Related papers: Heat equation on the hypergraph containing vertice…
We introduce and study an extension of the heat equation relevant to relativistic energy formula involving square root of differential operators. We furnish exact solutions of corresponding Cauchy (initial) problem using the operator…
By a probabilistic method we provide an explicit fundamental solution of the Cauchy problem associated to the heat equation on the half-line with constant drift and Dirichlet boundary condition at zero.
This paper considers the initial-boundary value problem for the heat equation with a dynamic type boundary condition. Under some regularity, consistency and orthogonality conditions, the existence, uniqueness and continuous dependence upon…
We study the Cauchy problem for the semilinear heat equation with the singular potential, called the Hardy-Sobolev parabolic equation, in the energy space. The aim of this paper is to determine a necessary and sufficient condition on…
In the present study examines the statistical structure of the generated randomized density of the normal distribution and the Cauchy distribution. The study put the allegation that a randomized probability density of the normal…
In this note we analyze, in terms of a simple example, the incompatibility of parabolic evolution and general covariance. For this we introduce a unit time-like four-vector and study the simplest heat flux equation with respect to it. In…
In Lipschitz domains, we study a Darcy-Forchheimer problem coupled with a singular heat equation by a nonlinear forcing term depending on the temperature. By singular we mean that the heat source corresponds to a Dirac measure. We establish…
Hypergraphs are important objects to model ternary or higher-order relations of objects, and have a number of applications in analysing many complex datasets occurring in practice. In this work we study a new heat diffusion process in…
We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by the J-self-adjoint linear operator L depending on a small parameter. The problem has been originated from the lubrication approximation…
Cheeger's inequality states that a tightly connected subset can be extracted from a graph $G$ using an eigenvector of the normalized Laplacian associated with $G$. More specifically, we can compute a subset with conductance…
We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted H\"older and Sobolev spaces with discrete…
We study the Cauchy problem in the hyperbolic space for the heat equation with a Fisher-KPP type forcing term. Depending on the relative strength of diffusion, measured by the infimum of the spectrum of the Laplace-Beltrami operator, as…
Developing an original idea of De Giorgi, we introduce a new and purely variational approach to the Cauchy Problem for a wide class of defocusing hyperbolic equations. The main novel feature is that the solutions are obtained as limits of…
We analyze the heat kernel associated to the Laplacian on a compact metric graph, with standard Kirchoff-Neumann vertex conditions. An explicit formula for the heat kernel as a sum over loops, developed by Roth and Kostrykin, Potthoff, and…
We consider the Cauchy problem for the nonstationary discrete p-Laplacian with inhomogeneous density \r{ho}(x) on an infinite graph which supports the Sobolev inequality. For nonnegative solutions when p > 2, we prove the precise rate of…
In this paper, we consider the global Cauchy problem for the $L^2$-critical semilinear heat equations $ \partial_t h=\Delta h\pm |h|^{\frac4d}h, $ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \R^+\times\R^d$. In…
We study partial data inverse problems for linear and nonlinear parabolic equations with unknown time-dependent coefficients. In particular, we prove uniqueness results for partial data inverse problems for semilinear reaction-diffusion…
We consider the Cauchy problem for the wave equation on a non-globally hyperbolic manifold of the special form (Minkowski plane with a handle) containing closed timelike curves (time machines). We prove that the classical solution of the…
Given a Hilbert space, we investigate the well-posedness of the Cauchy problem for the wave equation for operators with discrete non-negative spectrum acting on it. We consider the cases when the time-dependent propagation speed is regular,…
We address the inverse problem of identifying a time-dependent source coefficient in a one-dimensional heat equation with a fractional Laplacian subject to Dirichlet boundary conditions and an integral nonlocal data. An a priori estimate is…