Analysis of singular solutions for two nonlinear wave equations
Massimo Fonte
这是一篇为S.I.S.S.A.哲学博士学位提交的论文。国际安全标准协会
This is a Thesis submitted for the degree of Doctor Philosophiae at S.I.S.S.A./I.S.A.S.
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Singular phenomena in nonlinear elliptic problems. From blow-up boundary solutions to equations with singular nonlinearities
在这篇综述中，我们报告了一些最近的结果有关的各种奇异现象的研究中出现的一些类的非线性椭圆方程。我们建立定性结果的存在性，不存在性或唯一性的解决方案，我们专注于以下类型的问题：（i）blow-up边界解的Logistic方程;（ii）Lane-Emden-Fowler方程奇异非线性和次二次对流项。我们研究了这些问题中涉及的各种条款的综合影响：次线性或超线性非线性，奇异非线性项，对流非线性，以及符号变化的潜力。我们还考虑到分歧的非线性问题，我们建立了精确的速率衰减的解决方案，在一些具体的情况下。我们的方法结合了标准的技术基础上的最大值原理与非标准的参数，如Karamata定期变化理论。
Nonlocal problems with singular nonlinearity
研究了一类含有分数阶$p$ -Laplacian算子和奇异非线性项的非线性非局部问题解的存在唯一性.
We investigate the homogeneous Dirichlet problem in uniformly convex domains for a large class of degenerate elliptic equations with singular zero order term. In particular we establish sharp existence and uniqueness results of positive viscosity solutions.
Asymptotic stability of singular solution to nonlinear heat equation
本文讨论了非线性热方程奇异定态在加权Lebesgue范数下的渐近稳定性。
We prove sharp partial regularity criteria of nonlinear potential theoretic nature for the Lebesgue-Serrin-Marcellini extension of nonhomogeneous singular multiple integrals featuring $(p,q)$-growth conditions.
Singular standing-ring solutions of nonlinear partial differential equations
我们提出了一个一般的框架来构造奇异解的非线性发展方程，成为奇异的d维球，其中d>1。这些解的渐近分布和爆破速率与相应的一维方程在一点处变为奇异的解的渐近分布和爆破速率相同。我们对下列方程的这些新的奇异解进行了详细的数值研究：非线性薛定谔方程，双调和非线性薛定谔方程，非线性热方程和非线性双调和热方程.
Singular limit of a nonlinear fourth order inhomogeneous equation
本文利用匹配渐近展开的方法，得到一个几何运动作为非线性四阶非齐次方程的奇异极限。
In this paper, we prove the existence and regularity of weak positive solutions for a class of nonlinear elliptic equations with a singular nonlinearity, lower order terms and $L^{1}$ datum in the setting of variable exponent Sobolev spaces. We will prove that the lower order term has some regularizing effects on the solutions. This work generalizes some results given in \cite{1}.
Bifurcation and Asymptotics for Elliptic Problems with Singular Nonlinearity
本文研究了一类带奇异非线性项的椭圆型方程解的存在唯一性。我们考虑了以下几类问题：（i）具有次线性非线性和两个参数的奇异问题;（ii）渐近线性和奇异非线性在分歧问题中的联合作用;（iii）一类具有次二次对流项的奇异椭圆问题的分歧。在某些具体情况下，我们还建立了解在歧点附近的渐近性态。我们的分析依赖于椭圆型方程的最大值原理结合适当的估计。
A Nonlinear Singular Diffusion Equation with Source
讨论了一类带非线性边界条件的奇异扩散方程解的存在唯一性及其对初值的依赖性。证明了存在唯一的全局光滑解，且该解连续依赖于初值。
In this paper, we consider Kirchhoff-Schrodinger equations with singular exponential nonlinearities in R^4,using singular Adams inequality and variational techniques, we get the existence of ground state solutions. Moreover, we also get the same results without the Ambrosetti-Rabinowitz (AR) condition.
We establish an optimal C^{1,α}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.
On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlinearity
该研究利用变分法与逼近方法，考察兼具局部与非局部 Laplace 算子的奇异半线性椭圆方程，证明参数依赖情形下弱解存在及扰动情形下多解存在。
Logarithmic singularities of solutions to nonlinear partial differential equations
本文构造了一类非线性偏微分方程的奇异解族，这些奇异解族具有T.小林在我们的家庭解决方案的首项包含一个对数，可能乘以单项式。作为应用，我们研究了具有二次非线性项的非线性波动方程。证明是通过减少到一个Fuchsian方程的奇异系数。
Singularities of solutions of nonlocal nonlinear equations
本文研究非局部非线性方程弱解的局部奇异性，证明零容量集合在可积性条件下可去，并用基本解刻画孤立奇点附近的渐近行为。
Nonlinear Fokker-Planck equations with singular integral drifts and McKean-Vlasov SDEs
本文证明了Sobolev空间H^{-1}中具有奇异漂移的非线性Fokker-Planck方程的适定性，并给出了其在McKean-Vlasov方程强解存在性中的应用。
On a class of nonlinear elliptic, anisotropic singular perturbations problems
本文研究了柱域上非线性椭圆型各向异性奇异扰动问题解的渐近性态，给出了极限问题，证明了强收敛性，并给出了在积分微分问题中的应用.
