Tommaso Leonori, José Miguel Urbano
We study the Cauchy problem for the parabolic infinity Laplace equation. We prove a new comparison principle and obtain uniqueness of viscosity solutions in the class of functions with a polinomial growth at infinity, improving previous results obtained assuming a linear growth.
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We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We obtain decay estimates for large time in Lebesgue spaces.
Via Carleman estimates we prove uniqueness and continuous dependence results for lateral Cauchy problems for linear integro-differential parabolic equations without initial conditions. The additional information supplied prescribes the conormal derivative of the temperature on a relatively open subset of the lateral boundary of the space-time domain.
In Part I of the paper, we prove non-uniqueness of the solution to the Cauchy problem of the Euler equations of an ideal incompressible fluid in dimension two with vorticity in some Lebesgue space. The radially symmetric external force is locally integrable with values in the same Lebesgue space.
In Part II of the paper, we prove linear instability of a certain class of radially symmetric flows of an ideal incompressible fluid in dimension two used in Part I
In the present paper, we prove time decay estimates of solutions in weighted Sobolev spaces to the second order evolution equation with fractional Laplacian and damping for data in Besov spaces. Our estimates generalize the estimates obtained in the previous studies. The second aim of this article is to apply these estimates to prove small data global well-posedness for the Cauchy problem of the equation with power nonlinearities. Especially, the estimates obtained in this paper enable us to treat more general conditions on the nonlinearities and the spatial dimension than the results in the p…
Decay of Mass of the Solution to the Cauchy Problem of the p-Laplacian with Absorption on Infinite Graphs
研究了无限图上p-拉普拉斯方程解的质量衰减问题，通过能量不等式和嵌入结果，得出了时间稳定性的精确衰减率及通用上界。
Large time behavior of the solution to the Cauchy problem for the discrete p-Laplacian with density on infinite graphs
研究了无限图上离散p-拉普拉斯方程初值问题的大时间行为，证明了非负解的时间稳定化速率及密度趋于零时的上界。
We establish the existence and uniqueness of variational solution to the nonlinear Neumann boundary problem for the $p^{th}$-Sub-Laplacian associated to a system of Hörmander vector fields
In this paper we study a semilinear problem for the fractional laplacian that are the counterpart of the Neumann problems in the classical setting. We show uniqueness of minimal energy solutions for small domains.
On the uniqueness of eigenfunctions for the vectorial $p$-Laplacian
该研究针对向量值特征函数的非线性特征值问题，给出了对应瑞利商极小元的简洁唯一性证明，推进了向量值p-Laplacian相关理论研究。
The paper deals with the existence and uniqueness of a non-trivial solution to non-homogeneous $ p ( x ) -$laplacian equations, managed by non polynomial growth operator in the framework of variable exponent Sobolev spaces on Riemannian manifolds. The mountain pass Theorem is used.
Consider classical solutions to the following Cauchy problem in a punctured space: $ &u_t=Δu -u^p \text{in} (R^n-\{0\})\times(0,\infty); & u(x,0)=g(x)\ge0 \text{in} R^n-\{0\}; &u\ge0 \text{in} (R^n-\{0\})\times[0,\infty). $ We prove that if $p\ge\frac n{n-2}$, then the solution to \eqref{abstract} is unique for each $g$. On the other hand, if $p<\frac n{n-2}$, then uniqueness does not hold when $g=0$; that is, there exists a nontrivial solution with vanishing initial data.
We prove for a two dimensional bounded domain that the Cauchy data for the Schroedinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can determine uniquely the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results.
On the uniqueness of the quasi-geostrophic equation with the fractional Laplacian
本文研究了带分数阶拉普拉斯算子的表面准地转方程解的唯一性，证明在非齐次Besov空间中无需额外假设即可成立，并提取了方程非线性项良定性所需的最小正则性指标 $s=-1/2$。
We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-Δ_{p(x)}u = f(x,u)$ in $ (0,T)\timesΩ$; $u = 0$ on $(0,T)\times\partialΩ$; $u(0,x)=u_0(x)$ in $Ω$; involving the $p(x)$-Laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.
