Global,decaying solutions of a focusing energy-critical heat equation in |
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Authors: | Stephen Gustafson Dimitrios Roxanas |
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Affiliation: | 1. Department of Mathematics, University of British Columbia, V6T1Z2 Vancouver, Canada;2. Department of Mathematics, The University of Edinburgh, EH9 3FD Edinburgh, United Kingdom |
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Abstract: | We study solutions of the focusing energy-critical nonlinear heat equation in . We show that solutions emanating from initial data with energy and -norm below those of the stationary solution W are global and decay to zero, via the “concentration-compactness plus rigidity” strategy of Kenig–Merle [33], [34]. First, global such solutions are shown to dissipate to zero, using a refinement of the small data theory and the -dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza–Seregin–Sverak [17], [18] in an argument similar to that of Kenig–Koch [32] for the Navier–Stokes equations. |
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Keywords: | 35K05 35B40 35B65 Nonlinear heat equation Concentration compactness Regularity Asymptotic decay |
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