Abstract
In this paper, we deal with a class of semilinear parabolic problems related to a Hardy inequality with singular weight at the boundary.
More precisely, we consider the problem
where Ω is a bounded regular domain of \({\mathbbm{R}^N}\), \({d(x)=\text{dist}(x,\partial\Omega)}\), \({p > 0}\), and \({\lambda > 0}\) is a positive constant.
We prove that
-
1.
If \({0 < p < 1}\), then (P) has no positive very weak solution.
-
2.
If \({p=1}\), then (P) has a positive very weak solution under additional hypotheses on \({\lambda}\) and \({u_0}\).
-
3.
If \({p > 1}\), then, for all \({\lambda > 0}\), the problem (P) has a positive very weak solution under suitable hypothesis on \({u_0}\).
Moreover, we consider also the concave–convex-related case.
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Work partially supported by Project MTM2010-18128, MINECO, Spain.
B. Abdellaoui is also partially supported by a Grant from the ICTP centre of Italy.
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Abdellaoui, B., Biroud, K. & Primo, A. A semilinear parabolic problem with singular term at the boundary. J. Evol. Equ. 16, 131–153 (2016). https://doi.org/10.1007/s00028-015-0295-1
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DOI: https://doi.org/10.1007/s00028-015-0295-1