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Consider the nonlinear heat equation $$v_t -\Delta v=|v|^{p-1}v \qquad \qquad \qquad (NLH)$$ v t - Δ v = | v | p - 1 v ( N L H ) in the unit ball of $${\mathbb{R}^2}$$ R 2 , with Dirichlet boundary condition. Let $${u_{p,\mathcal{K}}}$$ u p , K be a radially symmetric, sign-changing stationary solution having a fixed number $${\mathcal{K}}$$ K of nodal regions. We prove that the solution of (NLH) with initial value $${\lambda u_{p,\mathcal{K}}}$$ λ u p , K blows up in finite time if | λ −1| > 0 is sufficiently small and if p is sufficiently large. The proof is based on the analysis of the asymptotic behavior of $${u_{p,\mathcal{K}}}$$ u p , K and of the linearized operator $${L= -\Delta - p | u_{p,\mathcal{K}} | ^{p-1}}$$ L = - Δ - p | u p , K | p - 1 .
Journal of Evolution Equations – Springer Journals
Published: Sep 1, 2014
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