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Existence to nonlinear parabolic problems with unbounded weights. (English) Zbl 1414.35112

The authors consider a weighted operator \(\Delta^{\omega}_p\) called in this paper \(\omega - p\) Laplacian, which is defined as \[ \Delta^{\omega}_p u = \operatorname{div}(\omega(x)|\nabla u|^{p-2}\nabla u) \] with a certain weight function \( \omega : \Omega\mapsto \mathbb{R}\) where \(\Omega\) is a bounded domain. The meaning of replacing \(\Delta_p\) by \(\Delta^{\omega}_p\) would describe space nonhomogeneity of the process.
In this paper, the authors provide a clear, self-contained theory of existence for nonlinear parabolic equations of the type \[ u_t-\Delta^{\omega_2}_pu = \lambda \omega_1(x)|u|^{p-2}u \] in \(\Omega \times T\) where \(\omega_1\), \(\omega_2\) are weights satisfying the Hardy-type inequality. The authors prove existence of a global weak solution in the weighted Sobolev spaces provided that \(\lambda > 0\) is smaller than the optimal constant in the inequality. The obtained solution is proven to belong to \[ L^p(\mathbb{R}_+; W^{1,p}_{(\omega_1,\omega_2),0}(\Omega))\cap L^{\infty}(\mathbb{R}_+; L^2(\Omega)). \]

MSC:

35K55 Nonlinear parabolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
47J35 Nonlinear evolution equations
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