Blowup stability of solutions of the nonlinear heat equation with a large life span

Abstract

We study the Cauchy problem for the nonlinear heat equation u t - ▵ u = | u | p - 1 u in R N . The initial data is of the form u 0 = λ ϕ , where ϕ ∈ C 0 ( R N ) is fixed and λ > 0 . We first take 1 < p < p f , where p f is the Fujita critical exponent, and ϕ ∈ C 0 ( R N ) ∩ L 1 ( R N ) 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 < p s , where p s is the Sobolev critical exponent, and ϕ ( 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 ϕ is not radial.