In this paper, we show global existence, in spatial dimensions greater than or equal to four, for semilinear wave equations with quadratic nonlinearities exterior to a nontrapping obstacle. This extends the previous work of Shibata-Tsutsumi and Hayashi. Shibata-Tsutsumi showed the result for spatial dimensions greater than or equal to 6, and Hayashi proved the result exterior to a ball.
We verify the three-dimensional Glassey conjecture for exterior domain (M, g), where the metric g is asymptotically Euclidean, provided that certain local energy assumption is satisfied. The radial Glassey conjecture exterior to a ball is also verified for dimension three or higher. The local energy assumption is satisfied for many important cases, including exterior domain with nontrapping obstacles and flat metric, exterior domain with star-shaped obstacles and small asymptotically Euclidean metric, as well as the nontrapping asymptotically Euclidean manifolds (R^n, g).
We study the global existence of solutions to semilinear damped wave equations in the scattering case with derivative power-type nonlinearity on (1+3) dimensional nontrapping asymptotically Euclidean manifolds. The main idea is to exploit local energy estimate, together with local existence to convert the parameter $μ$ to small one.
We study the global existence of solutions to semilinear wave equations with power-type nonlinearity and general lower order terms on $n$ dimensional nontrapping asymptotically Euclidean manifolds, when $n=3, 4$. In addition, we prove almost global existence with sharp lower bound of the lifespan for the four dimensional critical problem.
We prove almost global existence for semilinear wave equations outside of nontrapping obstacles. We use the vector field method, but only use the generators of translations and Euclidean rotations. Our method exploits 1/r decay of wave equations, as opposed to the much harder to prove 1/t decay.
We study the long time existence of solutions to nonlinear wave equations with power-type nonlinearity (of order $p$) and small data, on a large class of $(1+n)$-dimensional nonstationary asymptotically flat backgrounds, which include the Schwarzschild and Kerr black hole space-times. Under the assumption that uniform energy bounds and a weak form of local energy estimates hold forward in time, we give lower bounds of the lifespan when $n=3, 4$ and $p$ is not bigger than the critical one. The lower bounds for three dimensional subcritical and four dimensional critical cases are sharp in genera…
In this article, we prove that solutions to a problem in nonlinear elasticity corresponding to small initial displacements exist globally in the exterior of a nontrapping obstacle. The medium is assumed to be homogeneous, isotropic, and hyperelastic, and the nonlinearity is assumed to satisfy a null condition. The techniques contained herein would allow for more complicated geometries provided that there is a sufficient decay of local energy for the linearized problem.
The aim of this article is to prove an "almost" global existence result for some semilinear wave equations in the plane outside a bounded convex obstacle with the Neumann boundary condition.
We study two-dimensional semilinear strongly damped wave equation with mixed nonlinearity $|u|^p+|u_t|^q$ in an exterior domain, where $p,q>1$. Assuming the smallness of initial data in exponentially weighted spaces and some conditions on powers of nonlinearity, we prove global (in time) existence of small data energy solution with suitable higher regularity by using a weighted energy method.
We consider the Cauchy-Dirichlet problem for semilinear wave equations in a three space dimensional domain exterior to a bounded and non-trapping obstacle. We obtain a detailed estimate for the lower bound of the lifespan of classical solutions when the size of initial data tends to zero, in a similar spirit to that of the works of John and Hörmander where the Cauchy problem was treated. We show that our estimate is sharp at least for some special case.
We study the semilinear elliptic inequality $-Δu\geqφ(δ_K(x))f(u)$ in $R^N\setminus K,$ where $φ, f$ are non-negative and continuous functions, $K\subset R^N$ $(N\geq 2)$ is a compact set and $δ_K(x)={\rm dist}(x,\partial K)$. We obtain optimal conditions in terms of $φ$ and $f$ for the existence of $C^2$ positive solutions. Our analysis emphasizes the role played by the geometry of the compact set $K$.
In this paper, we consider the initial boundary value problem in an exterior domain for semilinear strongly damped wave equations with power nonlinear term of the derivative-type $|u_t|^q$ or the mixed-type $|u|^p+|u_t|^q$, where $p,q>1$. On one hand, employing the Banach fixed-point theorem we prove local (in time) existence of mild solutions. On the other hand, under some conditions for initial data and the exponents of power nonlinear terms, the blow-up results are derived by applying the test function method.
