Results 31 to 40 of about 152 (41)

Equivalence of viscosity and weak solutions for a $p$-parabolic equation [PDF]

arXiv, 2019
We study the relationship of viscosity and weak solutions to the equation \[ \smash{\partial_{t}u-\Delta_{p}u=f(Du)} \] where $p>1$ and $f\in C(\mathbb{R}^{N})$ satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions.
arxiv  

The concavity of $p$-entropy power and applications in functional inequalities [PDF]

Nonlinear Analysis Volume 179, February 2019, 2019
In this paper, we prove the concavity of $p$-entropy power of probability densities solving the $p$-heat equation on closed Riemannian manifold with nonnegative Ricci curvature. As applications, we give new proofs of $L^p$-Euclidean Nash inequality and $L^p$-Euclidean Logarithmic Sobolev inequality, moreover, an improvement of $L^p$-Logarithmic Sobolev
arxiv  

Singular p-Laplacian parabolic system in exterior domains: higher regularity of solutions and related properties of extinction and asymptotic behavior in time [PDF]

arXiv, 2017
We consider the IBVP in exterior domains for the p-Laplacian parabolic system. We prove regularity up to the boundary, extinction properties for p \in ( 2n/(n+2) , 2n/(n+1) ) and exponential decay for p= 2n/(n+1) .
arxiv  

Weak Harnack estimates for supersolutions to doubly degenerate parabolic equations [PDF]

arXiv, 2017
We establish weak Harnack inequalities for positive, weak supersolutions to certain doubly degenerate parabolic equations. The prototype of this kind of equations is $$\partial_tu-\operatorname{div}|u|^{m-1}|Du|^{p-2}Du=0,\quad p>2,\quad m+p>3.$$ Our proof is based on Caccioppoli inequalities, De Giorgi's estimates and Moser's iterative method.
arxiv  

A gradient flow for the p-elastic energy defined on closed planar curves [PDF]

arXiv, 2018
We study the evolution of closed inextensible planar curves under a second order flow that decreases the $p$-elastic energy. A short time existence result for $p \in (1,\infty)$ is obtained via a minimizing movements method. For $p = 2$, that is in the case of the classic elastic energy, long-time existence is retrieved.
arxiv  

Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion [PDF]

arXiv, 2012
This paper is concerned with weak solutions of the degenerate viscous Hamilton-Jacobi equation $$\partial_t u-\Delta_p u=|\nabla u|^q,$$ with Dirichlet boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^N$, where $p>2$ and $q>p-1$. With the goal of studying the gradient blow-up phenomenon for this problem, we first establish local well ...
arxiv  

L^{\infty} a priori bounds for gradients of solutions to quasilinear inhomogenous fast-growing parabolic systems [PDF]

J. Math. Anal. Appl. 393 (2012), no. 1, 222-230, 2012
We prove boundedness of gradients of solutions to quasilinear parabolic system, the main part of which is a generalization to p-Laplacian and its right hand side's growth depending on gradient is not slower (and generally strictly faster) than p - 1. This result may be seen as a generalization to the classical notion of a controllable growth of right ...
arxiv  

Boundedness of global solutions of a p-Laplacian evolution equation with a nonlinear gradient term [PDF]

arXiv, 2012
We investigate the boundedness and large time behavior of solutions of the Cauchy-Dirichlet problem for the one-dimensional degenerate parabolic equation with gradient nonlinearity: $$ u_t = (|u-x|^{p-2} u-x)_x+|u_x|^q \qquad \text{in}\quad (0, +\infty)\tiles(0, 1),\qquad q > p > 2.$$ We prove that: either $u_x$ blows up in finite time, or $u$ is ...
arxiv  

Complete quenching phenomenon for a parabolic p-Laplacian equation with a weighted absorption. [PDF]

J Inequal Appl, 2018
Zhu L.
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