We propose analogues of Green's and Picone's identities for the $p$-sub-Laplacian on stratified Lie groups. In particular, these imply a generalised Díaz-Saá inequality for the $p$-sub-Laplacian on stratified Lie groups. Using these we derive a comparison principle and uniqueness of a positive solution to nonlinear hypoelliptic equations on general stratified Lie groups extending to the setting of the general stratified Lie groups previously known results on Euclidean and Heisenberg groups.
We discuss the Cauchy problem for anisotropic wave equations. Precisely, we address the question to know which kind of Cauchy data on the lateral boundary are necessary to guarantee uniqueness of solutions of an anisotropic wave equation. In the case where the uniqueness holds, the natural problem that arise in this context is to estimate the solutions, in some appropriate space, in terms of the Cauchy data. We aim in this paper to transfer, via a reduced Fourrier-Bros-Iagolnitzer transform, the known stability estimates for the Cauchy problem for elliptic equations to that for waves equations…
On the Cauchy problem for logarithmic fractional Schr{ö}dinger equation
本文研究幂次在零到一之间、带对数非线性项的分数阶 Schrödinger 方程 Cauchy 问题，并在三类函数空间中建立整体存在性。
Uniqueness and some related estimates for Dirichlet problem with fractional Laplacian
该研究针对分数阶拉普拉斯方程基本Dirichlet型问题的非唯一性现象，建立了锐利的唯一性条件，推导了算子的$L^p$估计，引入加权分数阶Sobolev空间并证明其嵌入关系，为相关问题研究提供基础性质支撑。
Uniqueness of positive solutions for m-Laplacian equations with polynomial non-linearity
本文研究含多项式非线性的m-Laplacian方程正解的唯一性，利用第一特征函数的本质唯一性证明p趋于m时的唯一性，并进一步处理Laplacian方程扰动情形的唯一性结果。
In this paper we study the local behavior of a solution to the $l$th power of Laplacian with singular coefficients in lower order terms. We obtain a bound on the vanishing order of the nontrivial solution. Our proofs use Carleman estimates with carefully chosen weights. We will derive appropriate three-sphere inequalities and apply them to obtain doubling inequalities and the maximal vanishing order.
Existence, comparison principle and uniqueness for doubly nonlinear anisotropic evolution equation
本文证明了一类双重非线性各向异性演化方程在Cauchy-Dirichlet和Cauchy问题中的解的存在性，并在Caratheodory向量场假设下建立了比较原理，进而得到唯一性结果。
We prove a uniqueness theorem for the obstacle problem for linear equations involving the fractional Laplacian with zero Dirichlet exterior condition. The problem under consideration arises as the limit of some logistic-type equations. Our result extends (and slightly strengthens) the known corresponding results for the classical Laplace operator with zero boundary condition. Our proof, as compared with the known proof for the classical Laplace operator, is entirely new, and is based on the probabilistic potential theory. Its advantage is that it may be applied to a wide class of integro-diffe…
Inverse problems for the Bakry-Émery Laplacian on manifolds with boundary -- uniqueness and non-uniqueness
本文研究了带边界 Riemannian 流形上 Bakry-Émery Laplacian 两个相关反问题的唯一性与非唯一性，通过 Dirichlet-to-Neumann 映射恢复度量与权重在边界处的 Taylor 系数。
Existence and uniqueness of weak solutions to a parabolic nonlocal 1-Laplacian equation
本文研究抛物型非局部1-Laplacian方程，采用Rothe时间离散化方法证明了其弱解的存在性与唯一性，并得出弱解在时间上保持1/2-Hölder连续性的新结论。
In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: \begin{eqnarray*} (-Δ)^αu=λa(x)u-b(x)u^p&{\rm in}\,\,\R^N, \end{eqnarray*} where $ α\in(0,1) $, $ N\ge 2 $, $λ>0$, $a$ and $b$ are positive smooth function in $\R^N$ satisfying \[ a(x)\rightarrow a^\infty>0\quad {\rm and}\quad b(x)\rightarrow b^\infty>0\quad{\rm as}\,\,|x|\rightarrow\infty. \] Our proof is based on a comparison principle and existence, uniqueness and asymptotic behaviors of various boundary blow-up solutions for a class of elliptic equations invo…
We discuss the Cauchy problem for anisotropic wave equations. Precisely, we address the question to know which kind of Cauchy data on the lateral boundary are necessary to guarantee the uniqueness of continuation of solutions of an anisotropic wave equation. In the case where the uniqueness holds, the natural problem that arise naturally in this context is to estimate the solutions, in some appropriate space, in terms of norms of the Cauchy data. We aim in this paper to convert, via a reduced Fourrier-Bros-Iagolnitzer transform, the known stability results on the Cauchy problem for elliptic eq…
Existence, uniqueness, and decay results for singular $Φ$-Laplacian systems in $\mathbb{R}^N$
本文利用移位法、变分法及多种分析技术，研究了奇异 $\Phi$-Laplacian 系统解的存在性、先验估计、衰减性与唯一性，并给出了 Diaz-Saa 型结果。
We study the two-dimensional Navier-Stokes equations forced by random noise with a diffusive term generalized via a fractional Laplacian that has a positive exponent strictly less than one. Because intermittent jets are inherently three-dimensional, we instead adapt the theory of intermittent form of the two-dimensional stationary flows to the stochastic approach presented by Hofmanov$\acute{\mathrm{a}}$, Zhu $\&$ Zhu (2019, arXiv:1912.11841 [math.PR]) and prove its non-uniqueness in law.
