In this paper, the parabolic problem ut- div(ω(x) ∇ u) = h(t) f(u) + l(t) g(u) with non-negative initial conditions pertaining to Cb(RN) , will be studied, where the weight ω is an appropriate function that belongs to the Muckenhoupt class A1+2N 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)∼tr(r>-1), l(t)∼ts(s>-1), f(u) = up and 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 , l(t) = 0 and f(u) = up.
Bibliographical noteFunding Information:
We are very grateful for the time and valuable suggestions of the anonymous reviewers. Parts of this work are supported by ANID-FONDECYT Project No. 11220152 (Ricardo Castillo) and Universidad Nacional Mayor de San Marcos under Grant B21141451. O. Guzmán-Rea was supported by CNPq/Brazil, 166685/2020-8. Funding was provided by Universidad del Bío-Bío (Grant No. 2020139IF/R).
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- Degenerate coefficients
- Fujita exponent
- Global solution
- Heat equation
- Non-global solution