On Solvability of the Cauchy problem for the partial differential equations

A class of non-analytic functions for the global solvability of Kirchhoff equation

We consider the global solvability to the Cauchy problem of Kirchhoff equation with generalized classes of Manfrin's class. Manfrin's class is a subclass of Sobolev space, but we shall extend this class as a subclass of the ultradifferentiable class, and we succeed to prove the global solvability of Kirchhoff equation with large data in wider classes from the previous works.

Global solvability of massless Dirac-Maxwell systems

We consider the Cauchy problem of massless Dirac-Maxwell equations on an asymptotically flat background and give a global existence and uniqueness theorem for initial values small in an appropriate weighted Sobolev space. The result can be extended via analogous methods to Dirac-Higgs-Yang-Mills theories.

关于具有容量密度条件的领域上抛物型方程的全局Hölder可解性

Global Hölder Solvability of parabolic equations on domains with capacity density conditions

本文研究了具有容量密度条件的领域上抛物型方程的全局Hölder解的存在性，允许数据在边界附近具有临界奇异性。

Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations

We prove locally in time the existence of a smooth solution for multidimensional two-phase Stefan problem for degenerate parabolic equations of the porous medium type. We establish also natural Hölder class for the boundary conditions in the Cauchy-Dirichlet problem for a degenerate parabolic equation.

The Cauchy Problem for the Vibrating Plate Equation in modulation spaces

The local solvability of the Cauchy problem for the nonlinear vibrating plate equation is showed in the framework of modulation spaces. In the opposite direction, it is proved that there is no local wellposedness in Wiener amalgam spaces even for the solution to the homogeneous vibrating plate equation.

超临界分数阶热方程的局部可解性与伸缩临界奇异性

Local solvability and dilation-critical singularities of supercritical fractional heat equations

本文研究超临界分数阶半线性热方程的 Cauchy 问题，给出局部可解性的必要与充分条件，并提出初始数据的 DCS 概念，证明其广泛存在且得到精确公式。

具有幂型非线性的快速扩散方程的初始迹与可解性

Initial traces and solvability of the fast diffusion equation with power-type nonlinearity

本文研究幂型非线性快速扩散方程非负解的初始迹，刻画对应 Cauchy 问题解存在的必要条件，并借助一致局部 Morrey 空间及其推广给出尖锐充分条件。

Local Solvability For a Class of Partial Differential Operators With Double Characteristics

A necessary and sufficient condition for local solvability is presented for the linear partial differential operators $-X^2-Y^2+ia(x)[X,Y]$ in $\bold R^3=\{(x,y,t)\}$, where $X=\partial_x,\; Y=\partial_y+x^k\partial_t$, and $a\in C^{\infty}(\bold R^1)$ is real valued, for each positive integer $k$.

极弱框架下分数阶半线性热方程可解性的必要条件

Necessary conditions for the solvability of fractional semilinear heat equations in the very weak framework

本文针对半线性热方程Cauchy问题，提出新证明方法推导出极弱解框架下初始值的可解性必要条件，比积分解条件更具一般性。

A note on the Cauchy problem for $-D_0^2+2x_1D_0D_2+D_1^2+x_1^3D_2^2+\sum_{j=0}^2b_jD_j$

In this note, we improve a previously proven non-solvability result of the Cauchy problem for the Cauchy problem in the Gevrey class for a homogeneous second-order differential operator mentioned in the title. We prove that the Cauchy problem for this operator is not locally solvable at the origin for any lower order term in the Gevrey class of order greater than 5, lowering the previously obtained Gevrey order 6.

On local solvability of nonlinear elliptic partial differential systems of principle type: the second order

We prove results on solvability of nonlinear elliptic partial differential systems of principle type of second order. They are consequences of existence of non-radial solutions for nonlinear partial differential systems of Poisson type. As applications to geometry, we prove the exsitence of local harmonic maps with given tangent plane at a point between any Riemannan manifolds. More generally geometric objects defined by Beltrami-Laplace always exist locally.

Unique solvability of a non-local problem for mixed type equation with fractional derivative

In this work we investigate a boundary problem with non-local conditions for mixed parabolic-hyperbolic type equation with three lines of type-changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part we use solution of the first boundary problem with appropriate Green's function and in hyperbolic parts we use corresponding solutions of the Cauchy problem.

