Interior and Boundary Regularity of Solutions to a Plasma Type Equation

Abstract

ABSTRACT. In this paper, we will consider a plasma type equation with homogeneous boundary condition and nonnegative initial data such that there is a finite extinction T*. We will show that the solution is a positive classical solution in the interior of the parabolic cylinder and it decays to zero at the boundary at a certain explicit rate.

Harnack Estimates and Extinction Profile for Weak Solutions of Certain Singular Parabolic Equations

“…There is an analogy between (1.1) and (1.2) both in terms of results (regularity [6,8,13,15], extinction time [3, 6,13], growth conditions [8,20]) and techniques of proof. For example, when m > 1, the theory of an intrinsic Harnack inequality can be developed for (1.1) paralleling that of (1.2) for p > 2 (see [7]).…”

“…The notion of local weak solution (subsolution, supersolution) in the specified classes is standard and we refer to [6,10,13]. For these solutions we prove an intrinsic Harnack inequality, within the ranges (N -2)+/N < m < 1 for (1.1), and 2N/(N + 1) < p < 2 for (1.2),…”

Harnack estimates and extinction profile for weak solutions of certain singular parabolic equations

“…Here Q is an open set of R^, N > 1, 0 < T < oo, QT = Q x (0, T), and D = (d/dxx, ... , d/dxN). The notion of local weak solution (subsolution, supersolution) in the specified classes is standard and we refer to [6,10,13]. For these solutions we prove an intrinsic Harnack inequality, within the ranges (N -2)+/N < m < 1 for (1.1), and 2N/(N + 1) < p < 2 for (1.2),…”

Global existence and convergence of Yamabe flow

“…Assume that w( , T) is positive at a point p 0 , we are going to show that w( , T) is positive in a uniform neighborhood of p 0 . By the connectedness of M, this uniform positivity then implies that u is positive everywhere at time T. Our argument is inspired by [10] and [11].…”