Exterior Domain Research Articles

Blow-up of solutions to the semilinear wave equation with scale invariant damping on exterior domain

This paper is concerned with the blow-up of solutions to the initial boundary value problem for the wave equation with scale invariant damping term on the exterior domain, where the nonlinear terms are power nonlinearity |u|^{p} , derivative nonlinearity |u_{t}|^{p} and combined nonlinearities |u_{t}|^{p}+ |u|^{q} , respectively. Upper bound lifespan estimates of solutions to the problem are obtained by constructing suitable test functions and utilizing the test function technique. The main novelty is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent. To the best of our knowledge, the results in Theorems 1.1–1.3 are new.

Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity

Abstract We consider the asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity and fulfill previous results under faster convergence rate by Bao et al. [Monge-Ampère equation on exterior domains, Calc. Var PDE. 52 (2015), 39–63]. Different from known results, we obtain the limit of Hessian and/or gradient of solution at infinity relying on the convergence rate. The basic idea is to use a revised level set method, the spherical harmonic expansion, and the iteration method.

Existence of planar non-symmetric stationary flows with large flux in an exterior disk

This paper is concerned with the two-dimensional stationary Navier-Stokes system in the domain exterior to the unit disk. The existence of solutions with critical decay O(|x|−1) is established around some explicit flows with large flux. The solutions are obtained for non-symmetric external forces, and moreover, are unique in a certain class.

NONEXISTENCE OF GLOBAL SOLUTIONS FOR TIME FRACTIONAL WAVE EQUATIONS IN AN EXTERIOR DOMAIN

This paper concerns two classes of initial-boundary value problems of semilinear time fractional wave equations in an exterior domain. Based on a special technique of test functions, we determine some critical exponents in the sense of Fujita for a corresponding problem. We also obtain the existence and nonexistence of global weak solutions with certain appropriate conditions.

Normalized Solutions of Mass Subcritical Fractional Schrödinger Equations in Exterior Domains

Normalized Solutions of Mass Subcritical Fractional Schrödinger Equations in Exterior Domains

Existence of positive solutions for Kirchhoff-type problem in exterior domains

AbstractWe consider the following Kirchhoff-type problem in an unbounded exterior domain $\Omega\subset\mathbb{R}^{3}$: (*) \begin{align} \left\{ \begin{array}{ll} -\left(a+b\displaystyle{\int}_{\Omega}|\nabla u|^{2}\,{\rm d}x\right)\triangle u+\lambda u=f(u), & x\in\Omega,\\ \\ u=0, & x\in\partial \Omega,\\ \end{array}\right. \end{align}where a > 0, $b\geq0$, and λ > 0 are constants, $\partial\Omega\neq\emptyset$, $\mathbb{R}^{3}\backslash\Omega$ is bounded, $u\in H_{0}^{1}(\Omega)$, and $f\in C^1(\mathbb{R},\mathbb{R})$ is subcritical and superlinear near infinity. Under some mild conditions, we prove that if \begin{equation*}-\Delta u+\lambda u=f(u), \qquad x\in \mathbb R^3 \end{equation*}has only finite number of positive solutions in $H^1(\mathbb R^3)$ and the diameter of the hole $\mathbb R^3\setminus \Omega$ is small enough, then the problem (*) admits a positive solution. Same conclusion holds true if Ω is fixed and λ > 0 is small. To our best knowledge, there is no similar result published in the literature concerning the existence of positive solutions to the above Kirchhoff equation in exterior domains.

Global well-posedness to the compressible nematic liquid crystal flows with large oscillations and vacuum in 3D exterior domains

This paper aims at extending our previous work (Y. Liu and X. Zhong, arXiv:2204.06227) to the exterior problem. We show the global well-posedness of strong solutions to a simplified Ericksen–Leslie model of compressible nematic liquid crystal flows in a three-dimensional exterior domain when the initial total energy is sufficiently small. In order to obtain necessary a priori estimates on the boundary, some new analytic techniques are used.

Existence, nonexistence and uniqueness for Lane–Emden type fully nonlinear systems

We study existence, nonexistence, and uniqueness of positive radial solutions for a class of nonlinear systems driven by Pucci extremal operators under a Lane–Emden coupling configuration. Our results are based on the analysis of the associated quadratic dynamical system and energy methods. For both regular and exterior domain radial solutions we obtain new regions of existence and nonexistence. Besides, we show an exclusion principle for regular solutions, either in or in a ball, by exploiting the uniqueness of trajectories produced by the flow. In particular, for the standard Lane–Emden system involving the Laplacian operator, we prove that the critical hyperbola of regular radial positive solutions is also the threshold for existence and nonexistence of radial exterior domain solutions with Neumann boundary condition. As a byproduct, singular solutions with fast decay at infinity are also found.

