Infinite-time blow-up for the 3-dimensional energy-critical heat equation
By Manuel del Pino, Monica Musso, Jun-cheng Wei. Analysis & PDE, 2017
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We construct globally defined in time, unbounded positive solutions to the energy-critical heat equation in dimension three $$ u_t = \Delta u + u^5 , \quad {\mbox {in}} \quad \R^3 \times (0,\infty), \ \ u(x, 0)= u_0 (x)\inn \R^3. $$ For each $\gamma>1$ we find initial data (not necessarily radially symmetric) with $\lim\limits_{r \to \infty} |x|^\gamma u_0 (x) >0$ such that as $t \to \infty$ $$ \| u(\cdot ,t ) \|_\infty \sim t^{\gamma-1 \over 2} , \quad {\mbox {if}} \quad 1 2, \quad $$ and $$ \| u(\cdot , t)\|_\infty \sim \sqrt{t}\, (\ln t )^{-1} , \quad {\mbox {if}} \quad \gamma = 2. $$ Furthermore we show that this infinite time blow-up is co-dimensional one stable. The existence of such solutions was conjectured by Fila and King.
Subfields: Applied Mathematics, Control and Systems Engineering
View Paper: https://www.semanticscholar.org/paper/b29a8457e736b8fcff92c45962dfdb73f1d6a6bf
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In Applied Mathematics
In Control and Systems Engineering
Citations in the corpus
- Optimal Liouville theorems for superlinear parabolic problems (2020, background): “1] and [6] indicate that if n ≤ 6 and p = pS, then the behavior of so-called threshold solutions heavily depends on n.”