Singular solutions of the L^2-supercritical biharmonic Nonlinear Schrodinger equation
利用渐近分析和数值模拟研究了超临界双调和NLS的峰值型奇异解。这些解具有四次方根爆破速率，并以准自相似的普适轮廓坍缩，这是一个四阶非线性本征值问题的零Hamilton解.
Focusing Singularity in a Derivative Nonlinear Schrödinger Equation
数值研究了一类具有一般幂非线性项$|ψ|^{2σ}ψ_x$的非线性薛定谔方程.在$L^2$ -超临界状态，$σ>1$，我们的模拟表明，有一个有限的时间奇异性。我们得到一个精确的描述的局部结构的解决方案的爆破率和渐近的配置文件，在形式上类似的非线性薛定谔方程与超临界幂律非线性。
Nonlinear wave equations and singular solutions
我们构造解的非线性波动方程是沿着一个指定的非特征超曲面，这是一个函数的图形满足不Eikonal，但另一个偏微分方程的一阶奇异。采用Fuchsian约化方法。
Radial singular solutions of fully nonlinear equations in punctured balls
研究含逆二次势和p次幂吸收项的完全非线性一致椭圆方程，证明穿孔球内径向解在原点奇异并依p完整分类其渐近行为。
This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential.
Eigenfunctions and Very Singular Similarity Solutions of Odd-Order Nonlinear Dispersion PDEs
研究了奇阶非线性色散方程解的渐近性质。构造了全局时间相似解，从而得到了重标常微分方程的本征函数。
A theorem is proved on the uniform estimation of the residual term of the asymptotic expansion with respect to a small parameter of the solution of the initial problem for a singularly perturbed differential operator weakly nonlinear transport equation in the critical case.
We provide the Alexandroff-Bakelman-Pucci estimate and global $C^{1, α}$-regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem with the associated operator.
We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey space). Then, asymptotics of subcritical solutions is determined. These results are compared with conditions on the initial data leading to a finite time blowup.
We consider a class of singular weighted anisotropic $p$-Laplace equations. We provide sufficient condition on the weight function that may vanish or blow up near the origin to ensure the existence of at least one weak solution in the purely singular case and at least two different weak solutions in the purturbed singular case.
We consider a parabolic-type PDE with a diffusion given by a fractional Laplacian operator and with a quadratic nonlinearity of the 'gradient' of the solution, convoluted with a singular term b. Our first result is the well-posedness for this problem: We show existence and uniqueness of a (local in time) mild solution. The main result is about blow-up of said solution, and in particular we find sufficient conditions on the initial datum and on the term b to ensure blow-up of the solution in finite time.
Optimal bilinear control of nonlinear Schrödinger equations with singular potentials
本文考虑一类具有奇异位势的非线性薛定谔方程的最优双线性控制问题。我们证明了问题的适定性和最优控制的存在性。此外，一阶最优性系统严格推导。我们的结果在几个方面推广了\cite{Sp}中的结果。
Solutions to nonlinear Schrödinger equations with singular electromagnetic potential and critical exponent
研究了一类具有临界指数和奇异电磁势的非线性薛定谔方程解的存在性和定性行为。我们所研究的磁矢势是齐次的，包括Aharonov-Bohm类。特别是，变分参数，我们证明了一个结果的多重性的解决方案区分的对称性。
Structurally Stable Singularities for a Nonlinear Wave Equation
对于非线性波动方程$u_{tt} - c(u)\big(c(u) u_x\big)_x~=~0$，已知解在有限时间内会产生奇性。对于一个开的稠密初始数据集，本文给出了解在每个奇点邻域内的详细渐近描述，其中$|u_x|\to\infty$。分析了保守解和耗散解的不同结构。
Singular solutions of semilinear elliptic equations with exponential nonlinearities in 2-dimensions
本文通过新的指数增长率分类和模型非线性项方法，统一构造了二维指数型半线性椭圆方程的奇异解，并推广了Ibrahim等人的Moser-Trudinger型结果。
In this paper we study a singular Finsler double phase problem with a nonlinear boundary condition and perturbations that have a type of critical growth, even on the boundary. Based on variational methods in combination with truncation techniques we prove the existence of at least one weak solution for this problem under very general assumptions. Even in the case when the Finsler manifold reduces to the Euclidean norm, our work is the first one dealing with a singular double phase problem and nonlinear boundary condition.
