L ∞ estimates of solution for m -Laplacian parabolic equation with a nonlocal term

Pulun Hou; Caisheng Chen

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 2, page 389-400
  • ISSN: 0011-4642

Abstract

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In this paper, we consider the global existence, uniqueness and L ∞ estimates of weak solutions to quasilinear parabolic equation of m -Laplacian type u t - div ( | ∇ u | m - 2 ∇ u ) = u | u | β - 1 ∫ Ω | u | α d x in Ω × ( 0 , ∞ ) with zero Dirichlet boundary condition in ∂ Ω . Further, we obtain the L ∞ estimate of the solution u ( t ) and ∇ u ( t ) for t > 0 with the initial data u 0 ∈ L q ( Ω ) ( q > 1 ) , and the case α + β < m - 1 .

How to cite

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Hou, Pulun, and Chen, Caisheng. "$L^\infty $ estimates of solution for $m$-Laplacian parabolic equation with a nonlocal term." Czechoslovak Mathematical Journal 61.2 (2011): 389-400. <http://eudml.org/doc/196752>.

@article{Hou2011,
abstract = {In this paper, we consider the global existence, uniqueness and $L^\{\infty \}$ estimates of weak solutions to quasilinear parabolic equation of $m$-Laplacian type $u_\{t\}-\mathop \{\rm div\}(|\nabla u|^\{m-2\}\nabla u)=u|u|^\{\beta -1\}\int _\{\Omega \} |u|^\{\alpha \} \{\rm d\} x$ in $\Omega \times (0,\infty )$ with zero Dirichlet boundary condition in $\partial \Omega $. Further, we obtain the $L^\{\infty \}$ estimate of the solution $u(t)$ and $\nabla u(t)$ for $t>0$ with the initial data $u_0\in L^q(\Omega )$$(q>1)$, and the case $\alpha +\beta < m-1$.},
author = {Hou, Pulun, Chen, Caisheng},
journal = {Czechoslovak Mathematical Journal},
keywords = {$m$-Laplacian parabolic equations; global existence; uniqueness; $L^\{\infty \}$ estimates; -Laplacian parabolic equation; global existence; uniqueness; estimate},
language = {eng},
number = {2},
pages = {389-400},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$L^\infty $ estimates of solution for $m$-Laplacian parabolic equation with a nonlocal term},
url = {http://eudml.org/doc/196752},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Hou, Pulun
AU - Chen, Caisheng
TI - $L^\infty $ estimates of solution for $m$-Laplacian parabolic equation with a nonlocal term
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 2
SP - 389
EP - 400
AB - In this paper, we consider the global existence, uniqueness and $L^{\infty }$ estimates of weak solutions to quasilinear parabolic equation of $m$-Laplacian type $u_{t}-\mathop {\rm div}(|\nabla u|^{m-2}\nabla u)=u|u|^{\beta -1}\int _{\Omega } |u|^{\alpha } {\rm d} x$ in $\Omega \times (0,\infty )$ with zero Dirichlet boundary condition in $\partial \Omega $. Further, we obtain the $L^{\infty }$ estimate of the solution $u(t)$ and $\nabla u(t)$ for $t>0$ with the initial data $u_0\in L^q(\Omega )$$(q>1)$, and the case $\alpha +\beta < m-1$.
LA - eng
KW - $m$-Laplacian parabolic equations; global existence; uniqueness; $L^{\infty }$ estimates; -Laplacian parabolic equation; global existence; uniqueness; estimate
UR - http://eudml.org/doc/196752
ER -

References

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