Exterior Domain Research Articles

Lifespan estimates for semilinear damped wave equation in a two-dimensional exterior domain

Lifespan estimates for semilinear damped wave equations of the form ∂t2u-Δu+∂tu=|u|p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t^2u-\\Delta u+\\partial _tu=|u|^p$$\\end{document} in a two dimensional exterior domain endowed with the Dirichlet boundary condition are dealt with. For the critical case of the semilinear heat equation ∂tv-Δv=v2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _tv-\\Delta v=v^2$$\\end{document} with the Dirichlet boundary condition and the initial condition v(0)=εf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v(0)=\\varepsilon f$$\\end{document}, the corresponding lifespan can be estimated from below and above by exp(exp(Cε-1))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (\\exp (C\\varepsilon ^{-1}))$$\\end{document} with different constants C. This paper clarifies that the same estimates hold even for the critical semilinear damped wave equation in the exterior of the unit ball under the restriction of radial symmetry. To achieve this result, a new technique to control L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-type norm and a new Gagliardo–Nirenberg type estimate with logarithmic weight are introduced.

On thermally driven fluid flows arising in astrophysics

We investigate a potential model for an unbounded celestial bodies of finite mass composed of a solid core and a gaseous atmosphere. The system is governed by the Navier–Stokes–Fourier–Poisson equations, incorporating no-slip boundary conditions for velocity and a specified temperature distribution on the surface of the solid core. Additionally, a positive far-field condition is imposed on the temperature. This manuscript extends the mathematical theory of open fluid systems to unbounded exterior domains addressing these physically motivated yet highly challenging combination of boundary conditions. Notably, we establish the existence of global-in-time weak solutions and demonstrate the weak-strong uniqueness principle

Study of gravastar admitting Tolman IV spacetime in Rastall theory

In this paper, we present a novel solution representing the gravastar (or gravitational vacuum star) model within the framework of non-conservative Rastall gravity. As a viable alternative to black holes, the geometry of a gravastar comprises three distinct regions: the inner sector, the intermediate layer, and the exterior domain. Utilizing the temporal component of Tolman IV spacetime, we derive the singularity-free radial metric potentials for both the inner and intermediate regions. Further, by aligning the thin shell with the external Schwarzschild line element, we establish boundary conditions in order to determine unknown constants that appear due to the chosen ansatz and performing some integrations. Subsequently, we investigate various properties for the gravastar’s shell, including the equation of state parameter, proper length, energy, entropy and gravitational redshift for four different values of the Rastall parameter. It is concluded that the gravastar structure exists and provides a feasible substitute of black holes in the framework of Rastall theory.

Nonexistence for a system of time‐fractional diffusion equations in an exterior domain

A system of time‐fractional diffusion equations posed in an exterior domain of ( ) under homogeneous Dirichlet boundary conditions is investigated in this paper. The time‐fractional derivatives are considered in the Caputo sense. Using nonlinear capacity estimates specifically adapted to the nonlocal properties of the Caputo fractional derivative, the geometry of the domain, and the boundary conditions, we obtain sufficient conditions for the nonexistence of a weak solution to the considered system.

Multiplicity and symmetry breaking for supercritical elliptic problems in exterior domains

Abstract We deal with the following semilinear equation in exterior domains − Δ u + u = a ( x ) | u | p − 2 u , u ∈ H 0 1 ( A R ) , where A R := { x ∈ R N : | x | > R } , N ⩾ 3 , R > 0. Assuming that the weight a is positive and satisfies some symmetry and monotonicity properties, we exhibit a positive solution having the same features as a, for values of p > 2 in a suitable range that includes exponents greater than the standard Sobolev critical one. In the special case of radial weight a, our existence result ensures multiplicity of nonradial solutions. We also provide an existence result for supercritical p in nonradial exterior domains.

Quasiconformal mappings and a Bernstein type theorem over exterior domains in $$\mathbb {R}^2$$

Quasiconformal mappings and a Bernstein type theorem over exterior domains in $$\mathbb {R}^2$$

Viscosity solutions of the augmented K -Hessian equations in exterior domains

Viscosity solutions of the augmented K -Hessian equations in exterior domains

Comparison of finite element and discrete exterior calculus in computation of time-harmonic wave propagation with controllability

This paper discusses computation of time-harmonic wave problems using a mixed formulation and the controllability method introduced by Roland Glowinski. As an example, a scattering problem (in an exterior domain) is considered, and the continuous problem is first formulated in terms of differential forms. Based on the continuous formulation, we write the discrete problem and the controllability algorithm for methods based on both the finite element exterior calculus (FEEC) and the discrete exterior calculus (DEC). As the discrete exterior calculus method provides us with a diagonal “mass matrix”, time-stepping in the DEC approach is remarkably more efficient than in the FEEC approach. For the computations in this paper, we choose the lowest order Whitney elements (a.k.a. Raviart–Thomas elements) for the FEEC approach, and in the DEC discretization we use different diagonal approximations for the Hodge star. Especially, in the DEC approach, a “harmonic Hodge” approximation is used, the derivation of which is based on the time-harmonicity of the problem. Different type of grids are used to study the sensitivity of the solution to the quality of the grid. Putting an effort on meshes regular enough, the computed DEC-solution is as accurate as the FEEC-solution, but reached in the fraction of the time. Both methods seem to be able to keep the solution accuracy rather well in computations with a high wave number (corresponding to a high frequency and a small wave length).