First eigenvalue and Maximum principle for fully nonlinear singular operators
通过最大值原理，我们定义了一类完全非线性算子的主特征值，这类算子是p-Laplacian算子的非变分等价。我们也得到了一些先验的Holder估计的非负解低于该特征值。
This paper is concerned with multiplicity results for parametric singular double phase problems in $\mathbb{R}^N$ via the Nehari manifold approach. It is shown that the problem under consideration has at least two nontrivial weak solutions provided the parameter is sufficiently small. The idea is to split the Nehari manifold into three disjoint parts minimizing the energy functional on two of them. The third set turns out to be the empty set for small values of the parameter.
Dirichlet boundary conditions for degenerate and singular nonlinear parabolic equations
研究了一类非线性退化抛物型方程在有界区域上解的存在唯一性。我们证明了存在一个唯一的解决方案，满足可能的非齐次Dirichlet边界条件。为此，适当地引入和使用了一些屏障函数。
Ring-type singular solutions of the biharmonic nonlinear Schrodinger equation
给出了d维非线性指数为2σ +1的双调和非线性Schrodinger方程新的奇异解.这些解以准自相似环的轮廓坍塌，环的宽度L（t）在奇点处为零，半径与L^α成比例，其中α=（4-σ）/（σ（d-1））。当4/d \le σ<4时，这些解的爆破速率为1/（3+α），当σ=4时，爆破速率略快于1/4.这些解类似于非线性薛定谔方程的环型解。
Boundary Hölder gradient estimates for fully nonlinear degenerate or singular parabolic equations
本文针对完全非线性退化或奇异抛物型方程，利用紧性方法、障碍函数构造、正则化技术及小扰动解的边界正则性，建立了统一的边界Hölder梯度估计，推广了已有结果。
Mixed local and nonlocal semilinear elliptic equation with strongly singular and critical Choquard nonlinearity
本文研究混合局部与非局部椭圆方程，结合奇异与临界非线性，通过变分方法证明正解的存在性与多重性。
Multiplicity results for mixed local-nonlocal equations with singular and critical exponential nonlinearity in R^2
本文研究R^2中带可变奇异指数与临界指数非线性项的混合局部-非局部方程，利用下解-上解技巧与变分方法证明了至少两个正弱解的存在性。
We investigate a non-homogeneous nonlinear heat equation which involves degenerate or singular coefficients belonging to the $A_2$ class of functions. We prove the existence of a Fujita exponent and describe the dichotomy existence/non-existence of global in time solutions. The $A_2$ coefficient admits either a singularity at the origin or a line of singularities. In this latter case, the problem is related to the fractional laplacian, through the Caffarelli-Silvestre extension and is a first attempt to develop a parabolic theory in this setting.
We consider a nonlinear Neumann problem driven by the $p$-Laplacian. In the reaction term we have the competing effects of a singular and a convection term. Using a topological approach based on the Leray-Schauder alternative principle together with suitable truncation and comparison techniques, we show that the problem has positive smooth solutions.
We consider a nonlinear Dirichlet problem driven by the $(p,q)$-Laplacian with $1<q<p$. The reaction is parametric and exhibits the competing effects of a singular term and of concave and convex nonlinearities. We are looking for positive solutions and prove a bifurcation-type theorem describing in a precise way the set of positive solutions as the parameter varies. Moreover, we show the existence of a minimal positive solution and we study it as a function of the parameter.
We consider a Cauchy Dirichlet problem for a quasilinear second order parabolic equation with lower order term driven by a singular coefficient. We establish an existence result to such a problem and we describe the time behavior of the solution in the case of the infinite time horizon.
In this paper, we consider radial distributional solutions of the quasilinear equation $-Δ_N u=f(u)$ in the punctured open ball $ B_R\backslash\{0\}\subset \RR^N$, $N \geq 2$. We obtain sharp conditions on the nonlinearity $f$ for extending such solutions to the whole domain $B_R$ by preserving the regularity. For a certain class of noninearity $f$ we obtain the existence of singular solutions and deduce upper and lower estimates on the growth rate near the singularity.
In this paper, we consider a nonlinear Fuchsian type partial differential equation of the second order in the complex domain. Under a very weak assumption, we show the uniqueness of the solution. The result is applied to the problem of analytic continuation of local holomorphic solutions of this equation.
