An optimal Liouville theorem for the linear heat equation with a nonlinear boundary condition. (English) Zbl 07818485
Summary: Liouville theorems for scaling invariant nonlinear parabolic problems in the whole space and/or the halfspace (saying that the problem does not posses positive bounded solutions defined for all times \(t\in (-\infty,\infty))\) guarantee optimal estimates of solutions of related initial-boundary value problems in general domains. We prove an optimal Liouville theorem for the linear equation in the halfspace complemented by the nonlinear boundary condition \(\partial u/\partial \nu =u^q\), \(q>1\).
MSC:
| 35B53 | Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35B45 | A priori estimates in context of PDEs |
| 35K58 | Semilinear parabolic equations |
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