Sufficient conditions of local solvability for partial differential operators in the Colombeau context

We provide sufficient conditions of local solvability for partial differential operators with variable Colombeau coefficients. We mainly concentrate on operators which admit a right generalized pseudodifferential parametrix and on operators which are a bounded perturbation of a differential operator with constant Colombeau coefficients. The local solutions are intended in the Colombeau algebra $\G(\Om)$ as well as in the dual $\LL(\Gc(\Om),\wt{\C})$.

一类弱耗散Camassa-Holm方程的整体适定性与爆破的新结果

New results on the global solvability and blow-up for a class of weakly dissipative Camassa-Holm equations

本文研究非齐次Besov空间中弱耗散Camassa-Holm方程的柯西问题，通过转化为阻尼方程与非标准迭代法证明λ满足条件时的整体强解存在性（推广至Besov空间且无初始符号条件），并推导两类爆破准则揭示λ对奇异性的影响。

非齐次分数半线性热方程在洛伦兹-莫雷空间中的可解性

Solvability of inhomogeneous fractional semilinear heat equations in Lorentz--Morrey spaces

本文研究了分数半线性热方程在洛伦兹-莫雷空间中的可解性问题，通过引入洛伦兹-莫雷空间克服了传统方法的限制，并给出了不齐项的必要和充分条件。

On the solvability of degenerate stochastic partial differential equations in Sobolev spaces

Systems of parabolic, possibly degenerate parabolic SPDEs are considered. Existence and uniqueness are established in Sobolev spaces. Similar results are obtained for a class of equations generalizing the deterministic first order symmetric hyperbolic systems.

Existence of solutions for an inhomogeneous fractional semilinear heat equation

We obtain necessary conditions and sufficient conditions on the existence of solutions to the Cauchy problem for a fractional semilinear heat equation with an inhomogeneous term. We identify the strongest spatial singularity of the inhomogeneous term for the solvability of the Cauchy problem.

Optimal singularities of initial data for solvability of the Hardy parabolic equation

We consider the Cauchy problem for the Hardy parabolic equation $\partial_t u-Δu=|x|^{-γ}u^p$ with initial data $u_0$ singular at some point $z$. Our main results show that, if $z\neq 0$, then the optimal strength of the singularity of $u_0$ at $z$ for the solvability of the equation is the same as that of the Fujita equation $\partial_t u-Δu=u^p$. Moreover, if $z=0$, then the optimal singularity for the Hardy parabolic equation is weaker than that of the Fujita equation. We also obtain analogous results for a fractional case $\partial_t u+(-Δ)^{θ/2} u=|x|^{-γ}u^p$ with $0<θ<2$.

The unique global solvability and optimal time decay rates for a multi-dimensional compressible generic two-fluid model with capillarity effects

The present paper deals with the Cauchy problem of a compressible generic two-fluid model with capillarity effects in any dimension $N\geq2$. We first study the unique global solvability of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. Due to the presence of the capillary terms, we exploit the parabolic properties of the linearized system for all frequencies which enables us to apply contraction mapping principle to show the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mil…

Cauchy Problems for Parabolic Equations in Sobolev-Slobodeckii and Hölder Spaces on Uniformly Regular Riemannian Manifolds

In this paper we establish optimal solvability results, that is, maximal regularity theorems, for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless complete Riemannian manifolds with bounded geometry. We employ an anisotropic extension of the Fourier multiplier theorem for arbitrary Besov spaces introduced in earlier by the author. This allows for a unified treatment of Sobolev-Slobodeckii and little Hölder spaces. In the flat case we recover classical results for Petrowskii-parabolic Cauchy problems.

A Cauchy-Kowalevsky theorem for overdetermined systems of nonlinear partial differential equations and geometric applications

We consider a Cauchy problem for an overdetermined system of PDEs, and give necessary and sufficient conditions for solvability of this Cauchy problem for all data. As an application, we find all real tube hypersurfaces in complex space whose Levi number is maximal.

