A remark on well-posedness for hyperbolic equations with singular coefficients

Local Petrovskii lacunas at parabolic singular points of wavefronts of strictly hyperbolic PDE's

We enumerate the local Petrovskii lacunas (that is, the domains of local regularity of the principal fundamental solutions) of strictly hyperbolic PDE's with constant coefficients in $R^N$ at the parabolic singular points of their wavefronts. These points form the next difficult family of classes of the natural classification of singular points after the so-called simple singularities, studied previously. Also we promote a computer program counting for topologically different morsifications of critical points of smooth functions, and hence also for local components of the complement of a gener…

A remark on the Glimm scheme for inhomogeneous hyperbolic systems of balance laws

General hyperbolic systems of balance laws with inhomogeneous flux and source are studied. Global existence of entropy weak solutions to the Cauchy problem is established for small $BV$ data under appropriate assumptions on the decay of the flux and the source with respect to space and time. There is neither a hypothesis about equilibrium solution nor about the dependence of the source on the state vector as previous results have assumed.

Generalised Fourier integral operator methods for hyperbolic equations with singularities

This article addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalised functions. We employ the recently developed theory of generalised Fourier integral operators to construct parametrices for the solutions and to describe propagation of singularities in this setting. As required tools, the construction of generalised solutions to eikonal and transport equations is given and results on the microlocal regularity of the kernels of generalised Fourier integral operators …

二阶双曲方程的一种新准奇性形成机制

A New Quasi-Singularity Formation Mechanism for Second-order Hyperbolic Equations

该研究发现二维和三维线性及非线性双曲波动方程存在新准奇性形成机制，证明有限时间内可存在输入使波场Hölder范数超任意界，且该准奇性源于方程结构与输入特征的结合。

含法向双曲捕获的奇性传播

Propagation of Singularity with Normally Hyperbolic Trapping

该研究通过构造基于不稳定流形与纤维无穷远交集吹胀的新型符号类，证明了带法向双曲捕获的微局部估计，得出Kerr(-de Sitter)时空标量波动方程的该类估计较标准奇性传播的额外损失任意小的结论。

奇异扰动双曲型方程组Cauchy问题解的渐近展开。第2部分：初始条件

Asymptotics of the solution of the Cauchy problem for a singularly perturbed system of hyperbolic equations. Part 2. Initial conditions

本文针对描述两根刚性连接弦振动的奇异扰动双曲型方程组，构造了单个Cauchy问题的渐近小参数展开，确定了展开项的初始条件并估算了余项。

广义解法与超双曲系统中的非线性条件及强奇异数据

Generalized Solutions to Hyperbolic Systems with Nonlinear Conditions and Strongly Singular Data

本文基于Colombeau代数研究了具有非线性非局部边界条件的双曲系统广义解的存在性和唯一性，处理了强奇异数据和非Lipschitz非线性情况。

带漂移项的非散度型退化/奇异双曲方程的边界能控性

Boundary controllability for degenerate/singular hyperbolic equations in nondivergence form with drift

该研究针对带漂移项的非散度型退化/奇异波动方程的零能控问题，借助能量方法与边界能观性，给出了局域于非退化边界点的控制下边界能控性的相关条件。

Hyperbolic-parabolic singular perturbation for Kirchhoff equations with weak dissipation

We consider Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipative term whose coefficient may tend to 0 as t -> + infinity (weak dissipation). In this note we present some recent results concerning existence of global solutions, and their asymptotic behavior both as t -> + infinity and as epsilon -> 0. Since the limit equation is of parabolic type, this is usually referred to as a hyperbolic-parabolic singular perturbation problem. We show in particular that the equation exhibits hyperbolic or parabolic behavior depending on the value…

具有不可对角化主部分和可变重数的双曲系统，III：奇异系数

Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, III: singular coefficients

本文研究具有不可对角化主部分和可变重数的双曲系统，针对不连续或分布系数情形，证明了极弱意义下的适定性，并验证了与经典理论的一致性。

具有奇异系数的半线性双曲系统初/边界问题

Initial-Boundary Problems for Semilinear Hyperbolic Systems with Singular Coefficients

本文利用Colombeau代数研究了具有奇异系数的半线性双曲系统初/边界问题的全局广义解存在性和唯一性。

Hyperbolic--parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation

We consider the hyperbolic-parabolic singular perturbation problem for a nondegenerate quasilinear equation of Kirchhoff type with weak dissipation. This means that the dissipative term is multiplied by a coefficient b(t) which tends to 0 as t tends to +infinity. The case where b(t) behaves like (1+t)^{-p} with p<1 has recently been considered. The result is that the hyperbolic problem has a unique global solution, and the difference between solutions of the hyperbolic problem and the corresponding solutions of the parabolic problem converges to zero both as t tends to +infinity and as epsilon…

Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations with weak dissipation

We consider mildly degenerate Kirchhoff equations with a small parameter and a weak dissipation term. We prove the existence of global solutions when the parameter is small with respect to the size of initial data. Then we provide global-in-time error estimates on the difference between the solution of our problem and the solution of the corresponding first order problem.

delta波用于具有积分边界条件的强奇异初始-边界双曲问题

Delta Waves for a Strongly Singular Initial-Boundary Hyperbolic Problem with Integral Boundary Condition

本文研究了具有强奇异初始-边界条件的双曲方程的delta波解，通过构造分布极限确定了广义解的宏观行为，并将解分解为奇异和常规部分。

Remarks on Ruggeri's First Model of Dissipative Fluid Dynamics as a Symmetric Hyperbolic System

This note establishes properties of a model of dissipative fluid dynamics as a symmetric hyperbolic system by which Ruggeri once triggered the development of Rational Extended Thermodynamics. These properties, i.e., (1) convergence of solutions to those of the Navier-Stokes-Fourier equations and (2) nonlinear asymptotic stability of homogeneous reference solutions, also hold in the analogous situation for barotropic fluids.

Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients

This paper is devoted to strictly hyperbolic systems and equations with non-smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of generalised functions. Extending earlier results on symmetric hyperbolic systems, we introduce generalised strict hyperbolicity, construct symmetrisers, prove an appropriate Gårding inequality and establish existence, uniqueness and regularity of generalised solutions. Under additional regularity assumptions on the coefficients, when a classical solution of…

具有一般源项的非线性双曲组解的奇性形成

Formation of singularities in solutions to nonlinear hyperbolic systems with general sources

本文研究带一般源项的非线性双曲组，借助 John 与 Hörmander 的方法证明适当光滑初值下 $C^1$ 解必在有限时间内因导数爆破而形成奇性。

关于双曲守恒律解唯一性的一个注记

A remark on the uniqueness of solutions to hyperbolic conservation laws

本文针对严格双曲守恒律方程组，证明了任何满足 Liu 容许性条件且取值于半群定义域内的弱解必与唯一的 Lipschitz 半群轨迹重合，从而强化了解的唯一性结论。

Singular Bohr-Sommerfeld conditions for 1D Toeplitz operators: hyperbolic case

In this article, we state the Bohr-Sommerfeld conditions around a singular value of hyperbolic type of the principal symbol of a self-adjoint semiclassical Toeplitz operator on a compact connected Kähler surface. These conditions allow the description of the spectrum of the operator in a fixed size neighbourhood of the singularity. We provide numerical computations for three examples, each associated to a different topology.

Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions

We are interested in evolution phenomena on star-like networks composed of several branches which vary considerably in physical properties. The initial boundary value problem for singularly perturbed hyperbolic differential equation on a metric graph is studied. The hyperbolic equation becomes degenerate on a part of the graph as a small parameter goes to zero. In addition, the rates of degeneration may differ in different edges of the graph. Using the boundary layer method the complete asymptotic expansions of solutions are constructed and justified.

On the well posedness of a mathematical model for a singular nonlinear fractional pseudo-hyperbolic system with nonlocal boundary conditions and frictional damping

This paper is devoted to the study of the well-posedness of a singular nonlinear fractional pseudo-hyperbolic system. The fractional derivative is described in Caputo sense. The equations are supplemented by classical and nonlocal boundary conditions. Upon some a priori estimates and density arguments, we establish the existence and uniqueness of the strongly generalized solution for the associated linear fractional system in some Sobolev fractional spaces. On the basis of the obtained results for the linear fractional system, we apply an iterative process in order to establish the well-posedn…

Note on stability of an abstract coupled hyperbolic-parabolic system: singular case

In this paper we try to complete the stability analysis for an abstract system of coupled hyperbolic and parabolic equations $$ \left\{ \begin{array}{lll} \ds u_{tt} + Au - A^αw = 0, \\ w_t + A^αu_t + A^βw = 0,\\ u(0) = u_0, u_t(0) = u_1, w(0) = w_0, \end{array} \right. $$ where $A$ is a self-adjoint, positive definite operator on a complex Hilbert space $H$, and $(α, β) \in [0,1] \times [0,1]$, which is considered in \cite{Amk}, and after, in \cite{liu1}. Our contribution is to identify a fine scale of polynomial stability of the solution in the region $ S_3: = \left\{(α,β) \in [0,1] \times […

