Global Existence for Systems of Nonlinear Wave Equations in 3D with Multiple Speeds
Thomas C. Sideris, Shu-Yi Tu
在小初值和非线性项满足零条件的情况下，构造了具有多个传播速度的非线性波动方程组的初值问题的整体光滑解.
Global smooth solutions to the initial value problem for systems of nonlinear wave equations with multiple propagation speeds will be constructed in the case of small initial data and nonlinearities satisfying the null condition.
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We prove the existence of a new type of solutions to a nonlinear Schrödinger system. These solutions, which we call "multi-speeds solitary waves", are behaving at large time as a couple of scalar solitary waves traveling at different speeds. The proof relies on the construction of approximations of the multi-speeds solitary waves by solving the system backwards in time and using energy methods to obtain uniform estimates.
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Finite speed of propagations of the electromagnetic field in nonlinear isotropic dispersive mediums
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Global existence of solutions to multiple speed systems of quasilinear wave equations in exterior domains
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Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds
证明了时空周期平均零不可压平流中Kolmogorov-Petrovsky-Piskunov（KPP）型波前的存在性，并建立了最小速度的变分（极小化）公式.我们通过考虑一个正则化的波前方程的波前解序列的极限来讨论其存在性，其中非线性项是燃烧型的，具有点火截止。极限阵面方程是退化抛物型方程，不存在强解，但阵面的单调性和退化正则性使其具有必要的紧性。我们应用一个动态参数来证明所构造的KPP行进锋以最小速度传播，并推导出速度变分公式。动力学方法避免了波前方程的退化，并利用了反应扩散对流方程的抛物型最大值原理。动态方法不依赖于存在的旅行前。
Imaging nonlinearity coefficient and sound speed with the JMGT equation in frequency domain
本文利用JMGT方程在频域中通过两个源重建非线性系数和声速，并推导了正则化性质。
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本文给出了具有双幂非线性项的半线性椭圆型方程正解存在唯一性的等价条件。作为奖励，我们给出了一个更简单的证明，我们以前的结果，唯一性条件来自存在性条件。
In this paper, we use variational methods to prove the existence of positive solution for the following class of elliptic equation $$ -ε^{2}Δ{u}+V(z)u=f(u) \,\,\, \mbox{in} \,\,\, \mathbb{R}^{2}, $$ where $ε>0$ is a positive parameter, $V$ is a saddle-like potential and $f$ has an exponential critical growth.
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得到了非线性Schrodinger方程局部解和整体解的存在唯一性的新结果，包括自相似整体解。我们的分析是在Marcinkiewicz空间的框架。
Local existence and nonexistence of solutions to the Hardy parabolic equation with general nonlinearity
本文研究了Hardy抛物方程带一般非线性项的局部解存在性与不存在性，通过上解方法给出了初始函数的最优积分性条件。
In this note we study analytically and numerically the existence and stability of standing waves for one dimensional nonlinear Schrödinger equations whose nonlinearities are the sum of three powers. Special attention is paid to the curves of non-existence and curves of stability change on the parameter planes.
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本文研究了非紧连通度量图上的超临界非线性Schr{ö}dinger方程，在无小质量假设下证明了具预定质量的驻波解的存在性。
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The present paper is devoted to the study of the global solution and nonlinear stability to the coupled complex Ginzburg Landau and Burgers equations for sequential flames
We are concerned with the nonlinear problem $u_t=u_{xx}+f(u)$, where $f$ is of combustion type, coupled with the Stefan-type free boundary $h(t)$. According to [4,5], for some critical initial data, the transition solution $u$ locally uniformly converges to $θ$, which is the ignition temperature of $f$, and the free boundary satisfies $h(t)=C\sqrt{t}+o(1)\sqrt{t}$ for some positive constant $C$ and all large time $t$. In this paper, making use of two different approaches, we establish more accurate upper and lower bound estimates on $h(t)$ for the transition solution, which suggest that the no…
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Existence and finite speed of propagation of solutions for a multi-dimensional fractional thin-film equation
本文研究有界域上带Neumann边界条件的谱分数阶Laplacian薄膜方程初边值问题，在强滑移完全润湿条件下通过局部熵估计证明了解的存在性、支撑集有限传播速度及等待时间下界。
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本文引入包含奇性与无界低阶项的一般非强制非线性抛物问题的重整化解概念，并证明其存在性与唯一性。
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In this paper, we consider the nonlinear inhomogeneous compressible elastic waves in three spatial dimensions when the density is a small disturbance around a constant state. In homogeneous case, the almost global existence was established by Klainerman-Sideris [1996_CPAM], and global existence was built by Agemi [2000_Invent. Math.] and Sideris [1996_Invent. Math., 2000_Ann. Math.] independently. Here we establish the corresponding almost global and global existence theory in the inhomogeneous case.
Existence and blow up of small-amplitude nonlinear waves with a sign-changing potential
研究了任意维空间中具有变号位势的非线性波动方程。如果势很小且衰减很快，则小振幅解的存在性是由非线性项驱动的。然而，如果势导致线性化问题的增长，那么开始时很小的解可能会在有限时间内爆破。
Existence for noncoercive nonlinear elliptic equations with two lower-order terms
本文针对系数属于Marcinkiewicz和Lorentz空间的非强制非线性椭圆方程，证明了Dirichlet问题解的存在性，并研究了解的高阶可积性。
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