A Fictitious Domain Spectral Method for Solving the Helmholtz Equation in Exterior Domains

A Fictitious Domain Spectral Method for Solving the Helmholtz Equation in Exterior Domains

Decay property of solutions to the wave equation with space‐dependent damping, absorbing nonlinearity, and polynomially decaying data

We study the large time behavior of solutions to the semilinear wave equation with space‐dependent damping and absorbing nonlinearity in the whole space or exterior domains. Our result shows how the amplitude of the damping coefficient, the power of the nonlinearity, and the decay rate of the initial data at the spatial infinity determine the decay rates of the energy and the ‐norm of the solution. In the appendix, we also give a survey of basic results on the local and global existence of solutions and the properties of weight functions used in the energy method.

Lifespan estimate of solution to the semilinear wave equation with damping term and mass term

<abstract><p>This paper is mainly concerned with the initial boundary value problems of semilinear wave equations with damping term and mass term as well as Neumann boundary conditions on exterior domain in three dimensions. Blow-up and upper bound lifespan estimates of solutions to the problem with damping term and mass term are derived by applying test function technique and iterative method, where nonlinear terms are power nonlinearity $ |u|^p $, derivative nonlinearity $ |u_{t}|^p $, combined nonlinearities $ |u_t|^p+|u|^q $, respectively. Moreover, upper bound lifespan estimate of solution to the problem with scale invariant damping term, non-negative mass term and combined nonlinearities $ |u_t|^p+|u|^q $ is obtained. The proofs are based on the test function method and iterative approach. The main new contribution is that upper bound lifespan estimates of solutions are associated with the Strauss exponent and Glassey exponent. In addition, the variation trend of wave is achieved by taking advantage of numerical simulation.</p></abstract>

Existence and nonexistence of sign-changing solutions for singular superlinear equations on exterior domains

Existence and nonexistence of sign-changing solutions for singular superlinear equations on exterior domains

BLOW-UP OF SOLUTIONS TO THE SEMILINEAR WAVE EQUATIONS WITH FRICTIONAL AND VISCOELASTIC DAMPING TERMS

Our starting point in this paper is to investigate the weakly coupled system of semilinear wave equations with two types of damping terms and combined nonlinearities $|v_{t}|^{p_{1}}+|v|^{q_{1}}$, $|u_{t}|^{p_{2}}+|u|^{q_{2}}$ on exterior domain in $n\,(n\geq1)$ dimensions. Local existence and uniqueness of mild solutions to the problem are established. Moreover, non-existence of global solutions to the problem with power nonlinearities $|u|^{p},\,|v|^{q}$ in the case of coupled system and the problem with power nonlinearity $|u|^{p}$ in the case of single equation are verified. The proofs are based on the test function technique. It is worth noticing that the frictional damping $u_{t}$ is more dominant than the viscoelastic damping $\Delta u_{t}$ when the time trend to infinity. To the best of our knowledge, the blow-up results in Theorems 1.2--1.3 are new.

Pointwise estimates for systems of coupled $ p $-Laplacian elliptic equations

This work examines positive solutions of systems of inequalities$ \pm \Delta_{{\bf{p}}} {\bf{u}} \ge \rho(x) {\bf{f}}({\bf{u}}), \qquad \mbox{ in } \Omega, $where $ {\bf{p}} = (p_1,...,p_k) $, $ p_i > 1 $ and $ \Delta_{{\bf{p}}} $ is the diagonal-matrix $ \text{diag}(\Delta_{p_1},...,\Delta_{p_k}) $, $ \Delta_{p_i} $ is the $ p_i $-Laplace operator, $ \Omega $ is an arbitrary domain (bounded or not) in $ {\mathbb{R}}^N $ ($ N\ge 2 $), $ {\bf{u}} = (u_1,...,u_k)^T $ and $ {\bf{f}} = (f_1,...,f_k)^T $ are vector-valued functions and $ \rho(x) $ is a nonnegative function in $ \Omega $ which is locally bounded. Using a maximum principle-based argument we provide explicit estimates on positive solutions $ u $ at each point $ x\in\Omega $, and as applications we find Liouville type results in unbounded domains such as $ {\mathbb{R}}^N $, exterior domains or generally unbounded domains with the property that $ \sup_{x\in\Omega}dist (x,\partial\Omega) = \infty $, for various nonlinearities $ {\bf{f}} $ and weights $ \rho $. We also give explicit upper bounds on extremal parameters of related nonlinear multi-parameter eigenvalue problems in bounded domains.

