Equations In Exterior Domains Research Articles

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 behaviour of the entire solution u(t,x) emanating from a pulsating front for the equation in exterior domains. The proof of the asymptotic behaviour is relying on the application of the proof for the stability of the pulsating front.

Normalized bound states for the Choquard equations in exterior domains

Normalized bound states for the Choquard equations in exterior domains

Solvability of Hessian quotient equations in exterior domains

Abstract In this paper, we study the Dirichlet problem of Hessian quotient equations of the form $S_k(D^2u)/S_l(D^2u)=g(x)$ in exterior domains. For $g\equiv \mbox {const.}$ , we obtain the necessary and sufficient conditions on the existence of radially symmetric solutions. For g being a perturbation of a generalized symmetric function at infinity, we obtain the existence of viscosity solutions by Perron’s method. The key technique we develop is the construction of sub- and supersolutions to deal with the non-constant right-hand side g.

Exponential decay estimates for the damped defocusing Schrödinger equation in exterior domains

Exponential decay estimates for the damped defocusing Schrödinger equation in exterior domains

Positive solutions for Kirchhoff equation in exterior domains with small Sobolev critical perturbation

ABSTRACT The paper concerns with the existence of positive solutions for the following Kirchhoff problem where a, b>0, 4<p<6, is a parameter and is an exterior domain, that is, Ω is an unbounded domain with nonempty and bounded. After showing the nonexistence of ground state solutions, we prove the existence of one positive solution with higher energy when is contained in a small ball and is sufficiently small. The novelty of this paper is that we extend the result obtained in Alves and de Freitas [Existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth. Milan J Math. 2017;85:309–330] to nonlocal case and generalize the subcritical nonlinearity discussed in Chen and Liu [Positive solutions for Kirchhoff equation in exterior domains. J Math Phys. 2021;62:Article ID 041510] to small critical perturbation.

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), &amp; x\in\Omega,\\ \\ u=0, &amp; x\in\partial \Omega,\\ \end{array}\right. \end{align}where a &gt; 0, $b\geq0$, and λ &gt; 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 &gt; 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.

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

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 the Regularity of Weak Solutions to Time-Periodic Navier–Stokes Equations in Exterior Domains

Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity results for weak solutions to the corresponding initial-value problem cannot be transferred directly. The established regularity criterion demands a certain integrability of the purely periodic part of the velocity field or its gradient, but it does not concern the time mean of these quantities.

Global Well-Posedness of Second-Grade Fluid Equations in 2D Exterior Domain

In this article, we consider 2D second grade fluid equations in exterior domain with Dirichlet boundary conditions. For initial data $\boldsymbol{u}_0 \in \boldsymbol{H}^3(\Omega)$, the system is shown to be global well-posed. Furthermore, for arbitrary $T > 0$ and $s \geq 3$, we prove that the solution belongs to $L^\infty([0, T]; \boldsymbol{H}^s(\Omega))$ provided that $\boldsymbol{u}_0$ is in $\boldsymbol{H}^s(\Omega)$.

Ground state solutions for electromagnetic Schrödinger equations on unbounded domains

In this paper, we study the existence of ground state solutions for a electromagnetic Schrödinger equation in exterior domains. The main technical approach is based on the Nehari manifold method jointly with variational and topological methods.

Global well-posedness of 2D Euler-α equation in exterior domain

After casting Euler-α equations into vorticity-stream function formula, we obtain some very useful estimates from the properties of the vorticity formula in exterior domain. Basing on these estimates, one can have got the global existence and uniqueness of the solutions to Euler-α equations in 2D exterior domain provided that the initial data is regular enough.

On the Poisson equation in exterior domains

We construct a solution of the Poisson equation in exterior domains $\Omega \subset \mathbb R^n,\;n \ge 2,$ in homogeneous Lebesgue spaces $L^{2,q}(\Omega),;1 &amp;lt; q &amp;lt;\infty,$ with methods of potential theory and integral equations. We investigate the corresponding null spaces and prove that its dimensions is equal to $n+1$ independent of $q$.

Propagation Phenomena for Nonlocal Dispersal Equations in Exterior Domains

This paper is concerned with the spatial propagation of bistable nonlocal dispersal equations in exterior domains. We first obtain the existence and uniqueness of an entire solution which behaves like a planar traveling wave front for large negative time. Then, when the entire solution comes to the interior domain, the profile of the front will be disturbed. However, the disturbance is local in space for finite time, which means the disturbance disappears as its location is far away from the interior domain. Furthermore, we prove that the solution can gradually recover its planar wave profile uniformly in space and continue to propagate in the same direction for large positive time provided that the interior domain is compact and convex. Our work generalizes the local (Laplace) diffusion results obtained by Berestycki et al. (2009) to the nonlocal dispersal setting by using new known Liouville results and Lipschitz continuity of entire solutions due to Li et al. (2010).

Positive Solutions for a Class of Fractional Choquard Equation in Exterior Domain

Positive Solutions for a Class of Fractional Choquard Equation in Exterior Domain

Entire solutions of combustion reaction-diffusion equations in exterior domains

In this paper, we consider entire solutions of combustion reaction-diffusion equations in exterior domains. By constructing appropriate super- and sub-solutions, we show that any given transition front of homogeneous combustion reaction-diffusion equation can emanate an entire solution in exterior domains. In addition, we prove that the entire solution is a complete propagation in the exterior domain with some suitable spatial geometrical conditions.

Stability of entire solutions emanating from bistable planar traveling waves in exterior domains

This paper is devoted to the Lyapunov stability of entire solutions emanating from a planar front for bistable (bistable-type or multistable-type) reaction–diffusion equations in exterior domains Ω=RN∖K. Applying the super- and sub-solution method, we prove that the entire solution is Lyapunov stable under the small initial perturbations. In addition, the Lyapunov stability is independent of the geometrical conditions of the obstacle K.

Normalized solutions of mass subcritical Schrödinger equations in exterior domains

Normalized solutions of mass subcritical Schrödinger equations in exterior domains

Positive stationary solutions of convection-diffusion equations for superlinear sources

We investigate the existence and multiplicity of positive stationary solutions for acertain class of convection-diffusion equations in exterior domains. This problem leads to the following elliptic equation \[\Delta u(x)+f(x,u(x))+g(x)x\cdot \nabla u(x)=0,\] for \(x\in \Omega_{R}=\{ x \in \mathbb{R}^n, \|x\|\gt R \}\), \(n\gt 2\). The goal of this paper is to show that our problem possesses an uncountable number of nondecreasing sequences of minimal solutions with finite energy in a neighborhood of infinity. We also prove that each of these sequences generates another solution of the problem. The case when \(f(x,\cdot)\) may be negative at the origin, so-called semipositone problem, is also considered. Our results are based on a certain iteration schema in which we apply the sub and supersolution method developed by Noussair and Swanson. The approach allows us to consider superlinear problems with convection terms containing functional coefficient \(g\) without radial symmetry.