Global Well-posedness of a Class of Singular Hyperbolic Cauchy Problems

The goal of this paper is to establish a global well-posedness, cone condition and loss of regularity for singular hyperbolic equations with coefficients in { $L^1((0,T];C^\infty(\mathbb{R}^n)) \cap C^1((0,T];C^\infty(\mathbb{R}^n))$} and Cauchy data in an appropriate Sobolev space tailored to a metric on the phase space. The coefficients are unbounded near the singular hyperplane $t=0$ and polynomially growing as $|x| \to \infty.$ The singular behavior is characterized by the blow-up rate of the coefficients and their first $t$-derivatives near $t=0.$ In order to study the interplay of the si…

Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller--Segel type models

In this paper we deal with diffusive relaxation limits of nonlinear systems of Euler type modeling chemotactic movement of cells toward Keller--Segel type systems. The approximating systems are either hyperbolic--parabolic or hyperbolic--elliptic. They all feature a nonlinear pressure term arising from a \emph{volume filling effect} which takes into account the fact that cells do not interpenetrate. The main convergence result relies on compensated compactness tools and is obtained for large initial data under suitable assumptions on the approximating solutions. In order to justify such assump…

超几何空间上波方程的适定性与散射性

Well-posedness and scattering for wave equations on hyperbolic spaces with singular data

本文研究了超几何空间上波方程和克莱因-戈尔登方程的适定性与散射性，通过弱$L^p$空间框架建立了色散估计，并证明了解的全局适定性与指数渐近稳定性。

双曲弹性流的奇异性：收敛、量化与爆破

Singularities of the hyperbolic elastic flow: Convergence, quantization and blow-ups

研究双曲平面中闭曲线与开曲线的弹性流，在无初始能量小假设下证明收敛定理，量化奇异性能耗至少为8并分类爆破，显式揭示$λ$-8字图形的奇异极限。

Refined Asymptotics for Minimal Graphs in the Hyperbolic Space

We study the boundary behaviors of solutions $f$ to the Dirichlet problem for minimal graphs in the hyperbolic space with singular asymptotic boundaries and characterize the boundary behaviors of $f$ at the points strictly located in the tangent cones at the singular points on the boundary. For $n=2$, we also obtain a refined estimate of $f.$

偶数维空间中仿射双曲球面的全局解析性

Global analyticity of affine hyperbolic spheres in the even dimensional space

该研究针对一类源于几何问题的边界奇异蒙日-安培方程，给出解的各阶边界导数的最优估计，证明偶数维下解的全局解析性，进而得到偶数维仿射双曲球面的全局解析性结论。

临界弱双曲与奇异系统的渐近性与散射

Asymptotics and Scattering for Critically Weakly Hyperbolic and Singular Systems

本文研究含临界奇异系数的一阶线性双曲系统，改进弱双曲情形结果并量化解的渐近性、散射问题与正则性损失，还探讨其在多类方程及大爆炸奇点的应用。

Energy Estimates and Global Well-posedness for a Broad Class of Strictly Hyperbolic Cauchy Problems with Coefficients Singular in Time

The goal of this paper is to establish a global well-posedness for a broad class of strictly hyperbolic Cauchy problems with coefficients in $C^2((0,T];C^\infty(\mathbb{R}^n))$ growing polynomially in $x$ and singular in $t$. The problems we study are of strictly hyperbolic type with respect to a generic weight and a metric on the phase space. The singular behavior is captured by the blow-up of the first and second $t$-derivatives of the coefficients which allows the coefficients to be either logarithmic-type or oscillatory-type near $t=0$. To arrive at an energy estimate, we perform a conjuga…

Optimal decay-error estimates for the hyperbolic-parabolic singular perturbation of a degenerate nonlinear equation

We consider a degenerate hyperbolic equation of Kirchhoff type with a small parameter epsilon in front of the second-order time-derivative. In a recent paper, under a suitable assumption on initial data, we proved decay-error estimates for the difference between solutions of the hyperbolic problem and the corresponding solutions of the limit parabolic problem. These estimates show in the same time that the difference tends to zero both as epsilon -> 0, and as t -> +infinity. In particular, in that case the difference decays faster than the two terms separately. In this paper we consider the co…

Singular perturbation results for linear partial differential-algebraic equations of hyperbolic type

We consider constrained partial differential equations of hyperbolic type with a small parameter $\varepsilon>0$, which turn parabolic in the limit case, i.e., for $\varepsilon=0$. The well-posedness of the resulting systems is discussed and the corresponding solutions are compared in terms of the parameter $\varepsilon$. For the analysis, we consider the system equations as partial differential-algebraic equation based on the variational formulation of the problem. For a particular choice of the initial data, we reach first- and second-order estimates. For general initial data, lower-order es…

Some remarks on singular solutions of nonlinear elliptic equations. I

The paper concerns singular solutions of nonlinear elliptic equations.