Related papers: Backward uniqueness for parabolic operators with v…

We prove well-posedness in weighted tent spaces of weak solutions to the Cauchy problem $\partial_t u - \mathrm{div} A \nabla u = f, u(0)=0$, where the source $f$ also lies in (different) weighted tent spaces, provided the complex…

We consider the second order Cauchy problem $$u''+\m{u}Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\infty)\to[0,+\infty)$ is a continuous function, and $A$ is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is…

In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -\Delta u=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial \nu}} = V(x)u+b \vert…

We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form $u(R,\theta) = Rg(\theta)$, where $(R,\theta)$ are plane polar coordinates and $g: \mathbb{R}^{2} \to…

In this article we prove existence, uniqueness and regularity for the singular equation \begin{eqnarray*} \begin{cases} |\nabla u|^{\alpha}(F(D^{2}u)+h(x)\cdot\nabla u)+c(x)|u|^{\alpha}u+p(x)u^{-\gamma}=0 \ \mbox{ in } \ \Omega\\ u>0 \…

This paper is a generalization of the author's previous work [14]. We extend the argument [14] for any uniformly elliptic operator in divergence form $\mathcal{L}u=-div(A(x)\nabla u)$, more precisely, we study a fractional type degenerate…

We study the following ultraparabolic equation \[ \frac{\partial}{\partial t}u\left(t,s\right)+\frac{\partial}{\partial…

In this paper we present an improvement of [Math. Ann. 345 (2009), 213--243], where the authors proved a result concerning continuous dependence for backward parabolic operators whose coefficients are Log-Lipschitz in $t$ and $C^2$ in $x$.…

Consider the equation $$ u'(t)-\Delta u+|u|^\rho u=0, \quad u(0)=u_0(x), (1), $$ where $ u':=\frac {du}{dt}$, $ \rho=const >0, $ $x\in \mathbb{R}^3$, $t>0$. Assume that $u_0$ is a smooth and decaying function, $$\|u_0\|\:=\sup_{x\in…

We prove uniqueness results and Harnack inequality for Bessel operators \begin{align*} %\label{def L transf alpha} D_t-\Delta_{x} -2a\cdot\nabla_xD_y- D_{yy}- \frac cy D_y % \nonumber…

For $0<\lambda \leq 1$, let ${\mathcal U}(\lambda)$ denote the family of functions $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ analytic in the unit disk $\ID$ satisfying the condition $\left |\left (\frac{z}{f(z)}\right )^{2}f'(z)-1\right |<\lambda…

We consider degenerate differential operators $A = \displaystyle{\sum_{k,j=1}^d \partial_k (a_{kj} \partial_j)}$ on $L^2(\mathbb{R}^d)$ with real symmetric bounded measurable coefficients. Given a function $\chi \in…

We consider the semilinear problem \[ \Delta u = \lambda_+ \left(-\log u^+\right) 1_{\{u > 0\}} - \lambda_- \left(-\log u^- \right) 1_{\{u < 0\}} \qquad \hbox{ in } B_1, \] where $B_1$ is the unit ball in $\mathbb{R}^n$ and assume…

In the first part of this article we introduce the notion of a backward-forward conditioning (BFC) system that generalises the notion of zero-class admissibiliy introduced in [Xu,Liu,Yung]. We can show that unless the spectum contains a…

In this paper, we establish Liouville-type theorems for parabolic differential inequalities with $(p,q)-$Laplacian operator on Riemannian manifolds. By a test function argument, we establish nonexistence results under suitable weighted…

This article presents the convergence analysis of a sequence of piecewise constant and piecewise linear functions obtained by the Rothe method to the solution of the first order evolution partial differential inclusion…

We consider weak solutions $u:\Omega_{T}\rightarrow\mathbb{R}^{N}$ to parabolic systems of the type \[ u_{t}-\mathrm{div}\,A(x,t,Du)=f \qquad \mathrm{in}\ \Omega_{T}=\Omega\times(0,T), \] where $\Omega$ is a bounded open subset of…

For a Jacobi matrix J on l^2(Z_+) with Ju(n)=a_{n-1} u(n-1) + b_n u(n) + a_n u(n+1), we prove that \sum_{|E|>2} (E^2 -4)^{1/2} \leq \sum_n |b_n| + 4\sum_n |a_n -1|. We also prove bounds on higher moments and some related results in higher…

We present a general approach to the parametrix construction. We apply it to prove the uniqueness and existence of a weak fundamental solution for the equation $\partial_t =\mathcal{L}$ with non-symmetric non-local operators $$…

This paper examines the behavior of a positive solution $u\in C^{1,\alpha}(\Bar{\Omega})$ of the $(p,q)$ Laplace equation with a singular term and zero Dirichlet boundary condition. Specifically, we consider the equation: \begin{equation*}…