Related papers: Fundamental solution for super-critical non-symmet…

In this paper we study the fundamental solution $\varGamma(t,x;\tau,\xi)$ of the parabolic operator $L_{t}=\partial_{t}-\Delta+b(t,x)\cdot\nabla$, where for every $t$, $b(t,\cdot)$ is a divergence-free vector field, and we consider the case…

We study the existence of nontrivial nonlocal nonnegative solutions $u(x,t)$ of the nonlinear initial value problems \[ (\partial_t -\Delta)^\alpha u\geq u^\lambda \quad \text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq 1 \] \[ u=0…

The heat kernel associated with the setting of the classical Jacobi polynomials is defined by an oscillatory sum which cannot be computed explicitly, in contrast to the situation for the two other classical systems of orthogonal…

In this paper, we obtain an explicit formula for the heat kernel on the infinite Cayley graph of the modular group $\operatorname{PSL}_2\mathbb{Z}$, given by the presentation $\langle a,b\mid a^2=1, b^3=1\rangle$. Our approach extends the…

We study the heat kernel asymptotics for the Laplace type differential operators on vector bundles over Riemannian manifolds. In particular this includes the case of the Laplacians acting on differential p-forms. We extend our results…

Let $N\ge 1$ and let $f\in C[0,\infty)$ be a nonnegative nondecreasing function and $u_0$ be a possibly singular nonnegative initial function. We are concerned with existence and nonexistence of a local in time nonnegative solution in a…

The Zaremba boundary-value problem is a boundary value problem for Laplace-type second-order partial differential operators acting on smooth sections of a vector bundle over a smooth compact Riemannian manifold with smooth boundary but with…

Let $\mathscr{L}$ be either the Laplace--Beltrami operator, its shift without spectral gap, or the distinguished Laplacian on a symmetric space of noncompact type $\mathbb{X}$ of arbitrary rank. We consider the heat equation, the fractional…

The standard problem for the classical heat equation posed in a bounded domain $\Omega$ of $\mathbb R^n$ is the initial and boundary value problem. If the Laplace operator is replaced by a version of the fractional Laplacian, the initial…

We study a porous medium-type equation whose pressure is given by a nonlocal L\'{e}vy operator associated to a symmetric jump L\'{e}vy kernel. The class of nonlocal operators under consideration appears as a generalization of the classical…

In this paper, we consider the following type of non-local (pseudo-differential) operators $\LL $ on $\R^d$: $$ \LL u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} (a_{ij}(x) \frac{\partial}{\partial x_j}) + \lim_{\eps…

We consider nonnegative solutions of a parabolic equation in a cylinder $D \timesI$, where $D$ is a noncompact domain of a Riemannian manifold and $I =(0,T)$ with $0 < T \le \infty$ or $I=(-\infty,0)$. Under the assumption [SSP] (i.e., the…

We obtain Sobolev inequalities for the Schrodinger operator -\Delta-V, where V has critical behaviour V(x)=((N-2)/2)^2|x|^{-2} near the origin. We apply these inequalities to obtain pointwise estimates on the associated heat kernel,…

The problem of obtaining necessary and sufficient conditions for local existence of non-negative solutions in Lebesgue spaces for semilinear heat equations having monotonically increasing source term $f$ has only recently been resolved…

Let $G=(V,E)$ be a finite or locally finite connected weighted graph, $\Delta$ be the usual graph Laplacian. Using heat kernel estimate, we prove the existence and nonexistence of global solutions for the following semilinear heat equation…

In this paper, we establish gradient continuity for solutions to \[ (\partial_t - \operatorname{div}(A(x) \nabla u))^s =f,\ s \in (1/2, 1), \] when $f$ belongs to the scaling critical function space $L(\frac{n+2}{2s-1}, 1)$. Our main…

The purpose of this article is to establish regularity and pointwise upper bounds for the (relative) fundamental solution of the heat equation associated to the weighted dbar-operator in $L^2(C^n)$ for a certain class of weights. The…

We consider the problem of existence of a (unique) weak solution to the SDE describing symmetric $\alpha$-stable process with a locally unbounded drift $b:\mathbb R^d \rightarrow \mathbb R^d$, $d \geq 3$, $1<\alpha<2$. In this paper, $b$…

We establish a reduction principle to derive Li-Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension…

In this paper we show the H\"ormander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general H\"ormander's Lie bracket conditions, we show the regularization…