21,860 research outputs found
Existence of stable solutions to in with and
We consider the polyharmonic equation in
with and . We prove the existence of many entire stable
solutions. This answer some questions raised by Farina and Ferrero
Critical criteria of Fujita type for a system of inhomogeneous wave inequalities in exterior domains
We consider blow-up results for a system of inhomogeneous wave inequalities
in exterior domains. We will handle three type boundary conditions: Dirichlet
type, Neumann type and mixed boundary conditions. We use a unified approach to
show the optimal criteria of Fujita type for each case. Our study yields
naturally optimal nonexistence results for the corresponding stationary wave
system and equation. We provide many new results and close some open questions
A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science
In this paper, we study damped second-order dynamics, which are quasilinear
hyperbolic partial differential equations (PDEs). This is inspired by the
recent development of second-order damping systems for accelerating energy
decay of gradient flows. We concentrate on two equations: one is a damped
second-order total variation flow, which is primarily motivated by the
application of image denoising; the other is a damped second-order mean
curvature flow for level sets of scalar functions, which is related to a
non-convex variational model capable of correcting displacement errors in image
data (e.g. dejittering). For the former equation, we prove the existence and
uniqueness of the solution. For the latter, we draw a connection between the
equation and some second-order geometric PDEs evolving the hypersurfaces which
are described by level sets of scalar functions, and show the existence and
uniqueness of the solution for a regularized version of the equation. The
latter is used in our algorithmic development. A general algorithm for
numerical discretization of the two nonlinear PDEs is proposed and analyzed.
Its efficiency is demonstrated by various numerical examples, where simulations
on the behavior of solutions of the new equations and comparisons with
first-order flows are also documented
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