Related papers: Quantitative uniqueness for fractional heat type o…

In this paper, we consider fractional parabolic equation of the form $ \frac{\partial u}{\partial t}=-(-\Delta)^{\frac{\alpha}{2}}u+u\dot W(t,x)$, where $-(-\Delta)^{\frac{\alpha}{2}}$ with $\alpha\in(0,2]$ is a fractional Laplacian and…

Consider the linear stochastic fractional heat equation with vanishing initial condition: $$ \frac{\partial u (t,x)}{\partial t}=-(-\Delta)^{\frac{\alpha}2}u (t,x) + \dot{W}(t,x),\quad t> 0,\, x\in \mathbb R, $$ where…

For $\alpha \in (1,2)$ we consider the equation $\partial_t u = \Delta^{\alpha/2} u - r b \cdot \nabla u$, where $b$ is a divergence free singular vector field not necessarily belonging to the Kato class. We show that for sufficiently small…

In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-\Delta)^\frac{\theta}{2} u=|x|^{-\gamma} u^p ,\quad x\in{\bf R}^N, t>0,…

The purpose of this paper is to study and classify singular solutions of the Poisson problem $$ %\begin{equation}\label{eq 0.1} \left \{ \begin{aligned} {\mathcal L}^s_\mu u = f \quad\ {\rm in}\ \, \Omega\setminus \{0\},\\ u =0 \quad\ {\rm…

We consider an initial boundary value problem in a bounded domain $\Omega$ over a time interval $(0, T)$ for a time-fractional wave equation where the order of the fractional time derivative is between $1$ and $2$ and the spatial elliptic…

In this paper, we consider equations involving fully nonlinear nonlocal operators $$F_{\alpha}(u(x)) \equiv C_{n,\alpha} PV \int_{\mathbb{R}^n} \frac{G(u(x)-u(z))}{|x-z|^{n+\alpha}} dz= f(x,u).$$ We prove a maximum principle and obtain key…

This paper has two primary objectives. The first one is to demonstrate that the solutions of master equation \begin{equation*} (\partial_t-\Delta)^s u(x,t) =f(u(x, t)), \,\,(x, t)\in B_1(0)\times \mathbb{R}, \end{equation*} subject to the…

We obtain a vanishing result for solutions of the inequality $|\Delta u|\le q_1|u|+q_2|\nabla u|$ that decay to zero along a very general warped cylindrical end of a Riemannian manifold. The appropriate decay condition at infinity on $u$ is…

In this paper we derive Carleman estimates for the fractional relativistic operator. We consider changing-sign solutions to the heat equation for such operators. We prove monotonicity inequalities and convexity of certain energy functionals…

We prove strong unique continuation property for the differential inequality $|(\partial_t +\Delta)u(x,t)|\le V(x,t)|u(x,t)|$ with $V$ contained in weak spaces. In particular, we establish the strong unique continuation property for $V\in…

In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the Cauchy problem $$ \partial_t u+(-\Delta)^{\frac{\theta}{2}}u=u^p,\quad x\in{\bf R}^N,\,\,t>0, \qquad u(0)=\mu\ge…

We study the following fractional Schr\"{o}dinger equation \begin{equation*}\label{eq0.1} \varepsilon^{2s}(-\Delta)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} where $s\in(0,1)$. Under some conditions on $f(u)$, we show that…

We consider time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$. For such equations, we give an elementary proof of the weak maximum principle under no assumptions on the sign of the reaction…

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian $(-\Delta)^s$ with $s \in (0,1)$ for any space dimensions $N \geq 1$. By extending a monotonicity formula found by…

We consider nonlinear parabolic equations involving fractional diffusion of the form $\partial_t u + (-\Delta)^s \Phi(u)= 0,$ with $0<s<1$, and solve an open problem concerning the existence of solutions for very singular nonlinearities…

This paper investigates the higher pointwise regularity of nonnegative classical solutions for fully fractional parabolic equations $(\partial_t -\Delta)^{s} u = f,$ where $s\in(0,1)$. We establish $C^{k+\alpha+2s}$ or $C^{k+\alpha+2s,\ln}…

In this note we study the property of unique continuation for solutions of $|(-\Delta)^{\alpha/2}u|\leq|Vu|$, where $V$ is in a function class of potentials including $\bigcup_{p>n/\alpha}L^p(\mathbb{R}^n)$ for $n-1\leq\alpha<n$. In…

We study integrability conditions for existence and nonexistence of a local-in-time integral solution of fractional semilinear heat equations with rather general growing nonlinearities in uniformly local $L^p$ spaces. Our main results about…

In this article, we study a quantitative form of the Landis conjecture on exponential decay for real-valued solutions to second order elliptic equations with variable coefficients in the plane. In particular, we prove the following…