Green's function and infinite-time bubbling in the critical nonlinear heat equation

2016/04/25 by Cortazar, Carmen, del Pino, Manuel, Musso, Monica · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.07117

Abstract

Let Ω be a smooth bounded domain in \Rⁿ, n≥ 5. We consider the semilinear heat equation at the critical Sobolev exponent u_t = Δu + u^(n+2)/(n-2) \inn Ω× (0,∞), u =0 \onn \ppΩ× (0,∞). Let G(x,y) be the Dirichlet Green's function of -Δ in Ω and H(x,y) its regular part. Let q_j∈ Ω, j=1,…,k, be points such that the matrix [ \beginmatrix H(q₁, q₁) amp; -G(q₁,q₂) amp;⋯ amp; -G(q₁, q_k) -G(q₁,q₂) amp; H(q₂,q₂) amp; -G(q₂,q₃) ⋯ amp; -G(q₃,q_k) ⋮ amp; amp; \ddotsamp; ⋮ -G(q₁,q_k) amp;\cdotsamp; -G(q_k-1, q_k) amp; H(q_k,q_k) \endmatrix ] is positive definite. For any k≥ 1 such points indeed exist. We prove the existence of a positive smooth solution u(x,t) which blows-up by bubbling in infinite time near those points. More precisely, for large time t, u takes the approximate form u(x,t) ≈ ∑_j=1^k α_n ( \frac μ_j(t) μ_j(t)² + |x-ξ_j(t)|² )^\frac n-22 . Here ξ_j(t) → q_j and 0

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