\(L^{\infty }\) estimates of solution for \(m\)-Laplacian parabolic equation with a nonlocal term. (English) Zbl 1249.35177
Summary: In this paper, we consider the global existence, uniqueness and \(L^{\infty }\) estimates of weak solutions to quasilinear parabolic equation of \(m\)-Laplacian type \(u_{t}-\operatorname {div}(| \nabla u| ^{m-2}\nabla u)=u| u| ^{\beta -1}\int _{\Omega } | u| ^{\alpha }\, dx\) 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 \(\alpha +\beta < m-1\).
MSC:
| 35K65 | Degenerate parabolic equations |
| 35K92 | Quasilinear parabolic equations with \(p\)-Laplacian |
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