One Dimensional Behavior of Singular N Dimensional Solutions of Semilinear Heat Equations
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We consider u(x,t) a solution of u t =Δu+|u| p − 1 u that blows up at time T, where u:ℝ N ×[0, T)→ℝ, p>1, (N−2)p<N+2 and either u(0)≥ 0 or (3N−4)p<3N+8. We are concerned with the behavior of the solution near a non isolated blow-up point, as T−t→ 0. Under a non-degeneracy condition and assuming that the blow-up set is locally continuous and N−1 dimensional, we escape logarithmic scales of the variable T−t and give a sharper expansion of the solution with the much smaller error term (T−t)1, 1/2−η for any η>0. In particular, if in addition p>3, then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow-up set. Finally, we prove that the mere hypothesis that the blow-up set is continuous implies that it is C 1, 1/2−η for any η>0.
KeywordsError Term Heat Equation Logarithmic Scale Small Error Dimensional Solution
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