Singular parabolic equations with measures as initial data
type: Item
instance of: scholarly article
title: Singular parabolic equations with measures as initial data
zbMATH Open document ID: 1191.35158
author: Haifeng Shang
author: Fengquan Li
published in: Journal of Differential Equations
publication date: 2009-09-28T00:00:00Z
review text: This paper studies solving the Cauchy problem for singular \(p\)-Laplacian evolution equations. Specifically, for \(\mu\) a non-negative Radon measure, \( \frac{2N}{N+1}<p<2\), consider the following problem: \[ \begin{aligned} u_t-\text{div}(|Du|^{p-2}Du)= \lambda |Du|^l+ \lambda_0u^q &\quad\text{in }\mathbb R^N\times(0,T),\\ u(x,0)=\mu &\quad\text{on }\mathbb R^N, \end{aligned} \] with \(0<l<p,\) \(q\geq 1\) and \(\lambda,\lambda_0\geq 0\). The main theorem the authors obtain is that, if \[ [\mu]:= \sup_{x\in\mathbb R^N}\;\sup_{0<\rho <1} \left(\rho^\theta\frac{1}{|B_\rho(x)|}\int_{B_\rho(x)}\,d\mu \right) <\infty , \] with \(0\leq \theta \leq N\), \(\theta(l-p+1)<(p-l)\) and \(\theta(q-1)<p+\theta(p-2)\), then there exists a solution to the above initial value problem such that, for \(0<t<T_0\), \[ [u]_t:= \sup_{0<\tau<t}\;\sup_{x\in\mathbb R^N}\;\sup_{0<\rho<1} \left(\frac{\rho^\theta}{|B_\rho(x)|}\int_{B_\rho(x)}u(y,\tau) \,dy\right)\leq \gamma \left([\mu]^{\frac{\rho}{\kappa}}+1\right) \] and \[ \|u(\cdot ,t)\|_{\infty,\mathbb R^N}\leq \gamma t^{-(\frac{\theta}{p+\theta (p-2)})} \left([\mu]^{\frac{\rho}{\kappa}}+1\right), \] with \(T_0= T_0([\mu] ,N,p,q,\lambda, l,\theta,\lambda_0)\), \(\gamma=\gamma (N,p,q,\lambda ,l,\theta,\lambda_0)\) and \(\kappa =(p+N(p-2))\). The authors first establish a number of a priori estimates and then use these with an approximation method to prove the theorem. They conclude the paper by proving a second theorem for the supercritical case \(q>p-1+p/N\). The second theorem says for \(u\) a non-negative weak solution, for \(1<p<2\), and \(q>(1+\varepsilon)(p-1)+p/N\), there exist constants \(\rho_0\), \(0<\rho_0<1\), and \(\gamma \) so that \[ \rho^{\frac{p}{q-(1+\varepsilon)(p-1)}} \left(\frac{1}{|B_\rho(x)|} \int_{B_\rho(x)} u(y,t)\,dy\right) \leq \gamma, \] for all \(x\in\mathbb R^N\), \(0<t<T/2\), \(0<\varepsilon <p-1\) and \(0<\rho<\rho_0\). They discuss in detail how their results relate to other recent results, and what exte…
reviewed by: Caroline Sweezy
Mathematics Subject Classification ID: 35K67
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K91
zbMATH DE Number: 5608024
zbMATH Keywords: Cauchy problem
zbMATH Keywords: singular \(p\)-Laplace equation
zbMATH Keywords: gradient
zbMATH Keywords: source
zbMATH Keywords: measures as initial data
zbMATH Keywords: initial trace
MaRDI profile type: Publication
full work available at URL: j.jde.2009.06.003
OpenAlex ID: W2072185643
cites work: Weak solutions of quasilinear parabolic equations with initial measure data
cites work: Semilinear parabolic equations involving measures and low regularity data
cites work: Global existence for a degenerate nonlinear diffusion problem with nonlinear gradient term and source
cites work: Degenerate parabolic equations with initial data measures
cites work: Q4007944
cites work: The Initial Trace of a Solution of the Porous Medium Equation
cites work: Problems paraboliques semi-lineaires avec donnees measures
cites work: Initial trace of solutions of some quasilinear parabolic equations with absorption
cites work: Hölder estimates of solutions of singular parabolic equations with measurable coefficients
cites work: Degenerate parabolic equations
cites work: On the Cauchy Problem and Initial Traces for a Degenerate Parabolic Equation
cites work: The Cauchy problem for \(u_{t}=\Delta u+| \nabla u|^q\)
cites work: Q5562267
cites work: On the Cauchy problem for the evolution \(p\)-Laplacian equations with gradient term and source
cites work: Q5890172
cites work: On the Cauchy problem for the evolution \(p\)-Laplacian equations with gradient term and source and measures as initial data
cites work: On the Cauchy problem for a singular parabolic equation
cites work: On the Cauchy problem of evolution \(p\)-Laplacian equations with strongly nonlinear sources when \(1<p<2\)
DOI: 10.1016/J.JDE.2009.06.003
description: scientific article; zbMATH DE number 5608024