In this paper, we primarily consider the following semilinear elliptic equation \begin{eqnarray*} \arraycolsep=1pt\left\{ \begin{array}{lll} \displaystyle -Δu= h(x,u)\quad \ &{\rm in}\ Ω,\\[1.5mm] \phantom{ -Δ} \displaystyle u\ge 0\qquad &{\rm on}\ \partialΩ, \end{array}\right. \end{eqnarray*} where $Ω$ is an exterior domain in $R^N$ with $N\ge 3$, and derive optimal nonexistence results of positive supersolution. Our argument is based on a nonexistence result of positive supersolution of a linear elliptic problem with Hardy potential. We also establish sharp nonexistence results of positive s…
We consider the initial-boundary value problem of semilinear wave equation with nonlinearity $|u|^p$ in exterior domain in $\mathbf{R}^N$ $(N\geq 3)$. Especially, the lifespan of blowup solutions with small initial data are studied. The result gives upper bounds of lifespan which is essentially the same as the Cauchy problem in $\mathbf{R}^N$. At least in the case $N=4$, their estimates are sharp in view of the work by Zha--Zhou (2015). The idea of the proof is to use special solutions to linear wave equation with Dirichlet boundary condition which are constructed via an argument based on Waka…
In this paper we study the initial boundary value problem for two-dimensional semilinear wave equations with small data, in asymptotically Euclidean exterior domains. We prove that if $1<p\le p_c(2)$, the problem admits almost the same upper bound of the lifespan as that of the corresponding Cauchy problem, only with a small loss for $1<p\le 2$. It is interesting to see that the logarithmic increase of the harmonic function in $2$-D has no influence to the estimate of the upper bound of the lifespan for $2<p\le p_c(2)$. One of the novelties is that we can deal with the problem with flat metric…
This paper is concerned with resolvent estimates on the real axis for the Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet boundary conditions when the obstacle is trapping. There are two resolvent estimates for this situation currently in the literature: (i) in the case of elliptic trapping the general "worst case" bound of exponential growth applies, and examples show that this growth can be realised through some sequence of wavenumbers, (ii) in the prototypical case of hyperbolic trapping where the Helmholtz equation is posed in the exterior of two strictly conv…
We consider a class of semilinear nonlocal problems with vanishing exterior condition and establish a Ambrosetti-Prodi type phenomenon when the nonlinear term satisfies certain conditions. Our technique makes use of the probabilistic tools and heat kernel estimates.
In this paper we show a general Strichartz estimate for certain perturbed wave equation, and here we can drop the nontrapping hypothesis and handle trapping obstacles with some loss of derivatives for data in the local energy decay estimates. Then we give the obstacle version of sharp life span for semilinear wave equations when $n=3,p<p_c$, by using a real interpolation method, and by getting a corresponding finite time Strichartz estimates(see section 3). Finally, as an application of the general Strichartz estimates we have gained, we get the Strauss conjecture for semilinear wave equations…
Consider the equation u_t=Δu-Vu +au^p \text{in} R^n\times (0,T); u(x,0)=ϕ(x)\gneq0, \text{in} R^n, where $p>1$, $n\ge2$, $T\in(0,\infty]$, $V(x)\sim\fracω{|x|^2}$ as $|x|\to\infty$, for some $ω\neq0$, and $a(x)$ is on the order $|x|^m$ as $|x|\to\infty$, for some $m\in (-\infty,\infty)$. A solution to the above equation is called global if $T=\infty$. Under some additional technical conditions, we calculate a critical exponent $p^*$ such that global solutions exist for $p>p^*$, while for $1<p\le p^*$, all solutions blow up in finite time. We also show that when $V\equiv0$, the blow-up/global s…
In this paper, we discuss the global existence of weak solutions to the semilinear damped wave equation \begin{equation*} \begin{cases} \partial_t^2u-Δu + \partial_tu = f(u) & \text{in}\ Ω\times (0,T), \\ u=0 & \text{on}\ \partialΩ\times (0,T), \\ u(0)=u_0, \partial_tu(0)=u_1 & \text{in}\ Ω, \end{cases} \end{equation*} in an exterior domain $Ω$ in $\mathbb{R}^N$ $(N\geq 2)$, where $f:\mathbb{R}\to \mathbb{R}$ is a smooth function behaves like $f(u)\sim |u|^p$. From the view point of weighted energy estimates given by Sobajima--Wakasugi \cite{SoWa4}, the existence of global-in-time solutions wi…
We study a class of fractional semilinear elliptic equations and formulate the corresponding Calderón problem. We determine the nonlinearity from the exterior partial measurements of the Dirichlet-to-Neumann map by using first order linearization and the Runge approximation property.
In this paper we look for solutions of a semilinear Maxwell type equation, in even dimension, greater than four. These solutions are critical points of a functional which is strongly degenerate because of the presence of the exterior derivative. We prove that, assuming a suitable convexity condition on the nonlinearity, the equation possesses infinitely many finite energy solutions.