Infinite-Dimensional Linear Algebra and Solvability of Partial Differential Equations

We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called in the article \emph{regular}, acting on the algebraic dual $\mathcal D^*(Ω)$ of the space of test-functions $\mathcal D(Ω)$. The surjectivity of the partial differential operators guarantees solvability of the corresponding partial differential equations within $\mathcal D^*(Ω)$. We discuss our result in contrast to and compar…

Solvability and regularity for an elliptic system prescribing the curl, divergence, and partial trace of a vector field on Sobolev-class domains

We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field are prescribed in an open, bounded, Sobolev-class domain, and either the normal component or the tangential components of the vector field are prescribed on the boundary. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients.

Solvability of second-order equations with hierarchically partially BMO coefficients

By using some recent results for divergence form equations, we study the $L_p$-solvability of second-order elliptic and parabolic equations in nondivergence form for any $p\in (1,\infty)$. The leading coefficients are assumed to be in locally BMO spaces with suitably small BMO seminorms. We not only extend several previous results by Krylov and Kim [14]-[18] to the full range of $p$, but also deal with equations with more general coefficients.

Classical solvability of the multidimensional free boundary problem for the thin film equation in the case of partial wetting

We prove locally in time the existence of the unique smooth solution (including smooth interface) to the multidimensional free boundary problem for the thin film equation in the case of partial wetting. We also obtain the Schauder estimates and solvability for the Dirichlet and the Neumann problem for a linear degenerate parabolic equation of fourth order. The final expanded version of this paper is available at AIMS Journals, Discrete and Continuous Dynamical Systems - A at http://aimsciences.org/article/doi/10.3934/dcds.2017156

Elliptic regularity and solvability for partial differential equations with Colombeau coefficients

The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new notions of ellipticity and hypoellipticity, study their interrelation, and give a number of new examples and counterexamples. Using the concept of $\G^\infty$-regularity of generalized functions, we derive a general global regularity result in the case of operators with constant generalized coefficients, a more specialized result for second order operators, and a …

Exterior extension problems for strongly elliptic operators: solvability and approximation using fundamental solutions

In this work we study three exterior extension problems for strongly elliptic partial equations: the Cauchy problem (in a special statement), the "analytical" continuation problem and the so called "inner" Dirichlet problem in the scale of the Sobolev spaces over a domain with relatively smooth boundaries. We consider the existence of solutions to these problems, the dense solvability and conditional well-posedness of these problems for a wide class of strongly elliptic systems. We also consider the approximation of solutions to these problems by a single layer potential and by a linear combin…

具有$VMO_x$系数的抛物型方程拟线性Cauchy-Dirichlet问题

Quasilinear Cauchy-Dirichlet problem for parabolic equations with $VMO_x$ coefficients

研究具有不连续系数的抛物型拟线性方程Cauchy-Dirichlet问题，在适当结构条件下证明了强解的存在性、唯一性及Hölder连续性。

模型算子的Gevrey不可解性

Lack of Gevrey solvability for a model operator

本文证明在R^4中模型双曲算子的Cauchy问题在原点处对于Gevrey类s>6不是局部可解的。

Existence of solutions for a higher-order semilinear parabolic equation with singular initial data

We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.

Wave equations and symmetric first-order systems in case of low regularity

We analyse an algorithm of transition between Cauchy problems for second-order wave equations and first-order symmetric hyperbolic systems in case the coefficients as well as the data are non-smooth, even allowing for regularity below the standard conditions guaranteeing well-posedness. The typical operations involved in rewriting equations into systems are then neither defined classically nor consistently extendible to the distribution theoretic setting. However, employing the nonlinear theory of generalized functions in the sense of Colombeau we arrive at clear statements about the transfer …

时间分数阶广义库马诺-西瓦斯基方程的初值问题

A generalized time-fractional Kuramoto-Sivashinsky equation in the Schwartz space: global solvability and stability

本文研究了广义时间分数阶库马诺-西瓦斯基方程在Schwartz空间中的全局可解性，通过分离线性与非线性部分并应用分析方法，建立了拓扑函数空间中分数阶库马诺-西瓦斯基类型方程的严格分析框架。

On the Cauchy problem for the debar operator

We present new results concerning the solvability, of lack thereof, in the Cauchy problem for the debar operator, with initial values assigned on a weakly pseudoconvex hypersurface, and provide illustrative examples.

Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces

We consider a Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces. The equation generalizes the heat equation to the case of fractional power of the Laplace operator and the power of this operator can be different with respect to different groups of space variables. The time derivative can be either fractional Caputo - Jrbashyan derivative or usual derivative. Under some necessary conditions on the order of the time derivative we show that the operator of the whole problem is an isomorphism of appropriate anisotropic Hölder spaces. Under some another con…

The Cauchy problem for the Pavlov equation with large data

The Pavlov equation is one of the simplest integrable systems of vector fields arising from various problems of mathematical physics and differential geometry which are intensively studied in recent literature. In this report, solving a nonlinear Riemann-Hilbert problem via a Newtonian iteration scheme, we complete the inverse scattering theory and prove a short time unique solvability of the Cauchy problem of the Pavlov equation with large initial data.

Remark on the strong solvability of the Navier-Stokes equations in the weak $L^n$ space

The initial value problem of the incompressible Navier-Stokes equations with non-zero forces in $L^{n,\infty}(\mathbb{R}^n)$ is investigated. Even though the Stokes semigroup is not strongly continuous on $L^{n,\infty}(\mathbb{R}^n)$, with the qualitative condition for the external forces, it is clarified that the mild solution of the Naiver-Stokes equations satisfies the differential equations in the topology of $L^{n,\infty}(\mathbb{R}^n)$. Inspired by the conditions for the forces, we characterize the maximal complete subspace in $L^{n,\infty}(\mathbb{R}^n)$ where the Stokes semigroup is st…

具有边依赖导数阶的时间分数阶Airy型方程在星形图上的Cauchy问题

Cauchy Problem for the Time-Fractional Airy-Type Equation on a Star Graph with Edge-Dependent Order of Derivative

研究在无限度量星形图上具有边依赖Caputo导数的时间分数阶Airy型方程的Cauchy问题，通过构造分数阶Airy基本解生成的层势，将问题简化为广义Abel积分方程，利用Pskhu的可解性结果，建立了解的存在性、唯一性和显式积分表示。

Optimal singularities of initial functions for solvability of a semilinear parabolic system

Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system \[ \mbox{(P)} \qquad \cases{ \partial_t u=D_1Δu+v^p, & $x\in{\bf R}^N,\,\,\,t>0,$\\ \partial_t v=D_2Δv+u^q, & $x\in{\bf R}^N,\,\,\,t>0,$\\ (u(\cdot,0),v(\cdot,0))=(μ,ν), & $x\in{\bf R}^N,$ } \] where $D_1$, $D_2>0$, $0<p\le q$ with $pq>1$ and $(μ,ν)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${\bf R}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.

在$\u2202_t A$满足$L^1$ Carleson条件下Dirichlet问题与正则性问题可解性的等价性

Equivalence between solvability of the Dirichlet and Regularity problem under an $L^1$ Carleson condition on $\partial_t A$

本研究针对上半空间椭圆算子，在$|\u2202_t A|$满足$L^1$-Carleson条件下证明了Dirichlet问题与正则性问题可解性的等价性，扩展了$L^p$正则性问题可解的算子类。

A universal regularization method for ill-posed Cauchy problems for quasilinear partial differential equations

For the first time, a globally convergent numerical method is presented for ill-posed Cauchy problems for quasilinear PDEs. The key idea is to use Carleman Weight Functions to construct globally strictly convex Tikhonov-like cost functionals.

Global Existence and Large-time Behavior of Solutions to the Cauchy Problem of One-dimensional Viscous Radiative and Reactive Gas

Although there are many results on the global solvability and the precise description of the large time behaviors of solutions to the initial-boundary value problems of the one-dimensional viscous radiative and reactive gas in bounded domain with two typical types of boundary conditions, no result is available up to now for the corresponding problems in unbounded domain. This paper focuses on the Cauchy problem of such a system with prescribed large initial data and the main purpose is to construct its global smooth non-vacuum solutions around a non-vacuum constant equilibrium state and to stu…

Local solvability for a quasilinear wave equation with the far field degeneracy: 1D case

We study the Cauchy problem for the quasilinear wave equation $ \partial^2 _t u = u^{2a} \partial^2_x u + F(u) u_x $ with $a \geq 0$ and show a result for the local in time existence under new conditions. In the previous results, it is assumed that $u(0,x) \geq c_0>0$ for some constant $c_0$ to prove the existence and the uniqueness. This assumption ensures that the equation does not degenerate. In this paper, we allow the equation to degenerate at spacial infinity. Namely we consider the local well-posedness under the assumption that $u(0,x)>0$ and $u(0,x) \rightarrow 0$ as $|x| \rightarrow \…