Extreme Superposition: High-Order Fundamental Rogue Waves in the Far-Field Regime

This Memoir concerns fundamental rogue-wave solutions of the focusing nonlinear Schrödinger equation in the limit that the order of the rogue wave is large and the independent variables ( x , t ) (x,t) are proportional to the order (the far-field limit). We first formulate a Riemann-Hilbert representation of these solutions that allows the order to vary continuously rather than by integer increments. The intermediate solutions in this continuous family include also soliton solutions for zero boundary conditions spectrally encoded by a single complex-conjugate pair of poles of arbitrary order, as well as other solutions having nonzero boundary conditions matching those of the rogue waves albeit with far slower decay as x → ± ∞ x\to \pm \infty . The large-order far-field asymptotic behavior of the solution depends on which of three disjoint regions C \mathcal {C} (the “channels”), S \mathcal {S} (the “shelves”), and E \mathcal {E} (the “exterior domain”) contains the rescaled variables. On the region C \mathcal {C} , the amplitude is small and the solution is highly oscillatory, while on the region S \mathcal {S} , the solution is approximated by a modulated plane wave with a highly oscillatory correction term. The asymptotic behavior on these two domains is the same for all continuous orders. Assuming that the order belongs to the discrete sequence characteristic of rogue-wave solutions, the asymptotic behavior of the solution on the region E \mathcal {E} resembles that on S \mathcal {S} but without the oscillatory correction term. Solutions of other continuous orders behave quite differently on E \mathcal {E} .

Uncertainty quantification for locally resonant coated plates and shells

The effect of uncertainty in design parameters on the acoustic performance of coated plates and coated shells for maritime applications is presented. The locally resonant coatings are designed using a viscoelastic material with an impedance similar to water and embedded with layers of inclusions. Both voids and hard inclusions, which respectively exhibit monopole and dipole resonance scattering, are considered. Effective medium approximation theory is employed to characterise the layers of inclusions as homogenised layers with effective material and geometric properties. The coating is then modelled as a multilayered equivalent fluid composed of alternating layers of the homogeneous viscoelastic material and the homogenised layers of inclusions. The coating is bonded to a rigid backing plate and the acoustic response is calculated using the transfer matrix method. The coating is also externally applied to an elastic cylindrical shell. The acoustic response is calculated by expressing the shell displacements and acoustic pressures in the coating and exterior domain in terms of Fourier series expansions, and applying continuity equations at each interface between the shell surface, coating layers and the surrounding water. Efficient stochastic models based on the non-intrusive polynomial chaos expansion (PCE) method are developed by transforming the analytical models for the coated plates and shells into computationally efficient surrogate models using point collocation. Uncertainty in dominant design parameters associated with the geometry of the inclusions and material properties of the coating is examined. The influence of the design parameters for inclusions tuned to different resonance frequencies and for multiple layers of inclusions is also reported. Strong acoustic performance of coating designs occurs in a broad frequency range around local resonance of the inclusions. For all coating models, uncertainty in parameters which predominantly influence the local resonance of the inclusions were observed to yield the greatest variation in the acoustic responses of the coated plates and shells.

Evaluation of airborne transmission of respiratory contaminants in a passenger car cabin with window opening

Exhaled aerosols can be suspended in the air for long periods, facilitating the transmission of highly infectious respiratory diseases. The risk of exposure to occupants varies due to different usage conditions and driving scenarios of moving vehicles. Therefore, the present study investigates the airborne transmission of contaminants generated by coughing in a moving vehicle and explores the relationship between aerosol dispersion and fluid exchange between the interior and exterior domains of the car under the influence of window opening configurations, seating layouts and driving speeds. The results show that the cross-opening configurations provide a superior contaminant removal efficiency. The effect of seating layout on removal efficiency is significant, although it has a slight effect on air change rate (ACH). The best removal efficiency can be obtained when the passenger sits in the front seat with the right front window opening. Furthermore, the in-cabin ACH is almost linearly correlated with the driving speed. Higher speeds reduce contaminant concentrations faster. However, the risk of driver exposure does not depend solely on ACH, the peak cumulative contaminant concentration and residence time also need to be considered. This study offers data for the prevention and control of respiratory infectious diseases.

