Abstract.
In this work we study the problem
where \(\Omega \subset {\it \mathbb R}^{N} (N \geq 3)\) is a bounded regular domain such that \(0 \in \Omega\), \(\alpha \geq p-1, -\infty < \gamma < \frac{N-p}{p},\lambda > 0, f \in L^{1}(\Omega \times (0, T))\) and \(u_{0} \in L^{1}(\Omega)\) are positive functions. The main points under analysis are some nonexistence results and complete blow-up in the case \(p > 2\) and \(\gamma + 1 > 0\) and some examples of existence for \((\gamma +1) > 0\) and \(1 < p < 2\). These results are interesting as they prove the role of Harnack inequality in this kind of problems and allow to understand better the blow-up behavior.
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I. Peral Alonso: Partially supported by Project MTM2004-02223 of M.E.C. Spain
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Abdellaoui, B., Peral Alonso, I. The effect of Harnack inequality on the existence and nonexistence results for quasi-linear parabolic equations related to Caffarelli-Kohn-Nirenberg inequalities. Nonlinear differ. equ. appl. 14, 335–360 (2007). https://doi.org/10.1007/s00030-007-5048-6
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DOI: https://doi.org/10.1007/s00030-007-5048-6