$L^p$ solvability of the Stationary Stokes problem on domains with conical singularity in any dimension

The Dirichlet boundary value problem for the Stokes operator with $L^p$ data in any dimension on domains with conical singularity (not necessary a Lipschitz graph) is considered. We establish the solvability of the problem for all $p\in (2-\varepsilon,\infty]$ and also its solvability in $C(\overline{D})$ for the data in $C(\partial D)$

On solvability of the non-local problem for the fractional mixed-type equation with Bessel operator

The non-local problem is considered for the partial differential equation of mixed-type with Bessel operator and fractional order. An explicit solution is represented by Fourier-Bessel series in the given domain. It is established the connection between the given data and the unique solvability of the problem.

Some remarks on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem

The aim of the paper is to investigate on some questions of local regularity of a suitable weak solution to the Navier-Stokes Cauchy problem. The results are obtained in the wake of the ones, well known, by Caffarelli-Kohn-Nirenberg.

Determining a magnetic Schroedinger operator from partial Cauchy data

In this paper we show, in dimension n >=3, that knowledge of the Cauchy data for the Schroedinger equation in the presence of a magnetic potential, measured on possibly very small subsets of the boundary, determines uniquely the magnetic field and the electric potential.

A remark on unique continuation for the Cauchy-Riemann operator

In this note we obtain a unique continuation result for the differential inequality $|\bar{\partial}u|\leq|Vu|$, where $\bar{\partial}=(i\partial_y+\partial_x)/2$ denotes the Cauchy-Riemann operator and $V(x,y)$ is a function in $L^2(\mathbb{R}^2)$.

环面上带导数损失的全局亚椭圆性与可解性

Global Hypoellipticity and Solvability with Loss of Derivatives on the Torus

本文完整刻画了环面上傅里叶乘子算子带导数损失的全局亚椭圆性与可解性，并揭示其与数论的联系：向量场可解性的导数损失由无理测度决定，波动算子的损失显式依赖于参数。

On the Cauchy Problem for Differential Equations in a Banach Space over the Field of p-Adic Numbers. I

For the Cauchy problem for an operator differential equation of the form $y'(z) = Ay(z)$, where $A$ is a closed linear operator on a Banach space over the completion of an algebraic closure of the field of $p$-adic numbers, a criterion of correct solvability in the class of locally analytic vector-functions is established. It is shown how the Cauchy-Kovalevskaya theorem for $p$-adic partial differential equations may be obtained as a particular case from this criterion.

Uniqueness and stability result for Cauchy's equation of motion for a certain class of hyperelastic materials

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…

On the Cauchy problem for the periodic fifth-order KP-I equation

The aim of this paper is to investigate the Cauchy problem for the periodic fifth order KP-I equation \[\partial_t u - \partial_x^5 u -\partial_x^{-1}\partial_y^2u + u\partial_x u = 0,~(t,x,y)\in\mathbb{R}\times\mathbb{T}^2\] We prove global well-posedness for constant $x$ mean value initial data in the space $\mathbb{E} = \{u\in L^2,~\partial_x^2 u \in L^2,~\partial_x^{-1}\partial_y u \in L^2\}$ which is the natural energy space associated with this equation.

多调和热方程的不适定柯西问题

On the ill-posed Cauchy problem for the polyharmonic heat equation

针对多调和热方程的不适定柯西问题，提出了唯一性定理和基于抛物势实解析延拓的可解性判据。

Properties of Solutions of a Class of Hypocomplex Vector Fields

A Cauchy type integral operator is associated to a class of integrable vector fields with complex coefficients. Properties of the integral operator are used to deduce Holder solvability of semilinear equations and a strong similarity principle is established.

The global solvability of the Kirchhoff equation with Sobolev data

We consider linear and non-linear Cauchy equations in the context of Sobolev spaces. In particular, we show the global existence of solutions to the Kirchhoff equation with initial data in the Sobolev spaces, a problem that has been open for more than eighty years. Our proof is based on a new uniform estimate for solutions to the linear equation with time-dependent coefficient and a fixed point argument. As an immediate consequence of our result, the global solvability of the Kirchhoff equation with initial data in the Gevrey spaces is also obtained.