# American Institute of Mathematical Sciences

March  2021, 41(3): 1297-1318. doi: 10.3934/dcds.2020318

## Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium

 1 Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro - RJ, 21941-909, Brazil 2 Departamento de Matemática, Universidade de Brasília, Brasília - DF, 70910-900, Brazil

* Corresponding author: Juliana Fernandes

Received  December 2019 Revised  May 2020 Published  August 2020

Fund Project: The first author was partially supported by FAPERJ. The second author was partially supported by FAPDF, CAPES, and CNPq grant 308378/2017 -2

The present paper is on the existence and behaviour of solutions for a class of semilinear parabolic equations, defined on a bounded smooth domain and assuming a nonlinearity asymptotically linear at infinity. The behavior of the solutions when the initial data varies in the phase space is analyzed. Global solutions are obtained, which may be bounded or blow-up in infinite time (grow-up). The main tools are the comparison principle and variational methods. In particular, the Nehari manifold is used to separate the phase space into regions of initial data where uniform boundedness or grow-up behavior of the semiflow may occur. Additionally, some attention is paid to initial data at high energy level.

Citation: Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318
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