Gradient estimates for a quasilinear parabolic equation of the mean curvature type

A Quasilinear Parabolic Equation With Inhomogeneous Density and Absorption

Gradient estimates for a nonlinear parabolic equation under Ricci flow

Let (M,g(t)), 0 ≤ t ≤ T , be a n-dimensional complete noncompact manifold, n ≥ 2, with bounded curvatures and metric g(t) evolving by the Ricci flow ∂gij ∂t = −2Rij . We will extend the result of L. Ma and Y. Yang and prove a local gradient estimate for positive solutions of the nonlinear parabolic equation ∂u ∂t = ∆u − au log u − qu where a ∈ R is a constant and q is a smooth function on M × [...

Gradient Estimates for a Nonlinear Parabolic Equation on Riemannian Manifolds

Let (M, g) be a complete noncompact Riemannian manifold. In this paper, we derive a local gradient estimate for positive solutions to a simple nonlinear parabolic equation ∂u ∂t = ∆u+ au log u+ bu on M × [0,+∞), where a, b are two real constants. This equation is closely related to the gradient Ricci soliton. We extend the result of L. Ma (Journal of Functional Analysis 241 (2006) 374-382).

Global boundedness, interior gradient estimates, and boundary regularity for the mean curvature equation with boundary conditions

where ν is the outward pointing unit normal of ∂Ω, and where cosθ is a given function on ∂Ω. (Thus, in the capillarity problem, we are considering geometrically a function u in Ω̄ whose graph has the prescribed mean curvature H and which meets the boundary cylinder in the prescribed angle θ.) Here, H = H(x,t) is assumed to be a given locally Lipschitz function in Ω×R satisfying the structural co...

Quenching Profile for a Quasilinear Parabolic Equation

where 00, Z>0, and uq(x) >0, Vx G [—1, Z]. Without loss of generality, we may assume that uq(x) is smooth and bounded above by 1 such that uo(±Z) = 1. Since uo(x) is positive, the local (in time) existence and uniqueness of a classical solution of the problem (1.1)—(1.3) are trivial (see [8]). Many results in quenching, such as single point quenching and profiles, are similar to those b...