The existence and uniqueness in Hölder spaces of solutions of the Cauchy problem to parabolic integro-differential equation of the order α\in(0,2) is investigated. The principal part of the operator has kernel m(t,x,y)/|y|^{d+α} with a bounded nondegenerate m, Hölder in x and measurable in y. The result is applied to prove the uniqueness of the corresponding martingale problem.
In this paper we prove the existence and uniqueness of positive classical solution of the fractional Laplacian with singular nonlinearity in a smooth bounded domain with zero Drichlet boundary conditions. By the method of sub-supersolution, we derive the existence of positive classical solution to the approximation problems. In order to obtain the regularity, we first establish the existence of weak solution for the fraction Laplacian. Thanks to \cite{XY}, the regularity follows from the boundedness of weak solution.
The local existence and uniqueness of strong solutions for Cauchy problem of three-dimensional inhomogeneous incompressible Navier-Stokes-Vlasov equations
本文研究了三维不均匀不可压缩纳维-斯托克斯-维拉索方程柯西问题的强解的局部存在性和唯一性，通过线性化和近似解构造方法，丰富了强解的存在性结果。
Consider classical solutions to the parabolic reaction diffusion equation $$ &u_t =Lu+f(x,u), (x,t)\in R^n\times(0,\infty); &u(x,0) =g(x)\ge0, x\in R^n; &u\ge0, $$ where $$ L=\sum_{i,j=1}^na_{i,j}(x)\frac{\partial^2}{\partial x_i \partial x_j}+\sum_{i=1}^nb_i(x)\frac\partial{\partial x_i} $$ is a non-degenerate elliptic operator, $g\in C(R^n)$ and the reaction term $f$ converges to $-\infty$ at a super-linear rate as $u\to\infty$. We give a sharp minimal growth condition on $f$, independent of $L$, in order that there exist a universal, a priori upper bound for all solutions to the above Cauch…
We study the periodic Cauchy problem for an integrable equation with cubic nonlinearities introduced by V. Novikov. Like the Camassa-Holm and Degasperis-Procesi equations, Novikov's equation has Lax pair representations and admits peakon solutions, but it has nonlinear terms that are cubic, rather than quadratic. We show the local well-posedness of the problem in Sobolev spaces and existence and uniqueness of solutions for all time using orbit invariants. Furthermore we prove a Cauchy-Kowalevski type theorem for this equation, that establishes the existence and uniqueness of real analytic solu…
We consider Cauchy's equation of motion for hyperelastic materials. The solution of this nonlinear initial-boundary value problem is the vector field which discribes the displacement which a particle of this material perceives when exposed to stress and external forces. This equation is of greatest relevance when investigating the behaviour of elastic, anisotropic composites and for the detection of defects in such materials from boundary measurements. This is why results on unique solvability and continuous dependence from the initial values are of large interest in materials research and str…
To study the existence and uniqueness of solutions to Cauchy-type problems for fractional q-difference equations with the bi-ordinal Hilfer fractional q-derivative which is an extension of the Hilfer fractional q-derivative. An approach is based on the equivalence of the nonlinear Cauchy-type problem with a nonlinear Volterra q-integral equation of the second kind. Applying an analog of Banach fixed point theorem we prove the uniqueness and the existence of the solution. Moreover, we present an explicit solution to the q-analog of the Cauchy problem for the linear case.
