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Sign-changing stationary solutions and blowup for a nonlinear heat equation in dimension two

Sign-changing stationary solutions and blowup for a nonlinear heat equation in dimension two 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 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

Sign-changing stationary solutions and blowup for a nonlinear heat equation in dimension two

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References (26)

Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer Basel
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-014-0230-x
Publisher site
See Article on Publisher Site

Abstract

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

Journal of Evolution EquationsSpringer Journals

Published: Sep 1, 2014

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