Positive radial solutions for the problem with Minkowski-curvature operator on an exterior domain

<abstract><p>We are concerned with the problem with Minkowski-curvature operator on an exterior domain</p> <p><disp-formula> <label/> <tex-math id="FE46"> \begin{document} $ \begin{align} \left\{\begin{array}{ll} -\text{div}\Big(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\Big) = \lambda K(|x|)\frac{f(u)}{u^\gamma}\ \ \ &\text{in}\ B^c,\\[2ex] \frac{\partial u}{\partial n}|_{\partial B^c} = 0, \ \ \lim\limits_{|x|\to\infty}u(x) = 0, \end{array} \right. \end{align} \quad\quad\quad\quad\quad (P) $ \end{document} </tex-math> </disp-formula></p> <p>where $ 0\leq\gamma < 1 $, $ B^c = \{x\in \mathbb{R}^N: |x| > R\} $ is a exterior domain in $ \mathbb{R}^N $, $ N > 2 $, $ R > 0 $, $ K\in C([R, \infty), (0, \infty)) $ is such that $ \int_R^\infty rK(r)dr < \infty $, the function $ f:[0, \infty)\to (0, \infty) $ is a continuous function such that $ \lim\limits_{s\to\infty}\frac{f(s)}{s^{\gamma+1}} = 0 $ and $ \lambda > 0 $ is a parameter. We show that problem $ (P) $ has at least one positive radial solution for all $ \lambda > 0 $. The proof of our main result is based upon the method of sub and super solutions.</p></abstract>

Attractors for semigroups with multi-dimensional time and PDEs in unbounded domains

We develop the attractors theory for the semigroups with multidimensional time belonging to some closed cone in an Euclidean space and apply the obtained general results to partial differential equations (PDEs) in unbounded domains. The main attention is payed to elliptic boundary problems in general unbounded domains. In contrast to the previous works in this direction our theory does not require the underlying domain to be cylindrical or cone-like or to be shift semiinvariant with respect to some direction. In particular, the theory is applicable to the exterior domains.

Elliptic problem in an exterior domain driven by a singularity with a nonlocal Neumann condition

Elliptic problem in an exterior domain driven by a singularity with a nonlocal Neumann condition

A Computational Method for Acoustic Interaction with Large Complicated Underwater Structures Based on the Physical Mechanism of Structural Acoustics

A numerical coupling approach is proposed to fast predict the acoustic radiation from a vibrating large-complicated underwater structure. In this study, the physical mechanism of sound radiation from underwater large target is used for the first time to improve the efficiency and keep the accuracy of the numerical algorithm. Although the traditional coupled finite element method/boundary element method (FEM-BEM) is accurate, it contains a large number of boundary elements and thus requires a long computation time for large-complicated structures. The research on the physical mechanism of structural acoustics shows that when BEM is applied on the near-field artificial boundary at a proper distance away from the wet structural surface, large-size boundary elements are acceptable and the number of boundary elements and computation time are remarkably reduced. Thus, the fluid outside the structure is divided into the interior domain and the exterior domain by the artificial boundary. Then, the numerical method is realized by coupling structural finite element modelling with interior fluid finite element modelling and with exterior fluid boundary element modelling. Compared with the theoretical value, the experimental value and the results of the traditional FEM-BEM, the correctness of the proposed algorithm and its advantage of computational efficiency are verified. The computation time of the proposed method is over 99% shorter than that of FEM-BEM in the calculation example of a large-complicated structure. The proposed method can be further applied to multidomain acoustic and multibody acoustic calculations.

Lifespan estimates for critical semilinear wave equations and scale invariant damped wave equations in exterior domain in high dimensions

Lifespan estimates for critical semilinear wave equations and scale invariant damped wave equations in exterior domain in high dimensions

On Subsonic and Subsonic-Sonic Flows with General Conservatives Force in Exterior Domains

In this paper, we study the irrotational subsonic and subsonic-sonic flows with general conservative forces in the exterior domains. The conservative forces indicate the new Bernoulli law naturally. For the subsonic case, we introduce a modified cut-off system depending on the conservative forces which needs the varied Bers skill, and construct the solution by the new variational formula. Moreover, comparing with previous results, our result extends the pressure-density relation to the general case. Afterwards we obtain the subsonic-sonic limit solution by taking the extract subsonic solutions as the approximate sequences.