Solutions In Unbounded Domains Research Articles

On the Absence of Solutions of Differential Inequalities with the $$\infty $$-Laplacian

For differential inequalities with the $$\infty $$ -Laplacian in the principal part, we find conditions for the absence of solutions in unbounded domains. Examples are given that demonstrate the accuracy of these conditions.

On the Mixed Steklov—Neumann and Steklov-type Biharmonic Problems in Unbounded Domains

We study the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover, we study the unique solvability of the mixed Steklov–Neumann and Steklov-type biharmonic problems in unbounded domains with a compact and non-compact boundaries (in particular, the exterior of a compact set, half-space, the domain with conical points) under the assumption that generalized solutions of these problems has a bounded Dirichlet (energy) integral with weight |x|a. Depending on the value of the parameter a, we obtained uniqueness (non-uniqueness) theorems of these problems or present exact formulas for the dimension of the space of solutions.

On the Mixed Dirichlet--Farwig biharmonic problem in exterior domains

We study the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover, we study the unique solvability of the mixed Dirichlet--Farwig biharmonic problem in the exterior of a compact set under the assumption that generalized solutions of these problems has a bounded Dirichlet integral with weight $|x|^a$. Admitting different boundary conditions, we used the variation principle and depending on the value of the parameter $a$, we obtained uniqueness (non-uniqueness) theorems of the problem or present exact formulas for the dimension of the space of solutions.

On the Mixed Dirichlet–Steklov-Type and Steklov-Type Biharmonic Problems in Weighted Spaces

We studied the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover, we studied the unique solvability of the mixed Dirichlet–Steklov-type and Steklov-type biharmonic problems in the exterior of a compact set under the assumption that generalized solutions of these problems has a bounded Dirichlet integral with weight | x | a . Depending on the value of the parameter a, we obtained uniqueness (non-uniqueness) theorems of these problems or present exact formulas for the dimension of the space of solutions.

Stabilization of solutions of an anisotropic quasilinear parabolic equation in unbounded domains

The first initial-boundary value problem with the homogeneous Dirichlet boundary condition and a compactly supported initial function is considered for a model second-order anisotropic parabolic equation in a cylindrical domain D = (0,∞) × Ω. We find an upper bound that characterizes the dependence of the decay rate of solutions as t → ∞ on the geometry of the unbounded domain Ω ⊂ ℝn, n ≥ 3, and on nonlinearity exponents. We also obtain an estimate for the admissible decay rate of nonnegative solutions in unbounded domains; this estimate shows that the upper bound is sharp.

On calculation of hydrodynamic forces for steady flows in unbounded domains

This note addresses the question why the “impulse formula”, often employed to compute hydrodynamic forces in vortex-dominated time-dependent flows, is not applicable to steady flows in unbounded domains. By analyzing the asymptotic structure of steady and unsteady flow solutions in unbounded domains, it is demonstrated that one assumption made in the derivation of the impulse formula is in fact not satisfied in the steady case. This result also highlights the special character of steady flows in unbounded domains.

Periodic solutions in unbounded domains for the Boussinesq system

Assuming that the external forces of the system are small enough, the reference temperature being a periodic function, we study the existence, the uniqueness and the regularity of time-periodic solutions for the Boussinesq equations in several classes of unbounded domains of ℝn. Our analysis is based on the framework of weak-Lp spaces.

Symmetry and nonexistence of low Morse index solutions in unbounded domains

In this paper we prove symmetry results for classical solutions of semilinear elliptic equations in the whole R N or in the exterior of a ball, N ⩾ 2 , in the case when the nonlinearity is either convex or has a convex first derivative. More precisely we prove that solutions having Morse index j ⩽ N are foliated Schwarz symmetric, i.e. they are axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis. From this we deduce some nonexistence results for positive or sign changing solutions in the case when the nonlinearity does not depend explicitly on the space variable.

Analysis of a population model with strong cross-diffusion in unbounded domains

We study a parabolic population model in the full space and prove the global in time existence of a weak solution. This model consists of two strongly coupled diffusion equations describing the population densities of two competing species. The system features intrinsic growth, inter- and intra-specific competition of the species, as well as diffusion, cross-diffusion and self-diffusion, and drift terms related to varying environment quality. The cross-diffusion terms can be large, making the system non-parabolic for large initial data. For the first time, solutions in unbounded domains are studied. The method of our proof is a combination of a time semi-discretization, a transformation of the system guaranteeing the positivity of the solution, a special entropy symmetrizing the system, and compactness arguments. 2000 Mathematics Subject Classification: 35K55 (primary), 92D25, 35D05 (secondary).

