Existence and non-existence of global solutions for a heat equation with degenerate coefficients

Ricardo Castillo (), Omar Guzmán-Rea () and María Zegarra ()
Additional contact information
Ricardo Castillo: Universidad del Bío-Bío
Omar Guzmán-Rea: Universidade de Brasília
María Zegarra: Universidad Nacional Mayor de San Marcos

Partial Differential Equations and Applications, 2022, vol. 3, issue 6, 1-16

Abstract: Abstract In this paper, the parabolic problem $$u_t - div(\omega (x) \nabla u)= h(t) f(u) + l(t) g(u)$$ u t - d i v ( ω ( x ) ∇ u ) = h ( t ) f ( u ) + l ( t ) g ( u ) with non-negative initial conditions pertaining to $$C_b({\mathbb {R}}^N)$$ C b ( R N ) , will be studied, where the weight $$\omega $$ ω is an appropriate function that belongs to the Muckenhoupt class $$A_{1 + \frac{2}{N}}$$ A 1 + 2 N and the functions f, g, h and l are non-negative and continuous. The main goal is to establish the global and non-global existence of non-negative solutions. In addition, will be obtained both the so-called Fujita’s exponent and the second critical exponent in the sense of Lee and Ni (Trans Am Math Soc 333(1):365–378, 1992), in the particular case when $$h(t)\sim t^r \,(r>-1)$$ h ( t ) ∼ t r ( r > - 1 ) , $$l(t)\sim t^s \, (s>-1)$$ l ( t ) ∼ t s ( s > - 1 ) , $$f(u)=u^p$$ f ( u ) = u p and $$g(u)=(1+u)[\ln (1+u)]^p$$ g ( u ) = ( 1 + u ) [ ln ( 1 + u ) ] p . The results of this paper extend those obtained by Fujishima et al. (Calc Var Partial Differ Equ 58:62, 2019) that worked when $$h(t)=1$$ h ( t ) = 1 , $$l(t)=0$$ l ( t ) = 0 and $$f(u)=u^p $$ f ( u ) = u p .

Keywords: Heat equation; Non-global solution; Global solution; Degenerate coefficients; Fujita exponent; 35K05; 35A01; 35K58; 35K65; 35B33 (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s42985-022-00210-2 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:pardea:v:3:y:2022:i:6:d:10.1007_s42985-022-00210-2

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/42985/

DOI: 10.1007/s42985-022-00210-2

Access Statistics for this article

Partial Differential Equations and Applications is currently edited by Zhitao Zhang

More articles in Partial Differential Equations and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().