A parabolic variational inequality in unbounded domains

Necessary conditions of optimality for the time optimal problem for nonlinear monotone parabolic equations are presented. For some problems governed by a parabolic variational inequality, these conditions are applied to a penalty problem associated with the inequality. Optimality conditions for the time optimal problem for the variational inequality are derived by passing to the limit in the penalizing term.

We show the existence of weak solutions of nonlinear parabolic partial differential equations in unbounded domains, provided that a variant of the Leray-Lions conditions is satisfied.

Efficient space-time Legendre rational spectral method for parabolic problems in unbounded domains

This chapter presents the application of the boundary element method to high-frequency Helmholtz problems in unbounded domains. Based on a standard combined integral equation approach for sound-hard scattering problems we discuss the discretization, preconditioning and fast evaluation of the involved operators. As engineering problem, the propagation of high-intensity focused ultrasound fields into the human rib cage will be considered. Throughout this chapter we present code snippets using the open-source Python boundary element software BEM++ to demonstrate the implementation.

SynopsisIn this paper, we prove various existence and non-existence results for semilinear elliptic problems in unbounded domains. In particular we prove for general classes of unbounded domains that there exists no solution distinct from 0 offor any smooth f satisfying f(0) = 0. This result is obtained by the use of new identities that solutions of semilinear elliptic equations satisfy.

Solution of the Stochastic control problem in unbounded domains

This paper provides an existence theorem for the Dirichlet problem for a quasilinear partial differential equation of elliptic type in an unbounded domain. The principal new feature of this work is that the results are obtained under weaker monotonicity conditions on the coefficients of the equation than those employed in earlier work in this area.

We give a stability and error analysis of linearly implicit one-step methods for time discretization of non-linear parabolic equations. We derive precise error bounds for Rosenbrock and W-methods, and we explain the error reduction by Richardson extrapolation of the linearly implicit Euler method which occurs in spite of the breakdown of asymptotic expansions. The parabolic equations are studied in a Hilbert space framework that includes semilinear and quasilinear parabolic equations, and also stiff reaction-diffusion equations with reactions at different time scales.

An analysis of wave-motion problems in unbounded or infinite domains is presented. Analytical solutions that express outgoing waves from an artificial boundary such as a circle or a sphere can be obtained easily. These solutions may be transformed into discretized relations between values of velocity potential and volumetric flux on the artificial boundary. These relations are in the same forms as the FEM equations. They can be joined directly to finite elements within the artificial boundary in order to construct an equation system for the whole domain. Some problems of radiation and scattering are calculated numerically by this method. The so-called fictitious interior eigenvalue failure does not arise in this method.

A Palais–Smale approach to Lane–Emden equations

This contribution deals with dual-primal boundary element tearing and interconnecting methods for Poisson-type problems in unbounded domains. To the best of our knowledge, no rigorous analysis of the corresponding preconditioners has been done yet. In the present paper we fill this theoretical gap and generalize the method to unbounded domains. Furthermore, we discuss implementational issues.

Approximations by orthonormal mapped Chebyshev functions for higher-dimensional problems in unbounded domains

We prove the existence and uniqueness of a solution of a nonlinear parabolic variational inequality in an unbounded domain without conditions at infinity. In particular, the initial data may infinitely increase at infinity, and a solution of the inequality is unique without any restrictions on its behavior at infinity.

CONTENTS § 1. Introduction § 2. An estimate of the Hölder constant for § 3. An estimate of the Hölder norm for on the boundary § 4. Estimates of the Hölder norms for in domains § 5. Estimates of the and norms for solutions of quasi-linear elliptic and parabolic equations § 6. Estimates of the Hölder norms of higher derivatives of solutions of quasi-linear elliptic and parabolic equations § 7. Existence theorems § 8. References

In this paper we study the Cauchy problem for a class of semi-linear parabolic type equations with weak data in the homogeneous spaces. We give a method which can be used to construct local mild solutions of the abstract Cauchy problem inĊσ,s,p andLq([0, T);Hs,p) by introducing the concept of both admissible quintuplet and compatible space and establishing time-space estimates for solutions to the linear parabolic type equations. For the small data, we prove that these results can be extended globally in time. We also study the regularity of the solution to the abstract Cauchy problem for nonlinear parabolic type equations in Ċσ,s,p. As an application, we obtain the same result for Navier-Stokes equations with weak initial data in homogeneous Sobolev spaces.