Markus Keel, Hart Smith, Christopher Sogge
We establish global existence in 3+1 dimensions of small-amplitude solutions of quasilinear Dirichlet-wave equations satisfying the null condition outside of star-shapped obstacles.
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Normalized solutions of quasilinear Schrödinger equation with Sobolev critical exponent on star-shaped bounded domains
本文研究星形有界域上具Sobolev临界指数的拟线性Schrödinger方程，通过双偶方法证明了基态与山路解两类正归一化解的存在性。
We study long time existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. In particular, we obtain exterior domain analogs of the four dimensional results of Hörmander where the nonlinearity is permitted to depend on the solution not just its first and second derivatives. Previous proofs in exterior domains omitted this dependence as it did not mesh well with the energy methods in use.
We study the initial-boundary value problem of quasilinear wave equations outside of a star-shaped obstacle in four space dimensions, in which the nonlinear term under consideration may explicitly depend on the unknown function itself. By some new $L^{\infty}_{t}L^{2}_{x}$ and weighted $L^{2}_{t,x}$ estimates for the unknown function itself, together with energy estimates and KSS estimates, for the quasilinear obstacle problem we obtain a lower bound of the lifespan $T_{\varepsilon}\geq \exp{(\frac{c}{\varepsilon^2})}$, which coincides with the sharp lower bound of lifespan estimate for the co…
Long-Time Existence of Quasilinear Wave Equations Exterior to Star-shaped Obstacle in $2\mathbf{D}$
本文研究二维星形障碍外部拟线性波动方程小数据解的长时间存在性，通过建立Morawetz型能量估计和加权L^2乘积估计，得到几乎尖锐的存在寿命下界。
Liouville-type results for some quasilinear anisotropic elliptic equations
本文证明了拟线性各向异性椭圆方程稳定解的Liouville型定理，并涵盖纯Finsler p-Laplacian情形。
We prove global existence for quasilinear wave equations outside of a wide class of obstacles. The obstacles may contain trapped hyperbolic rays as long as there is local exponential energy decay for the associated linear wave equation. Thus, we can handle all non-trapping obstacles. We are also able to handle non-diagonal systems satisfying the appropriate null condition.
In this paper we prove global existence for certain multispeed Dirichlet-wave equations with quadratic nonlinearities outside of obstacles. We assume the natural null condition for systems of quasilinear wave equations with multiple speeds. The null condition only puts restrictions on the self-interactions of each wave family. We use the method of commuting vector fields and weighted space-time $L^2$ estimates.
We explore the global existence of solutions to systems of quasilinear wave equations satisfying the null condition when the initial data are sufficiently small. We adapt an approach of Keel, Smith, and Sogge, which relies on integrated local energy estimates and a weighted Sobolev estimate that yields decay in $|x|$, by using the $r^p$-weighted local energy estimates of Dafermos and Rodnianski. One advantage of this approach is that all time-dependent vector fields can be avoided and the proof can be readily adapted to address wave equations exterior to star-shaped obstacles.
The authors show that bilinear estimates for null forms hold for Dirichlet-wave equations outside of convex obstacle. This generalizes results for the Euclidean case of Klainerman and Machedon, and of Sogge for the variable coefficient boundaryless case. The estimates are used to prove a local existence theorem for semilinear wave equations satisfying the null condition.
Global well-posedness for a system of quasilinear wave equations on a product space
研究乘积空间上拟线性波动方程组的全局适定性，通过结合光锥内外能量估计，证明小正则初始数据的全局存在性。
We investigate an initial-boundary value problem for a quasilinear nonhomogeneous, anisotropic Maxwell system subject to an absorbing boundary condition of Silver & Müller type in a smooth, bounded, strictly star-shaped domain of $\mathbb{R}^{3}$. Imposing usual smallness assumptions in addition to standard regularity and compatibility conditions, a nonlinear stabilizability inequality is obtained by showing nonlinear dissipativity and observability-like estimates enhanced by an intricate regularity analysis. With the stabilizability inequality at hand, the classic nonlinear barrier method is …
This article focuses on long-time existence for quasilinear wave equations with small initial data in exterior domains. The nonlinearity is permitted to fully depend on the solution at the quadratic level, rather than just the first and second derivatives of the solution. The corresponding lifespan bound in the boundaryless case is due to Lindblad, and Du and Zhou first proved such long-time existence exterior to star-shaped obstacles. Here we relax the hypothesis on the geometry and only require that there is a sufficiently rapid decay of local energy for the linear homogeneous wave equation,…
Hörmander proved global existence of solutions for sufficiently small initial data for scalar wave equations in $(1+4)-$dimensions of the form $\Box u = Q(u, u', u'')$ where $Q$ vanishes to second order and $(\partial_u^2 Q)(0,0,0)=0$. Without the latter condition, only almost global existence may be guaranteed. The first author and Sogge considered the analog exterior to a star-shaped obstacle. Both results relied on writing the lowest order terms $u\partial_αu = \frac{1}{2}\partial_αu^2$ and as such do not immediately generalize to systems. The current study remedies such and extends both re…
