Related papers: Uniqueness Theorems for Ordinary Differential Equa…

We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value $\partial_t^{\alpha} u(x,t) = -Au(x,t)$, where $-A = \sum}{i,j=1}^d \partial_i(a_{ij}(x)\partial_j) +…

We develop a theory of existence, uniqueness and regularity for a porous medium equation with fractional diffusion, $\frac{\partial u}{\partial t} + (-\Delta)^{1/2} (|u|^{m-1}u)=0$ in $\mathbb{R}^N$, with $m>m_*=(N-1)/N$, $N\ge1$ and $f\in…

We prove that for almost complex structures of H\"older class at least 1/2, any J-holomorphic disc, that is constant on some non empty open set, is constant. This is in striking contrast with well known, trivial, non-uniqueness results. We…

We study the Schr\"{o}dinger equation: \begin{eqnarray} - \Delta u+V(x)u+f(x,u)=0,\qquad u\in H^{1}(\mathbb{R}^{N}),\nonumber \end{eqnarray} where $V$ is periodic and $f$ is periodic in the $x$-variables, $0$ is in a gap of the spectrum of…

We derive monotonicity formulae for solutions of the fractional H\'{e}non-Lane-Emden equation \begin{equation*} (-\Delta)^{s} u=|x|^a |u|^{p-1} u \ \ \ \text{in } \ \ \mathbb{R}^n, \end{equation*} when $0<s<2$, $a>0$ and $p>1$. Then, we…

Under general conditions we show that the solution of a stochastic parabolic partial differential equation of the form \[ \partial_t u = \mathrm{div} (A \nabla u) + f(t,x, u) + g_i (t,x,u) \dot{w}^i_t \] is almost surely H\"older continuous…

We study the unconditional uniqueness of solutions to the Benjamin-Ono equation with initial data in $H^{s}$, both on the real line and on the torus. We use the gauge transformation of Tao and two iterations of normal form reductions via…

In this paper we study the existence and uniqueness of a solution and propose an iterative method for solving a beam problem which is described by the fully fourth order equation $$u^{(4)}(x)=f(x,u(x),u'(x),u'''(x),u'''(x)), \quad 0 < x <…

This short, self-contained article seeks to introduce and survey continuous-time deep learning approaches that are based on neural ordinary differential equations (neural ODEs). It primarily targets readers familiar with ordinary and…

For the $p$-Laplace Dirichlet problem (where $\varphi (t)=t|t|^{p-2}$, $p>1$) \[ \varphi(u'(x))'+ f(u(x))=0 \;\;\;\; \mbox{for $-1<x<1$}, \;\; u(-1)=u(1)=0 \] assume that $f'(u)>(p-1)\frac{f(u)}{u}>0$ for $u>\gamma>0$, while $\int_u^\gamma…

Let $2\le n\le 5$. We establish an apriori interior H\"older regularity of $C^2$-stable solutions to the semilinear equation $-\Delta u=f(u)$ in any domain of $R^n$ for any nonlinearity $f\in C^{0,1}(R) $.If $f $ is nondecreasing and convex…

This article is concerned with the existence and uniqueness of solutions to some fractional order boundary value problems. Our results are based on some fixed point theorems. For the applicability of our results, we provide an example.

Supposing only that $\displaystyle\lim_{t \to 0} \frac{f(t)}{t} = 0$ and $\displaystyle\lim_{t \to \infty} \frac{f(t)}{t^{p}} = 0$, for some $p \in \left(1,\frac{N+1}{N-1}\right)$, we prove that solutions to the extension problem…

The conditions under which, the continuity equation can be substituted by an ordinary non differential equation, will be discussed. Since continuity equation is a fundamental equation, this result will be applicable in a vast area of…

For given non-consistent initial conditions, we study the stability of a class of generalised linear systems of difference equations with constant coefficients and taking into account that the leading coefficient can be a singular matrix.…

This work deals with a Skorokhod problem driven by a maximal operator: \begin{aligned} &du(t)+Au(t)(dt)\ni f(t)dt+dM(t), \; 0<t<T,\\ &u(0)=u_{0}, \end{aligned} which is a multivalued deterministic differential equation with a singular…

The main purpose of this paper is to obtain necessary and sufficient conditions under which a nonautonomous, finite-dimensional and two-sided dynamics generated by a sequence of matrices or a linear ODE exhibits Hyers-Ulam stability.…

We study stable solutions to fractional semilinear equations $(-\Delta)^s u = f(u)$ in $\Omega \subset \mathbb{R}^n$, for convex nonlinearities $f$, and under the Dirichlet exterior condition $u=g$ in $\mathbb{R}^n \setminus \Omega$ with…

We give two conditions that are necessary and sufficient for the uniqueness of Filippov solutions of scalar, autonomous ordinary differential equations with discontinuous velocity fields. When only one of the two conditions is satisfied, we…

Normality arguments are applied to study the oscillation of solutions of $f''+Af=0$, where the coefficient $A$ is analytic in the unit disc $\mathbb{D}$ and $\sup_{z\in\mathbb{D}} (1-|z|^2)^2|A(z)| < \infty$. It is shown that such…