Electron. J. Differential Equations, Vol. 2025 (2025), No. 67, pp. 1-11.
Existence and non-existence of solutions for Hardy parabolic equations with singular initial data
Aldryn Aparcana, Brandon Carhuas-Torre, Ricardo Castillo, Miguel Loayza
Abstract:
We establish the existence, non-existence and uniqueness of the local solutions of
the Hardy parabolic equation \(u_t - \Delta u = h(t)|\cdot |^{-\gamma}g(u)\) on
\(\Omega \times (0,T) \) with Dirichlet boundary conditions.
We assume that \(\Omega\) with \(0\in \Omega\) is a smooth domain bounded or unbounded,
\(h \in C(0,\infty)\), \(g \in C([0,\infty))\) is a non-decreasing function,
\(0<\gamma<\min\{2,N\}\), and the initial data have a singularity at the origin.
Submitted January 21, 2025. Published July 4, 2025.
Math Subject Classifications: 35A01, 35A02, 35B33, 35D30, 35K58.
Key Words: Local existence; Hardy parabolic equation; Lebesgue spaces; critical values; uniqueness.
DOI: 10.58997/ejde.2025.67
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Aldryn Aparcana
Facultad de Ciencias
Universidad Nacional San Luis Gonzaga
Ica, Perú
email: aldryn.aparcana@unica.edu.pe
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Brandon Carhuas-Torre
Departamento de Matemática
Universidade Federal de Pernambuco
Av. Jornalista Anibal Fernandes
Cidade Universitáa
Recife, Pernambuco, Brasil
email: brandon.carhuas@ufpe.br
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Ricardo Castillo
Facultad de Ciencias, Departamento de Matemática
Universidad del Bío-Bío
Avenida Collao 1202, Casilla 5-C
Concepci\'on, Bío-Bío, Chile
email: rcastillo@ubiobio.cl
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Miguel Loayza
Departamento de Matemática
Universidade Federal de Pernambuco
Av. Jornalista Anibal Fernandes
Cidade Universitáa
Recife, Pernambuco, Brasil
email: miguel.loayza@ufpe.br
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