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arXiv · 1707.07644

Global, decaying solutions of a focusing energy-critical heat equation in $\mathbb{R}^4$

Stephen Gustafson·Dimitrios Roxanas
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Abstract

We study solutions of the focusing energy-critical nonlinear heat equation $u_t = \Delta u - |u|^2u$ in $\mathbb{R}^4.$ We show that solutions emanating from initial data with energy and $\dot{H}^1-$norm below those of the stationary solution $W$ are global and decay to zero, via the "concentration-compactness plus rigidity" strategy of Kenig-Merle. First, such global solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza, Seregin and Sverak in an argument similar to that of Kenig and Koch for the Navier-Stokes equations.

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Stephen Gustafson, Dimitrios Roxanas. 2017-07-24. Global, decaying solutions of a focusing energy-critical heat equation in $\mathbb{R}^4$. https://arxiv.org/abs/1707.07644

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