Geodesic
$L^\infty$ estimates of solution for $m$-Laplacian parabolic equation with a nonlocal term
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 2, pp. 389-400
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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 $\alpha +\beta m-1$.
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 $\alpha +\beta m-1$.
DOI : 10.1007/s10587-011-0083-1
Classification : 35A01, 35A02, 35B45, 35D30, 35K20, 35K65, 35K92
Keywords: $m$-Laplacian parabolic equations; global existence; uniqueness; $L^{\infty }$ estimates 
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     title = {$L^\infty$ estimates of solution for $m${-Laplacian} parabolic equation with a nonlocal term},
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     year = {2011},
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Hou, Pulun; Chen, Caisheng. $L^\infty$ estimates of solution for $m$-Laplacian parabolic equation with a nonlocal term. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 2, pp. 389-400. doi: 10.1007/s10587-011-0083-1

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