We prove global existence of solutions to quasilinear wave equations with quadratic nonlinearities exterior to nontrapping obstacles in spatial dimensions four and higher. This generalizes a result of Shibata and Tsutsumi in spatial dimensions greater than or equal to six. The technique of proof would allow for more complicated geometries provided that an appropriate local energy decay exists for the associated linear wave equation.
This paper is concerned with the forward and inverse problems for the fractional semilinear elliptic equation $(-Δ)^s u +a(x,u)=0$ for $0<s<1$. For the forward problem, we proved the problem is well-posed and has a unique solution for small exterior data. The inverse problems we consider here consists of two cases. First we demonstrate that an unknown coefficient $a(x,u)$ can be uniquely determined from the knowledge of exterior measurements, known as the Dirichlet-to-Neumann map. Second, despite the presence of an unknown obstacle in the media, we show that the obstacle and the coefficient ca…
In this paper, we give a full classification of the nonexistence of positive weak solutions to the semilinear elliptic inequality involving the fractional Hardy potential in punctured and in exterior domains. Our methods are self-contained and new. The main ideas and key ingredients will be discussed in the next section after Theorem 1.1 for punctured domains, and after Theorem 1.4 for exterior domains. We will also explain why all the previous methods and techniques do not apply to our general setting. Let us give here a foretaste of our line of attack: Based on the imbalance between the Hard…
We consider the initial boundary value problem in exterior domain for semilinear wave equations with power-type nonlinearity |u| p. We will establish blow-up results when p is less than or equal to Strauss' exponent which is the same one for the whole space case R n .
The purpose in this paper is to determine the global behavior of solutions to the initial-boundary value problems for energy-subcritical and critical semilinear heat equations by initial data with lower energy than the mountain pass level in energy spaces associated with self-adjoint operators satisfying Gaussian upper bounds. Our self-adjoint operators include the Dirichlet Laplacian on an open set, Robin Laplacian on an exterior domain, and Schrödinger operators, etc.
We consider a semilinear elliptic problem with a nonlinear term which is the product of a power and the Riesz potential of a power. This family of equations includes the Choquard or nonlinear Schroedinger--Newton equation. We show that for some values of the parameters the equation does not have nontrivial nonnegative supersolutions in exterior domains. The same techniques yield optimal decay rates when supersolutions exists.
We investigate the initial-value problem for the semilinear plate equation containing localized strong damping, localized weak damping and nonlocal nonlinearity. We prove that if nonnegative damping coefficients are strictly positive almost everywhere in the exterior of some ball and the sum of these coefficients is positive a.e. in $% \mathbb{R} ^{n}$, then the semigroup generated by the considered problem possesses a global attractor in $H^{2}\left( \mathbb{R} ^{n}\right) \times L^{2}\left( \mathbb{R} ^{n}\right) $. We also establish boundedness of this attractor in $ H^{3}\left( \mathbb{R} …
In this paper, we consider exterior problem of the critical semilinear wave equation in three space dimensions with variable coefficients and prove global existence of smooth solutions. Similar to the constant coefficients case, we show that the energy cannot concentrate at any point $(t,x)\in(0,\infty)\timesΩ$. For that purpose, following Ibrahim and Majdoub \cite{Ibrahim}, we use a geometric multiplier close to the well-known Morawetz multiplier used in the constant coefficients case. Then we use comparison theorem from Riemannian Geometry to estimate the error terms. Finally, using Strichar…
We study semilinear problems in general bounded open sets for non-local operators with exterior and boundary conditions. The operators are more general than the fractional Laplacian. We also give results in case of bounded $C^{1,1}$ open sets.
