe graphics environment provides the same user interface to all of the underlying files. The processor where the model is solved is transparent to the user. In aU cases, the user has easy access to plots ofresults and to the animation ofthe model diagra

Supersymmetric version of a hydrodynamic system in Riemann invariants and its solutions

In this paper, a supersymmetric extension of a system of hydrodynamic type equations involving Riemann invariants is formulated in terms of a superspace and superfield formalism. The symmetry properties of both the classical and supersymmetric versions of this hydrodynamical model are analyzed through the use of group-theoretical methods applied to partial differential equations involving both bosonic and fermionic variables. More specifically, we compute the Lie superalgebras of both models and perform classifications of their respective subalgebras. A systematic use of the subalgebra structures allow us to construct several classes of invariant solutions, including travelling waves, centered waves and solutions involving monomials, exponentials and radicals.Comment: 30 page

Supersymmetric version of a Gaussian irrotational compressible fluid flow

The Lie point symmetries and corresponding invariant solutions are obtained for a Gaussian, irrotational, compressible fluid flow. A supersymmetric extension of this model is then formulated through the use of a superspace and superfield formalism. The Lie superalgebra of this extended model is determined and a classification of its subalgebras is performed. The method of symmetry reduction is systematically applied in order to derive special classes of invariant solutions of the supersymmetric model. Several new types of algebraic, hyperbolic, multi-solitonic and doubly periodic solutions are obtained in explicit form.Comment: Expanded introduction and added new section on classical Gaussian fluid flow. Included several additional reference

Abstract basins of attraction

Abstract basins appear naturally in different areas of several complex variables. In this survey we want to describe three different topics in which they play an important role, leading to interesting open problems

Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For the semilinear heat equation $u_t= \Delta u+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure

Asymptotic behavior of solutions to the σk\sigma_k-Yamabe equation near isolated singularities

$\sigma_k$-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $\sigma_k$-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $\sigma_k$-Yamabe equation is asymptotic to a radial solution to the same equation on $\mathbb R^n \setminus \{0\}$. These results generalize earlier pioneering work in this direction on the classical Yamabe equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli et al, we formulate and prove a general asymptotic approximation result for solutions to certain ODEs which include the case for scalar curvature and $\sigma_k$ curvature cases. An alternative proof is also provided using analysis of the linearized operators at the radial solutions, along the lines of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.Comment: 55 page

Definitions of quasiconformality

We establish that the infinitesimal “ H -definition” for quasiconformal mappings on Carnot groups implies global quasisymmetry, and hence the absolute continuity on almost all lines. Our method is new even in R n where we obtain that the “limsup” condition in the H -definition can be replaced by a “liminf” condition. This leads to a new removability result for (quasi)conformal mappings in Euclidean spaces. An application to parametrizations of chord-arc surfaces is also given.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46582/1/222_2005_Article_BF01241122.pd