On the Blow-Up Set for Non-Newtonian Equation with a Nonlinear Boundary Condition
We identify the blow-up set of solutions to the problem 𝑢𝑡=(|𝑢𝑥|𝑝−2𝑢𝑥)𝑥, 𝑥>0, 𝑡∈(0,𝑇), −|𝑢𝑥|𝑝−2𝑢𝑥(0,𝑡)=𝑢𝑝−1(0,𝑡), 𝑡∈(0,𝑇), and 𝑢(𝑥,0)=𝑢0(𝑥)≥0, 𝑥>0, where 𝑝>2. We obtain that the blow up set 𝐵(𝑢) satisfies [0,𝑝(𝑝−1)/(𝑝−2))⊂𝐵(𝑢)⊂[0,𝑝(𝑝−1)/(𝑝−2)]. The proof is based on the analysis of the a...
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2010-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2010/680572 |
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| Summary: | We identify the blow-up set of solutions to the problem 𝑢𝑡=(|𝑢𝑥|𝑝−2𝑢𝑥)𝑥, 𝑥>0, 𝑡∈(0,𝑇), −|𝑢𝑥|𝑝−2𝑢𝑥(0,𝑡)=𝑢𝑝−1(0,𝑡), 𝑡∈(0,𝑇), and 𝑢(𝑥,0)=𝑢0(𝑥)≥0, 𝑥>0, where 𝑝>2. We obtain that the blow up set 𝐵(𝑢) satisfies [0,𝑝(𝑝−1)/(𝑝−2))⊂𝐵(𝑢)⊂[0,𝑝(𝑝−1)/(𝑝−2)]. The proof is based on the analysis of the asymptotic behavior of self-similar representation and on the comparison methods. |
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| ISSN: | 1085-3375 1687-0409 |