Blowup stability of solutions of the nonlinear heat equation with a large life span |
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Authors: | Flá vio Dickstein |
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Affiliation: | Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21944-970 Rio de Janeiro, Brazil |
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Abstract: | We study the Cauchy problem for the nonlinear heat equation ut-?u=|u|p-1u in RN. The initial data is of the form u0=λ?, where ?∈C0(RN) is fixed and λ>0. We first take 1<p<pf, where pf is the Fujita critical exponent, and ?∈C0(RN)∩L1(RN) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<ps, where ps is the Sobolev critical exponent, and ?(x) decaying as |x|-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial. |
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Keywords: | 35B40 35K55 35K57 |
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