In this paper we consider the initial-boundary value problem for the heat, damped wave, complex-Ginzburg-Landau and Schr"odinger equations with the power type nonlinearity $|u|^p$ with $p in (1,2]$ in a two-dimensional exterior domain. Remark that $2=1+2/N$ is well-known as the Fujita exponent. If $p>2$, then there exists a small global-in-time solution of the damped wave equation for some initial data small enough (see Ikehata'05), and if $p<2$, then global-in-time solutions cannot exist for any positive initial data (see Ogawa-Takeda'09 and Lai-Yin'17). The result is that for given initial d…
The global existence for semilinear wave equations with space-dependent critical damping $\partial_t^2u-Δu+\frac{V_0}{|x|}\partial_t u=f(u)$ in an exterior domain is dealt with, where $f(u)=|u|^{p-1}u$ and $f(u)=|u|^p$ are in mind. Existence and non-existence of global-in-time solutions are discussed. To obtain global existence, a weighted energy estimate for the linear problem is crucial. The proof of such a weighted energy estimate contains an alternative proof of energy estimates established by Ikehata--Todorova--Yordanov [J.\ Math.\ Soc.\ Japan (2013), 183--236] but this clarifies the prec…
We are concerned with the nonexistence of sign-changing global weak solutions for a class of semilinear parabolic differential inequalities with convection terms in exterior domains. A weight function of the form $t^α|x|^σ$ is considered in front of the power nonlinearity. Two types of non-homogeneous boundary conditions are investigated: Neumann-type and Dirichlet-type boundary conditions. Using a unified approach, for each case, we establish sufficient criteria for the nonexistence of global weak solutions. When $α=0$, the critical exponent in the sense of Fujita is obtained. This exponent i…
We examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the blow up part of the John theorem, with $F_p(u)=|u|^{p}$, on the Schwarzschild and Kerr black hole backgrounds. Concerning the case of Schwarzschild, we construct a class of small data, so that the solution blows up along the outgoing null cone, which applies for both $F_p(u)=|u|^{p}$ and the focusing nonlinearity $F_p(u)=|u|^{p-1}u$. The proof suggests that the black hole does not have any essential influence on the formation of singularity, in the region away from the Cauchy horizo…
We find solutions $E:Ω\to\mathbb{R}^3$ of the problem \[ \left\{\begin{aligned} &\nabla\times(\nabla\times E) + λE = \partial_E F(x,E) &&\quad \text{in}Ω\\ &ν\times E = 0 &&\quad \text{on}\partialΩ\end{aligned} \right. \] on a simply connected, smooth, bounded domain $Ω\subset\mathbb{R}^3$ with connected boundary and exterior normal $ν:\partialΩ\to\mathbb{R}^3$. Here $\nabla\times$ denotes the curl operator in $\mathbb{R}^3$, the nonlinearity $F:Ω\times\mathbb{R}^3\to\mathbb{R}$ is superquadratic and subcritical in $E$. The model nonlinearity is of the form $F(x,E)=Γ(x)|E|^p$ for $Γ\in L^\inft…
This paper is devoted to studying the following two initial-boundary value problems for semilinear wave equations with variable coefficients on exterior domain with subcritical exponent in $n$ space dimensions: u_{tt}-partial_{i}(a_{ij}(x)\partial_{j}u)=|u|^{p}, (x,t)\in Ω^{c}\times(0,+\infty), n\geq 3 and u_{tt}-\partial_{i}(a_{ij}(x)\partial_{j}u)=|u_{t}|^{p}, (x,t)\in Ω^{c}\times (0,+\infty), n\geq 1, where $a_{ij}(x)=δ_{ij}, when |x|\geq R. The exponents $p$ satisfies $ 1<p<p_{1}(n)$ in (0.1), and $p \leq p_{2}(n)$ in (0.2), where $p_{1}(n)$ is the larger root of the quadratic equation (n-…
We study the well-posedness of a semilinear fractional diffusion equation and formulate an associated inverse problem. We determine fractional power type nonlinearities from the exterior partial measurements of the Dirichlet-to-Neumann map. Our arguments are based on a first order linearization as well as the parabolic Runge approximation property.
In this paper we prove symmetry results for classical solutions of semilinear cooperative elliptic systems in R^N, or in the exterior of a ball. We consider the case of fully coupled systems and nonlinearities which are either convex or have a convex derivative. The solutions are shown to be foliated Schwarz symmetric if a bound on their Morse index holds. As a consequence of the symmetry results we also obtain some nonexistence theorems.
Recently, several works have been carried out in attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In this paper, we study the Dirichlet problem for superlinear equation (E) ${\mathbb L} u = u^p +λμ$ in a bounded domain $Ω$ with homogeneous boundary or exterior Dirichlet condition, where $p>1$ and $λ>0$. The operator ${\mathbb L}$ belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum $μ$ is taken in the optimal …
We prove well-posedness results and a priori bounds on the solution of the Helmholtz equation $\nabla\cdot(A\nabla u) + k^2 n u = -f$, posed either in $\mathbb{R}^d$ or in the exterior of a star-shaped Lipschitz obstacle, for a class of random $A$ and $n,$ random data $f$, and for all $k>0$. The particular class of $A$ and $n$ and the conditions on the obstacle ensure that the problem is nontrapping almost surely. These are the first well-posedness results and a priori bounds for the stochastic Helmholtz equation for arbitrarily large $k$ and for $A$ and $n$ varying independently of $k$. These…