The global attractor and numerical simulation of a forced weakly damped MKdV equation

In the paper long-time behavior of solutions derived from the dissipative MKdV equation in the unbounded domain is studied. The Sobolev interpolation inequality and prior estimate on time t are applied to show the existence of solution in unbounded domain. Moreover, operator decomposition techniques and Kuratowskii α -noncompacted measures are applied to study the smooth property of the solution. Existence of global attractor for dissipative MKdV equation in H 2 ( R 1 ) is proved. Then, the computational stability of universal double-time differential format is proved by using nature analysis approach and dissipative conservative scheme theory. Finally, accordant conclusion with theoretical proof is drawn by numerical simulation.

Transport, reaction, and delay in mathematical biology, and the inverse problem for traveling fronts

Reaction-diffusion equations are the standard models for reacting and moving particles in ecology, cell biology, and other fields of biology. In many situations, a more detailed description of the movements of particles or individuals is required. Then reaction-transport systems, reaction Cattaneo systems, and Kramers-Langevin approaches can be used. The typical limit solutions in unbounded domains are traveling fronts which pose new mathematical problems in the case of transport equations. The inverse problem for traveling fronts is a novel problem, which is investigated here in great detail. Additional features are delays, which typically lead to oscillations, and quiescent phases, which can be shown to stabilize against the onset of oscillations. In particular, neutral delay equations can be rigorously derived from first-order hyperbolic equations with appropriate boundary conditions modeling age structure. Multi-species systems lead to studying various phenomena such as Turing instability, interaction of diffusion and delay, and cross diffusion.

Uniqueness classes for solutions in unbounded domains of the first mixed problem for the equation with quasi-elliptic operator

In a cylindrical domain , where is an unbounded subdomain of , one considers the evolution equation the right-hand side of which is a quasi-elliptic operator with highest derivatives of orders with respect to the variables . For the mixed problem with Dirichlet condition at the lateral part of the boundary of a uniqueness class of the Täcklind type is described. For domains tapering at infinity another uniqueness class is distinguished, a geometric one, which is broader than the Täcklind-type class. It is shown that for domains with irregular behaviour of the boundary this class is wider than the one described for a second-order parabolic equation by Oleĭnik and Iosif'yan (Uspekhi Mat. Nauk, 1976 [17]). In a wide class of tapering domains non-uniqueness examples for solutions of the first mixed problem for the heat equation are constructed, which supports the exactness of the geometric uniqueness class. Bibliography: 33 titles.

Topology-Based Inequalities and Inverse Problems for Near Force-Free Magnetic Fields

We review a conjecture characterizing the knotting of current paths arising as solutions to an inverse problem involving force-free magnetic fields. Results about the nonexistence of solutions involving force-free fields supported in a finite domain are then considered, as are explicit constructions of force-free solutions in unbounded domains. This shows why truncating solutions defined on unbounded domains has proven ineffective in the literature, and why the solution to the inverse problem involves a near force-free magnetic field. Solutions are then characterized by inequalities involving the current distribution's mean asymptotic linking and crossing numbers. The inequalities are related to the invariants involved in the conjecture of Crager and Kotiuga.

Bounded Solutions for Nonlinear Elliptic Equations in Unbounded Domains

We prove here the existence of a bounded, radial solution in unbounded domain of the nonlinear elliptic problem

On properties of solutions of semilinear second-order parabolic equations

We consider semilinear second-order parabolic equations whose principal parts may have either divergence or nondivergence form and whose nonlinear terms satisfy conditions of Bernstein-Dini type. We study the qualitative properties of the classical solutions of nondivergence equations and generalized solutions of equations with divergent principal parts: the behavior of solutions in various unbounded domains and near the boundaries of domains, removability of singularities of solutions, vanishing of solutions in unbounded domains, in particular solutions of compact support and uniqueness and continuous dependence on the boundary conditions for solutions of the exterior initial/boundary problem.

Nature of the growth of positive solutions of a parabolic equation

For a second-order uniformly parabolic equation with uniformly oscillating coefficients we obtain a power law estimate of the growth of positive solutions in unbounded domains in the lower half-plane.