According to Gurarii and Matsaev [5], these properties are equivalent, but no proof was ever published. Here is a summary of known results. When n = 1, both properties hold if and only if D is a bounded interval. For PII this is evident, for PI this follows from the results of Tychonov [7]. When n = 2, and D is an angular sector {z : | arg z| < α}, the property PII holds if and only if α < 45◦....

In this paper we shall study the evolution of the zero set of the solution of the heat equation perturbed by a potential c and Neumann boundary conditions in one and two dimensions. 1. Introduction Let IR d be a bounded connected domain with a smooth boundary @ (we assume throughout that is of class C 3) and let c be a continuous function on , the closure of. Consider the heat equation with a p...

This paper shows how the enclosure method which was originally introduced for elliptic equations can be applied to inverse initial boundary value problems for parabolic equations. For the purpose a prototype of inverse initial boundary value problems whose governing equation is the heat equation is considered. An explicit method to extract an approximation of the value of the support function a...

This paper concentrates on analysis and discussion of the heat equation as it pertains to smoothing of geometric shapes and its relationship to the problem of surface fairing. The geometric heat equation distorts a given shape in order to obtain scale-space representation of a shape. This scale-space provides a complete description of the original shape in terms of small to large scale structur...

Keywords: Stefan problem Non-classical heat equation Free boundary problem Similarity solution Nonlinear heat sources Volterra integral equation a b s t r a c t We prove the existence and uniqueness, local in time, of the solution of a one-phase Stefan problem for a non-classical heat equation for a semi-infinite material with a convective boundary condition at the fixed face x = 0. Here the he...

We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: ut = J ∗u−u , where J is a symmetric continuous probability density. Depending on the tail of J , we give a rather complete picture of the problem in optimal classes of data by: (i) estimating the initial trace of (possibly unbounded) solutions; (ii) showing existence and uniqueness results in a su...

In this paper we study the Gross heat equation perturbed by noises with the initial condition being a generalized function. The noises are given by either a white noise or a space-time white noise. The main technique we use is the representation of the Gross Laplacian as a convolution operator. It enables us to apply the convolution calculus on a suitable distribution space to obtain the explic...

1. Let a certain body, capable of absorbing heat, occupy a region of n dimensional Riemannian space V, whose metric is defined by the positive quadratic differential form ds2 = gjj dxl dxJ. Suppose that this body is heated by conduction in any manner. The situation can be visualized as a flow of heat from the warmer to the cooler parts of the body. The rate of flow of heat within the body can b...