Electron. J. Differential Equations, Vol. 2023 (2023), No. 72, pp. 1-21.
Qualitative properties of solutions to a reaction-diffusion equation with weighted strong reaction
Razvan Gabriel Iagar, Ana I. Muñoz, Ariel Sánchez
Abstract:
We study the existence and qualitative properties of solutions to the
Cauchy problem associated to the quasilinear reaction-diffusion equation
$$
\partial_tu=\Delta u^m+(1+|x|)^{\sigma}u^p,
$$
posed for \((x,t)\in\mathbb{R}^N\times(0,\infty)\), where \(m>1\), \(p\in(0,1)\)
and \(\sigma>0\). Initial data are taken to be bounded, non-negative and compactly
supported. In the range when \(m+p\geq 2\), we prove existence of local solutions
with a finite speed of propagation of their supports for compactly supported
initial conditions. We also show in this case that, for a given compactly
supported initial condition, there exist infinitely many solutions
to the Cauchy problem, by prescribing the evolution of their interface.
In the complementary range \(m+p< 2\), we obtain new Aronson-Benilan estimates
satisfied by solutions to the Cauchy problem, which are of independent interest
as a priori bounds for the solutions. We apply these estimates to establish
infinite speed of propagation of the supports of solutions if \(m+p< 2\),
that is, \(u(x,t)>0\) for any \(x\in\mathbb{R}^N\), \(t>0\), even in the case when
the initial condition \(u_0\) is compactly supported.
Submitted June 13, 2023. Published October 23, 2023.
Math Subject Classifications: 35B44, 35B45, 35K57, 35K59.
Key Words: Reaction-diffusion equations; weighted reaction; strong reaction; Aronson-Benilan estimates.
DOI: 10.58997/ejde.2023.72
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Razvan Gabriel Iagar
Departamento de Matemática Aplicada
Ciencia e Ingenieria de Materiales y Tecnologia Electrónica
Universidad Rey Juan Carlos
Móstoles, 28933, Madrid, Spain
email: razvan.iagar@urjc.es
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Ana I. Muñoz
Departamento de Matemática Aplicada
Ciencia e Ingenieria de Materiales y Tecnologia Electrónica
Universidad Rey Juan Carlos
Móstoles, 28933, Madrid, Spain
email: anaisabel.munoz@urjc.es
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Ariel Sánchez
Departamento de Matemática Aplicada
Ciencia e Ingenieria de Materiales y Tecnologia Electrónica
Universidad Rey Juan Carlos
Móstoles, 28933, Madrid, Spain
email: ariel.sanchez@urjc.es
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