Nonlocal elliptic equations involving logarithmic Laplacian: Existence, non-existence and uniqueness results
本文研究涉及对数拉普拉斯算子的非局部椭圆方程的存在性、非存在性与唯一性，证明Pohozaev恒等式和Díaz-Saa型不等式，并揭示加权非局部问题的极小能量解在超线性和次线性增长下分别收敛于Brézis-Nirenberg型和逻辑型极限问题的解。
Multidimensionnel Borg-Levinson uniqueness and stability results for the Robin Laplacian with unbounded potential
针对三维及以上有界区域中带 Robin 边界条件的 Schrödinger 算子，本文利用特征值与完整或部分边界 Dirichlet 测量组成的谱数据，研究无界势反演的 Borg-Levinson 唯一性与稳定性。
Cauchy problem for 3D incompressible Hall-magnetohydrodynamics (Hall-MHD) system with fractional Laplacians is studied. First, global well-posedness of small-energy solutions with general initial data in $H^s$, $s>\frac{5}{2}$, is proved. Second, a special class of large-energy initial data is constructed, with which the Cauchy problem is globally well-posed. The proofs rely upon a new global bound of energy estimates involving Littlewood-Paley decomposition and Sobolev inequalities, which enables one to overcome the $\frac{1}{2}$-order derivative loss of the magnetic field.
This paper, which is the follow-up to part I, concerns the equation $(-Δ)^{s} v+G'(v)=0$ in $\mathbb{R}^{n}$, with $s \in (0,1)$, where $(-Δ)^{s}$ stands for the fractional Laplacian ---the infinitesimal generator of a Lévy process. When $n=1$, we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits $\pm 1$ at $\pm \infty$) if and only if the potential $G$ has only two absolute minima in $[-1,1]$, located at $\pm 1$ and satisfying $G'(-1)=G'(1)=0$. Under the additional hypothesis $G"(-1)>0$ and $G"(1)>0$, we also establish its uniqueness and asympt…
Uniqueness of positive solutions for finsler p-Laplacian equations with polynomial non-linearity
本文研究Finsler p-Laplacian方程在多项式非线性下正解的唯一性，利用线性化方法推广了Brasco和Lindgren的结论。
In this paper we derive a sufficient condition for the existence of a unique solution of a Cauchy type q-fractional problem (involving the fractional q-derivative of Riemann-Liouville type) for some nonlinear differential equations. The key technique is to first prove that this Cauchy type q-fractional problem is equivalent to a corresponding Volterra q-integral equation. Moreover, we define the $q$-analogue of the Hilfer fractional derivative or composite fractional derivative operator and prove some similar new equivalence, existence and uniqueness results as above. Finally, some examples ar…
Initial-boundary value problems for second order fully nonlinear PDEs with Caputo time fractional derivatives of order less than one are considered in the framework of viscosity solution theory. Associated boundary conditions are Dirichlet and Neumann, and they are considered in the strong sense and the viscosity sense, respectively. By a comparison principle and Perron's method, unique existence for the Cauchy-Dirichlet and Cauchy-Neumann problems are proved.
For the non-local space-time reaction-diffusion equation involving fractional $p$-Laplacian \begin{equation*} \begin{cases} \frac{\partial^{α}u}{\partial t^{α}}+(-Δ)_{p}^{s} u=μu^{2}(1-kJ*u)-γu,&(x,t)\in\mathbb{R}^{N}\times(0,T)\\ u(x,0)=u_{0}(x),& x\in\mathbb{R}^{N} \end{cases} \end{equation*} $μ>0 ,k>0,γ\geq 1,α\in(0,1),s\in(0,1),1<p$, we consider for $N\leq2$ the problem of finding a global boundedness of the weak solution by virtue of Gagliardo-Nirenberg inequality and fractional Duhamel's formula. Moreover, we prove such weak solution converge to $0$ exponentially or locally uniformly as …
The Cauchy problem in $\mathbb{R}^d,$ $d\geq 1,$ for a non-local in time p-Laplacian equations is considered. The nonexistence of nontrivial global weak solutions by using the test function method is obtained.
