Related papers: Self-similar solution for Hardy operator
This paper concerns the global in time existence of solutions for a semilinear heat equation \begin{equation} \tag{P} \label{eq:P} \begin{cases} \partial_t u = \Delta u + f(u), &x\in \mathbb{R}^N, \,\,\, t>0, \\[3pt] u(x,0) = u_0(x) \ge 0,…
In this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional Laplacians and the critical exponents. This work can be seen…
It is shown that the fourth-order parabolic thin film equation with critical absorption admits the large time behaviour, where special logarithmic perturbations enter the standard source-type asymptotics. Such asymptpotic phenomena on…
We investigate the existence of positive solutions to fractional equations presenting a double criticality: a multi-polar Hardy-type potential and a Sobolev critical nonlinearity. The nonlocal nature of the operator and the absence of…
We compute the first 5 terms in the short-time heat trace asymptotics expansion for an operator of Laplace type with transfer boundary conditions using the functorial properties of these invariants.
We study the regularity of the bounded self-similar solution to the one-phase Stefan problem with fractional diffusion posed on the whole line. In terms of the enthalpy $h(x,t)$, the evolution problem reads \[ \begin{cases} \partial_t h +…
We consider a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. In a previous paper we have found mass-preserving, nonnegative weak solutions of the equation satisfying energy…
We study Cauchy problem for the Hardy-H\'enon parabolic equation with an inverse square potential, namely, \[\partial_tu -\Delta u+a|x|^{-2} u= |x|^{\gamma} F_{\alpha}(u),\] where $a\ge-(\frac{d-2}{2})^2,$ $\gamma\in \mathbb R$, $\alpha>1$…
In this note we analyse \emph{quantitative} approximation properties of a certain class of \emph{nonlocal} equations: Viewing the fractional heat equation as a model problem, which involves both \emph{local} and \emph{nonlocal}…
This paper extends results of M. van den Berg on two-term asymptotics for the trace of Sch\"odinger operators when the Laplacian is replaced by non-local (integral) operators corresponding to rotationally symmetric stable processes and…
We consider certain constant-coefficient differential operators on $\mathbb{R}^d$ with positive-definite symbols. Each such operator $\Lambda$ with symbol $P$ defines a semigroup $e^{-t\Lambda}$ , $t>0$ , admitting a convolution kernel…
This paper introduces a generalized fractional Halanay-type coupled inequality, which serves as a robust tool for characterizing the asymptotic stability of diverse time fractional functional differential equations, particularly those…
In this short paper, we derive an alternative proof for some known [van den Berg & Gilkey 2015] short-time asymptotics of the heat content in compact full-dimensional submanifolds $S$ with smooth boundary. This includes formulae like…
This paper establishes a Feynman-Kac formula to represent the solution to general time inhomogeneous stochastic parabolic partial differential equations driven by multiplicative fractional Gaussian noises in bounded domain where L_t is a…
We analyze the long-time asymptotics for the Degasperis--Procesi equation on the half-line. By applying nonlinear steepest descent techniques to an associated $3 \times 3$-matrix valued Riemann--Hilbert problem, we find an explicit formula…
We prove heat kernel estimates for the $\bar\partial$-Neumann Laplacian acting in spaces of differential forms over noncompact, strongly pseudoconvex complex manifolds with a Lie group symmetry and compact quotient. We also relate our…
We give local in time sharp two sided estimates of the heat kernel associated with the relativistic stable operator perturbed by a critical (Hardy) potential.
We construct solutions of Schr\"odinger equations which are asymptotically self-similar solutions as time goes to infinity. Also included are situations with two bubbles. These solutions are global, with non-zero $L^2$ norms, and are…
Upper bounds are obtained for the heat content of an open set D in a geodesically complete Riemannian manifold M with Dirichlet boundary condition on bd(D), and non-negative initial condition. We show that these upper bounds are close to…
We study a semilinear PDE generalizing the Fujita equation whose evolution operator is the sum of a fractional power of the Laplacian and a convex non-linearity. Using the Feynman-Kac representation we prove criteria for asymptotic…