Electron. J. Differential Equations, Vol. 2026 (2026), No. 31, pp. 1-18.
Global self-similar solutions for Hardy-Henon equations with linear and quasilinear diffusion
Razvan Gabriel Iagar, Ariel Sanchez, Erik Sarrion-Pedralva
Abstract:
Global self-similar solutions to the parabolic Hardy-Henon equation
$$
u_t=\Delta u^m+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty),
$$
are classified in the range of exponents \(m\geq1\), \(p >m\) and \(\sigma >\max\{-2,-N\}\).
The classification varies strongly with respect to the celebrated \emph{Fujita} and
\emph{Sobolev critical exponents}
$$
p_F(\sigma)=m+\frac{\sigma+2}{N}, \quad p_S(\sigma)=
\begin{cases}
\frac{m(N+2\sigma+2)}{N-2}, & \text{if } N\geq3, \\
\infty, & \text{if } N\in\{1,2\}.
\end{cases}
$$
Indeed, if \(p\in(p_F(\sigma),p_S(\sigma))\), both equations admit self-similar
solutions with either compact support (if \(m >1\)) or Gaussian-like tail as
\(|x|\to\infty\) (if \(m=1\)), as well as a one-parameter family satisfying
$$
u(x,t)\sim C|x|^{-(\sigma+2)/(p-m)}, \quad \text{as }|x|\to\infty.
$$
If \(p\geq p_S(\sigma)\), there are only self-similar solutions with the latter
algebraic tail, while for \(m< p\leq p_F(\sigma)\) no global solutions exist.
The results open the way for a deeper study of the role of these solutions in the
dynamics of the Hardy-Henon equations.
Submitted February 24, 2026. Published April 28, 2026.
Math Subject Classifications: 35A24, 35B33, 35B36, 35C06, 35K57, 35K59.
Key Words: Global solutions; spatially inhomogeneous source; Sobolev critical exponent;
Fujita critical exponent; self-similar solutions; Hardy-Henon equations
DOI: 10.58997/ejde.2026.31
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Razvan Gabriel Iagar
Departamento de Matemática Aplicada
Ciencia e Ingenieria de los Materiales y Tecnologia Electrónica
Universidad Rey Juan Carlos
Móstoles, 28933, Madrid, Spain
email: razvan.iagar@urjc.es
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Ariel Sánchez
Departamento de Matemática Aplicada
Ciencia e Ingenieria de los Materiales y Tecnologia Electrónica
Universidad Rey Juan Carlos
Móstoles, 28933, Madrid, Spain
email: ariel.sanchez@urjc.es
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Erik Sarrión-Pedralva
Departamento de Matemática Aplicada
Ciencia e Ingenieria de los Materiales y Tecnologia Electrónica
Universidad Rey Juan Carlos
Móstoles, 28933, Madrid, Spain
email: erik.sarrion@urjc.es
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