We study the behaviour of global solutions to the quasilinear heat equation with a reaction localized $$ u_t=(u^m)_{xx}+a(x) u^p, $$ $m, p>0$ and $a(x)$ being the characteristic function of an interval. we prove that there exists $p_0=\max\{1,\frac{m+1}2\}$ such that all global solution are bounded if $p>p_0$, while for $p\le p_0$ all the solution are global and unbounded. In the last case, we prove that if $p<m$ the grow-up rate is different to the one obtained when $a(x)\equiv1$, while if $p>m$ the grow-up rate coincides with that rate, but only inside the support of $a$; outside the interva…
We prove almost global existence for multiple speed quasilinear wave equations with quadratic nonlinearities in three spatial dimensions. We prove new results both for Minkowski space and also for nonlinear Dirichlet-wave equations outside of star shaped obstacles. The results for Minkowski space generalize a classical theorem of John and Klainerman. Our techniques only uses the classical invariance of the wave operator under translations, spatial rotations, and scaling. We exploit the $O(|x|^{-1})$ decay of solutions of the wave equation as opposed to the more difficult $O(|t|^{-1})$ decay. A…
This article focuses on almost global existence for quasilinear wave equations with small initial data in 4-dimensional exterior domains. The nonlinearity is allowed to depend on the solution at the quadratic level as well as its first and second derivatives. For this problem in the boundaryless setting, Hörmander proved that the lifespan is bounded below by $\exp(c/ε)$ where $ε>0$ denotes the size of the Cauchy data. Later Du, the second author, Sogge, and Zhou showed that this inequality also holds for star-shaped obstacles. Following up on the authors' work in the 3-dimensional case, we wea…
Solutions stable outside a compact set for anisotropic quasilinear equations with negative exponent or exponential nonlinearity
研究Finsler p-Laplacian下负幂和指数非线性方程在紧集外稳定解的不存在性，给出尖锐临界阈值，并修复了早期证明的缺陷。
The paper is devoted to investigating long time behavior of smooth small data solutions to 3-D quasilinear wave equations outside of compact convex obstacles with Neumann boundary conditions.
Stability of the Catenoid for the Hyperbolic Vanishing Mean Curvature Equation Outside Symmetry
本文证明了 n≥5 维悬链面作为闵可夫斯基空间中双曲消失平均曲率方程稳态解的非线性渐近稳定性，通过新的轮廓构造、调制分析及向量场方法适配，处理了无对称性假设下余维一初值扰动的拟线性波动方程稳定性问题。
We consider the elliptic equation $-Δu = f(u)$ in the whole $\R^{2m}$, where $f$ is of bistable type. It is known that there exists a saddle-shaped solution in $\R^{2m}$. This is a solution which changes sign in $\R^{2m}$ and vanishes only on the Simons cone ${\mathcal C}=\{(x^1,x^2)\in\R^m\times\R^m: |x^1|=|x^2|\}$. It is also known that these solutions are unstable in dimensions 2 and 4. In this article we establish that when $2m=6$ every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavi…
We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation $-Δu = f(u)$ in the whole $\R^{2m}$, where $f$ is of bistable type. It is known that in dimension $2m=2$ there exists a saddle-shaped solution. This is a solution which changes sign in $\R^2$ and vanishes only on $\{|x_1|=|x_2|\}$. It is also known that this solution is unstable. In this article we prove the existence of saddle-shaped solutions in every even dimension, as well as their instability in the case of dimension $2m=4$. More precisely, our main result establishes that if $…
The purpose of this expository note is to revisit Morawetz's method for obtaining a lower bound on the rate of exponential decay of waves for the Dirichlet problem outside star-shaped obstacles, and to discuss the uniqueness of the sphere as the extremizer of the sharp lower bound proved by Ralston.
We prove that the quintic Schrodinger equation with Dirichlet boundary conditions is locally well posed for H^{1}_{0} data on any smooth, non-trapping domain of R^3. The key ingredient is a smoothing effect in L^{5}_{x}L^{2}_{t} for the linear equation. We also derive scattering results for the whole range of defocusing sub-quintic Schrodinger equations outside star-shaped domains.
We study quasilinear degenerate singular elliptic equation of type -Delta_p u = \frac{u^{p^*(s)-1}}{|y|^t}$ in a smooth bounded domain Ωin R^n=R^k \times R^{N-k}$, x=(y,z) in R^k \times R^{N-k}, 2 \leq k<N and N \geq 3, 1<p<2, 0\leq s\leq p, 0\leq t\leq s, p^*(s)=\frac{p(n-s)}{n-p}. We study existence of solution for t<s, non-existence in a star-shaped domain for t=s and s<k(\frac{p-1}{p}). We also show that solution is in C^{1,\al}(Ω) for some 0<\al<1 provided t<\frac{k}{N}(\frac{p-1}{p}). The regularity of solution can be improved to the class W^{2,p}(Ω) when t<k(\frac{p-1}{p}). We also stud…
The good commutator approach to global asymptotics for the Schrödinger equation with variable coefficients
该研究提出良好对易子方法结合波包测试、两尺度椭圆分析，建立变系数薛定谔方程解的时间衰减与全局渐近性，还获准线性薛定谔方程小数据全局存在性等新结果，拓宽了物理空间方法的适用范围。
Nonlinear Schrödinger equation in an exterior domain
该研究针对非捕获障碍物外区域的三次、五次非线性薛定谔方程，利用局部光滑效应与Strichartz型估计，证明了其在二维、三维中围绕零解及非平凡轨迹的局部精确可控性，并讨论了相关开放问题。
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