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- W2337282543 abstract "In this paper, we study the blow-up and/or global existence of the following porous medium equation with variable source power, u t =Δ u m + u p ( x ) , in Ω×(0, T ), where Ω is either a bounded domain or the whole space R N , and p ( x ) is a positive and continuous bounded function defined in Ω satisfying 0 p - =inf p ( x )≤ p ( x )≤ p + =sup p ( x ) p ( x ) and the structure of spatial domain Ω. Roughly speaking, for the Cauchy problem, if p ( x )≤1, uniqueness of the solutions might not hold, and all nontrivial solutions exist globally. If p + > m , there are blow-up solutions. If 1 p ( x )≤ m +2/ N , every nontrivial solution blows up. If p ( x )> m +2/ N , there exist nontrivial global solutions. When p - m +2/ N , the examples we constructed suggest that for some p ( x ) all nontrivial solutions blow up and for others there exist global solutions. The situation is totally different in bounded domains. There is only one critical exponent p = m , while there are three factors, p ( x ); m and properties of the domain, interacting with each other. If p + > m , there are blow-up solutions as in R N . If p + m , all solutions exist globally in time. If p ( x )> m or the domain is small enough, there are global solutions. The most interesting part is that for some p ( x ) satisfying p - m p + , we find the Fujita phenomenon in the bounded domain." @default.
- W2337282543 created "2016-06-24" @default.
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- W2337282543 date "2016-03-01" @default.
- W2337282543 modified "2023-09-26" @default.
- W2337282543 title "Blow-up of porous medium equations with variable source power" @default.
- W2337282543 doi "https://doi.org/10.1360/012015-17" @default.
- W2337282543 hasPublicationYear "2016" @default.
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