Positive solutions of Kirchhoff type problems with critical growth on exterior domains

Positive solutions of Kirchhoff type problems with critical growth on exterior domains

Local and global existence for the Ericksen - Leslie problem in unbounded domains

The work deals with the Ericksen-Leslie System for nematic liquid crystals on the space RN with N≥3, on R+3 and on Ω⊆R3 exterior domain with sufficiently smooth boundary. The crystal orientation is described by unit vector v that is a small perturbation of a fixed constant vector η. We prove through a combination of energy method with dispersive a priori estimates a local existence and global existence for small initial data by a contraction argument. In particular, we obtain the following regularity of the liquid velocity u and of the crystal orientation vu∈L∞((0,T);Hs(Ω)),∇u∈L2((0,T);Hs(Ω)),∇v∈L∞((0,T);Hs(Ω)),∇2v∈L2((0,T);Hs(Ω)) for s>N2−1 if Ω=RN and s∈(12,1] if Ω=R+3 or in the exterior case, asking low regularity assumptions on u0 and v0.

Determining coefficients for a fractional p-Laplace equation from exterior measurements

We consider an inverse problem of determining the coefficients of a fractional p -Laplace equation in the exterior domain. Assuming suitable local regularity of the coefficients in the exterior domain, we offer an explicit reconstruction formula in the region where the exterior measurements are performed. This formula is then used to establish a global uniqueness result for real-analytic coefficients. In addition, we also derive a stability estimate for the unique determination of the coefficients in the exterior measurement set.

Parabolic Hessian Quotient Equation in Exterior Domain

This study mainly focuses on the parabolic Hessian quotient equation in the exterior domain. The existence and uniqueness of generalized parabolically symmetric solutions with generalized asymptotic behavior are proven using Perron’s method.

Mesh‐free Monte Carlo method for electrostatic problems with floating potentials

AbstractNumerical simulation plays a crucial role in the analysis and design of power equipment, such as lightning protection devices, which may become inefficient using traditional grid‐based methods when handling complex geometries of large problems. The authors propose a grid‐free Monte Carlo method to handle electrostatic problems of complex geometry for both the interior and exterior domains, which is governed by the Poisson equation with a floating potential boundary condition that is neither a pure Dirichlet nor a Neumann condition. The potential and gradient at any given point can be expressed in terms of integral equations, which can be estimated recursively within the walk‐on‐sphere algorithm. Numerical examples have been demonstrated, including the evaluation of the mutual capacitance matrix of multi‐conductor structures and lighting striking near real fractal trees. The proposed method shows advantages in terms of geometric flexibility and robustness, output sensitivity, and parallelism, which may become a candidate for game‐changing numerical methods and exhibit great potential applications in high‐voltage engineering.

On Estimation of the Decay of Solutions of an Initial-Boundary-Value Problem for a System of Semilinear Equations of Magnetoelasticity with Dissipative Term in Exterior Domains

On Estimation of the Decay of Solutions of an Initial-Boundary-Value Problem for a System of Semilinear Equations of Magnetoelasticity with Dissipative Term in Exterior Domains

Blow-Up of Solutions to Boundary Value Problems of Coupled Wave Equations with Damping Terms on Exterior Domain

Blow-Up of Solutions to Boundary Value Problems of Coupled Wave Equations with Damping Terms on Exterior Domain

Azer Sign-changing radial solutions for a semilinear problem on exterior domains with nonlinear boundary conditions

In this paper we are interested to the existence and multiplicity of sign changing radial solutions of problem of elliptic equations $\Delta U(x)+\varphi(|x|)f(U)=0$ with a nonlinear boundary conditions on exterior of the unite ball centered at the origin in $\mathbb{R}^{N}$ such that $ u(x) \rightarrow 0$ as $ |x|\to \infty $, with any given number of zeros where the nonlinearity $ f(u) $ is odd, superlinear for $ u $ lager enough and $ f0 $ on $(0,\beta)$, $ f0$ on $(\beta,\infty) $. The function $\varphi0$ is $ C^{1} $ on $ [R,\infty) $ where $ 0\varphi(|x|)\leq c_0\,|x|^{-\alpha}$ with $ \alpha2(N-1) $ and $ N2 $ for large $ |x| $.

Stability of pulsating fronts for bistable reaction-diffusion equations in spatially periodic media

The first part of the paper is concerned with the asymptotic stability of pulsating fronts in RN for spatially periodic bistable reaction-diffusion equations with respect to decaying perturbations. Precisely, we show that the solution u(t,x) of the initial value problem converges to the pulsating front as t→+∞ uniformly in RN. In the second part, we investigate the existence and asymptotic behavior of the entire solution u(t,x) emanating from a pulsating front for the equation in exterior domains. The proof of the asymptotic behavior is relying on the application of the proof for the stability of the pulsating front.