C^{1,α} regularity for a class of singular/degenerate fully nonlinear elliptic equations with oblique boundary conditions
本文研究了一类具有斜边界条件的奇异/退化全非线性椭圆方程的粘性解的全局 $C^{1, α}$ 正则性，扩展了 BKO25 的结果至更广的方程类，包括奇异情形。
Singular Phenomena of Solutions for Nonlinear Diffusion Equations involving $p(x)$-\hbox{Laplacian} Operator
研究了含$p(x)$ - \hbox{Laplacian}算子和非线性源的非线性扩散方程齐次$\hbox{Dirichlet}$边值问题解的奇异现象（在有限时间内消失和爆破）.讨论了变指数$p(x)$和初始能量（数据）对解的性质的影响。同时，我们得到了解的临界熄灭指数和爆破指数。
We study a class of elliptic problems, involving a $k$-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. Furthermore, we study the multiplicity of regular solutions and bifurcation diagrams. An essential ingredient of this study is analyzing the number of intersection points between the singular and regular solutions for rescaled problems. In the particular case of the exponential nonlinearity, we obtain the convergence of regular solutions to the singular …
Symmetry of solutions of some semilinear elliptic equations with singular nonlinearities
考虑有界光滑区域上奇异半线性椭圆型方程的正解，Dirichlet边界条件为零.我们通过移动平面方法给出了解的H^1_0部分的一些弱最大值原理和强最大值原理，从而可以推导出解的对称性和单调性。
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。
Singularities of solutions to nonlinear Schrödinger equations
该研究借助波包变换分析非线性薛定谔方程初值问题解的波前集，提出了判定解在特定时刻特定方向属于s阶索伯列夫空间的充分条件，推进了相关方程解的正则性研究。
The Neumann problem for singular fully nonlinear operators
本文研究了二阶椭圆齐次算子在$C^2$有界区域上的Neumann问题。受\cite{bnv}的启发，我们定义了主特征值的概念，并通过最大值原理刻画了它。此外，还给出了Neumann问题解的Lipschitz正则性、唯一性和存在性结果.
The Dirichlet problem for singular fully nonlinear operators
本文证明了完全非线性椭圆算子狄利克雷问题的（粘性）解的存在性，当解的梯度为零时，这些解是退化的或奇异的.对于这类算子，可以推广本征值的概念，本文讨论了当主本征值的inf为正时，即最大值原理和最小值原理都成立时的情形。
Sharp gradient estimates for a class of singular/degenerate fully nonlinear elliptic equations with oblique boundary conditions and Hamiltonian terms
研究一类具斜边界条件和哈密顿项的奇异/退化完全非线性椭圆方程，在一般条件下，于统一框架内建立直至边界的精确 \(C^{1,\alpha}\) 正则性。
Removing singularities for fully nonlinear PDEs
本文针对具有经典密度和雅可比不等式的完全非线性椭圆偏微分方程粘性解，结合Savin小扰动定理与雅可比不等式，提出快速加倍证明以移除半直线及满足单侧条件的奇异性，覆盖蒙日-安培等方程。
Smooth solutions and singularity formation for the inhomogeneous nonlinear wave equation
研究一维非线性非齐次波动方程：$v_{tt} - T(v,x)_{xx} = 0$ .通过构造“解耦”Riccati型方程的光滑解，得到了一个对未知量的全变差没有限制的奇异性形成结果，推广了Lax和第一作者的奇异性结果.这些结果被应用到几个一维双曲线模型，如可压缩欧拉流与一般的压力法，弹性在非均匀介质中，横向MHD流，和可压缩流在变面积管道。
We show that the derivative nonlinear Schrödinger (DNLS) equation is globally well-posed in the weighted Sobolev space $H^{2,2}(\mathbb{R})$. Our result exploits the complete integrability of DNLS and removes certain spectral conditions on the initial data, thanks to Xin Zhou's analysis on spectral singularities in the context of inverse scattering.
We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator plus an indefinite potential. In the reaction we have the competing effects of a singular term and of concave and convex nonlinearities. In this paper the concave term is parametric. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter $λ$ varies. This work continues our research published in arXiv:2004.12583, where $ξ\equiv 0 $ and in the reaction the parametric term is the singular one.
We consider an Allen-Cahn equation with nonlinear diffusion, motivated by the study of the scaling limit of certain interacting particle systems. We investigate its singular limit and show the generation and propagation of an interface in the limit. The evolution of this limit interface is governed by mean curvature flow with a novel, homogenized speed in terms of a surface tension-mobility parameter emerging from the nonlinearity in our equation.
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