In this work, we prove the existence, uniqueness and smoothing properties of the solution to the Cauchy problem for the spatially homogeneous Boltzmann equation with Debye-Yukawa potential for probability measure initial datum.
We study the Cauchy problem for Fokker--Planck--Kolmogorov equations with unbounded and degenerate coefficients. Sufficient conditions for the existence and uniqueness of solutions are indicated.
Ultrasound modulation of electrical or optical properties of materials offers the possibility to devise hybrid imaging techniques that combine the high electrical or optical contrast observed in many settings of interest with the high resolution of ultrasound. Mathematically, these modalities require that we reconstruct a diffusion coefficient $σ(x)$ for $x\in X$, a bounded domain in $\Rm^n$, from knowledge of $σ(x)|\nabla u|^2(x)$ for $x\in X$, where $u$ is the solution to the elliptic equation $-\nabla\cdotσ\nabla u=0$ in $X$ with $u=f$ on $\partial X$. This inverse problem may be recast as …
We prove regularity results such as interior Lipschitz regularity and boundary continuity for the Cauchy-Dirichlet problem associated to a class of parabolic equations inspired by the evolutionary $p$-Laplacian, but extending it at a wide scale. We employ a regularization technique of viscosity-type that we find interesting in itself.
This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $$ \partial_t u (x,t) = \sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) \qquad (x,t) \in \mathbb{R}^N \times\, ]- \infty ,T[,$$ proved by a functional analytic approach based on Choquet theory. As a consequence, we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.
We study existence and uniqueness of bounded solutions to a fractional nonlinear porous medium equation with a variable density, in one space dimension.
In this paper, the existence, uniqueness and regularity properties, Strichartz type estimates for solution of multipoint Cauchy problem for linear and nonlinear Schrödinger equations with general elliptic leading part is obtained.
In this paper, we investigate the asymptotic behavior of solutions toward a multiwave pattern of the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the flux function is convex or concave but linearly degenerate on some interval, and also the viscosity is a nonlinearly degenerate one (p-Laplacian type viscosity). When the corresponding Riemann problem admits a Riemann solution which consists of rarefaction waves and contact discontinuity, it is proved that the solution of the Cauchy problem tends toward th…
Uniqueness and non-uniqueness pairs for the fractional Laplacian
研究分数拉普拉斯算子在离散集上的唯一性和非唯一性条件，通过分析方程解的唯一性与非唯一性，提出相关数学条件及示例。
A fractional Laplacian problem with mixed singular nonlinearities and nonregular data
本文研究了具有非正则数据和混合奇异非线性的分数阶拉普拉斯问题，证明了正解及熵解的存在性与唯一性，并通过放宽假设条件完善了存在性结果。
Given that the terminal condition is of at most linear growth, it is well known that a Cauchy problem admits a unique classical solution when the coefficient multiplying the second derivative (i.e., the volatility) is also a function of at most linear growth. In this note, we give a condition on the volatility that is necessary and sufficient for a Cauchy problem to admit a unique solution.
We study uniqueness of flows of probability measures solving the Cauchy problem for nonlinear Fokker-Planck-Kolmogorov equation with unbounded coefficients. Sufficient conditions for uniqueness are indicated and examples of non-uniqueness are constructed.
Degenerate parabolic $p$-Laplacian equations: existence, uniqueness and asymptotic behavior of solutions
本文研究由矩阵 Q 和权函数 v 控制的退化抛物型 p-Laplacian 方程，在温和可积性条件下证明了解的存在唯一性，并进一步在退化 Sobolev 不等式条件下建立了有限时间熄灭性、超压缩界及其与加权 Sobolev 不等式的等价关系。
We establish the existence and uniqueness of mild solutions for the polyatomic ellipsoidal BGK model, which is a relaxation type kinetic model describing the evolution of polyatomic gaseous system at the mesoscopic level.
We study the well-posedness of the Cauchy problem for a fractional porous medium equation with a varying density. We establish existence of weak energy solutions; uniqueness and nonuniqueness is studied as well, according with the behavior of the density at infinity.
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