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

Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source $$ |x|^{-2}\partial_tu=\Delta u+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ are…

We study the one-loop covariant effective action of Lifshitz theories using the heat kernel technique. The characteristic feature of Lifshitz theories is an anisotropic scaling between space and time. This is enforced by the existence of a…

Recently, great attention has been focused on the study of fractional and non-local operators of elliptic type, both for pure mathematical research and in view of concrete real-world applications. Our problem is related to the fractional…

The \textit{method of semigroups} is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators $L$ and their regularity properties in related functional spaces. The…

We obtain $L^q$--$L^p$ decay estimates, $1\le q<p<\infty$ for solutions of nonlocal heat equations of the form $\partial_tu+\mathcal{L} u=0$. Here $\mathcal{L}$ is an integral operator given by a symmetric nonnegative kernel of L\'evy type.…

Heat-kernel expansion and zeta function regularisation are discussed for Laplace type operators with discrete spectrum in non compact domains. Since a general theory is lacking, the heat-kernel expansion is investigated by means of several…

We study the low-energy approximation for calculation of the heat kernel which is determined by the strong slowly varying background fields in strongly curved quasi-homogeneous manifolds. A new covariant algebraic approach, based on taking…

For $n\geq 3$, we study the existence and asymptotic properties of the fundamental solution for elliptic operators in nondivergence form, ${\mathcal L}(x,\partial_x)=a_{ij}(x)\partial_i\partial_j+b_k(x)\partial_k$, where the $a_{ij}$ have…

For an elliptic differential operator $D$ of order $h$ in $n$ dimensions, the spectral $\zeta$-function $\zeta_D(s)$ for $\Re s > \frac{n}{h}$ can be evaluated as an integral over the heat kernel $e^{-t D}$. Here, alternative expressions…

In this work we address some questions concerning the Cauchy problem for a generalized nonlinear heat equations considering as functional framework the variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$. More precisely, by mixing some…

We propose a novel derivation of the non-local heat kernel expansion, first studied by Barvinsky, Vilkovisky and Avramidi, based on simple diagrammatic equations satisfied by the heat kernel. For Laplace-type differential operators we…

In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators $$ \partial_{t}u(t,x) = \mathcal{L}^{a}u(t,x) + f(t,x), \quad t>0 $$ in $L_{q}(L_{p})$ spaces. Our spatial…

We study the following time-fractional heat equation: \begin{equation*} ^{C}\partial_{t}^{\alpha}u(t)+\mathscr{L}u(t)=0,\quad u(0)=u_0\in X, \quad t\in[0,T],\quad T>0,\quad 0<\alpha<1, \end{equation*} where $^{C}\partial_{t}^{\alpha}$ is…

We give new sufficient conditions for comparability of the fundamental solution of the Schr\"odinger equation $\partial_t=\Delta+V$ with the Gauss-Weierstrass kernel and show that local $L^p$ integrability of $V$ for $p> 1$ is not necessary…

We determine a fundamental solution for the differential operator (Delta - lambda_z)^n on the Riemannian symmetric space G/K, where G is any complex semi-simple Lie group, and K is a maximal compact subgroup. We develop a global zonal…

One of the principal topics of this paper concerns the realization of self-adjoint operators $L_{\Theta, \Om}$ in $L^2(\Om; d^n x)^m$, $m, n \in \bbN$, associated with divergence form elliptic partial differential expressions $L$ with…

For $d\geq 1$ and $0<\beta<\alpha<2$, consider a family of pseudo differential operators $\{\Delta^{\alpha} + a^\beta \Delta^{\beta/2}; a \in [0, 1]\}$ that evolves continuously from $\Delta^{\alpha/2}$ to $ \Delta^{\alpha/2}+…

We prove Calder\'on-Zygmund type estimates of weak solutions to non-homogeneous nonlocal parabolic equations under a minimal regularity requirement on kernel coefficients. In particular, the right-hand side is presented by a sum of…

We consider here convolution operators, in the Caputo sense, with non-singular kernels. We prove that the solutions to some integro-differential equations with such operators (acting on the space variable) coincide with the transition…

We present an ab initio approach to integration theory for nonunital spectral triples. This is done without reference to local units and in the full generality of semifinite noncommutative geometry. The main result is an equality between…