Let P be a principal U(1)-bundle over a closed manifold M . On P , one can define a modified version of the Ricci flow called the Ricci Yang-Mills flow, due to these equations being a coupling of Ricci flow and the Yang-Mills heat flow. We use maximal regularity theory and ideas of Simonett concerning the asymptotic behavior of abstract quasilinear parabolic partial differential equations to st...

An efficient scheme for solving boundary integral equations of the heat equation based on the Galerkin method is introduced. The parabolic fast multipole method (pFMM) is applied to accelerate the evaluation of the thermal layer potentials. In order to remain attractive for a wide range of applications, a key issue is to ensure efficiency for a big variety of temporal to spatial mesh ratios. Wi...

Elliptic partial differential equations (PDEs) are frequently used to model a variety of engineering phenomena, such as steady-state heat conduction in a solid, or reaction-diffusion type problems. However, computing a solution can sometimes be difficult or inefficient using standard solvers. Techniques have been developed, including the method of lines (Schiesser, 1991), which can solve parabo...

The concept of the iterative parabolic approximation based on the multiscale technique is discussed. This approach is compared with the traditional ways to derive the wide-angle parabolic equation. While the latter fail in the nonlinear case, the multiscale derivation technique leading to iterative parabolic equations can be easily adapted to handle it. The nonlinear iterative parabolic approxi...

Hölder estimates for spatial second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions.

We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates.

Abstract In this paper, we consider the sign of solutions to Cauchy problems linear and semilinear polyharmonic heat equations. for higher order parabolic equations have no positivity preserving property in general, however, it is expected that these are eventually globally positive if initial data decay slowly enough. We first show existence threshold rate datum which separates whether corresp...

It is well known that the parabolic partial differential equations in two or more space dimensions with overspecified boundary data, feature in the mathematical modeling of many phenomena. In this article, an inverse problem of determining an unknown time-dependent source term of a parabolic equation in general dimensions is considered. Employing some transformations, we change the inverse prob...

In this paper, a nonlinear inverse problem of parabolic type, is considered. By reducing this inverse problem to a system of Volterra integral equations the existence, uniqueness, and stability of the solution will be shown.

Fick’s law expresses the proportionality of solute flux with respect to concentration gradient. Similar relations are Darcy’s law for the fluid flow in porous media, Ohm’s law for the electric flux and Fourier’s law for heat transfers. When introduced in the corresponding balance equations, these flux laws yield diffusion equations of parabolic character. Different attempts have been made to ob...