An Optimal Liouville Theorem for the Linear Heat Equation with a Nonlinear Boundary Condition
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An Optimal Liouville Theorem for the Linear Heat Equation with a Nonlinear Boundary Condition Pavol Quittner1 Dedicated to the memory of Pavol Brunovský Received: 14 September 2020 / Revised: 14 September 2020 / Accepted: 21 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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 ∈ (−∞, ∞)) 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 ∂u/∂ν = u q , q > 1. Keywords Liouville theorem · Heat equation · Nonlinear boundary condition · Blow-up rate Mathematics Subject Classification 35K60 · 35B45 · 35B40
1 Introduction and Main Results Liouville theorems for scaling invariant superlinear 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 ∈ (−∞, ∞)) guarantee optimal estimates of solutions of related initial-boundary value problems in general domains, including estimates of singularities and decay, see [11] or [16] and the references therein. In the case of the model problem u t − u = u p ,
x ∈ Rn , t ∈ R,
where p > 1, n ≥ 1 and u = u(x, t) > 0, an optimal Liouville theorem (i.e. a Liouville theorem valid in the full subcritical range) has been recently proved in [14]. Its proof was inspired by [4] and it was based on refined energy estimates for suitably rescaled solutions. In this paper we adapt the arguments in [14] to prove an optimal Liouville theorem for the
B 1
Pavol Quittner [email protected] Department of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava, Slovakia
123
Journal of Dynamics and Differential Equations
problem u t − u = 0 uν = uq
in Rn+ × R,
on ∂Rn+ × R,
(1)
where u = u(x, t) > 0, Rn+ := {(x = (x1 , x2 , . . . , xn ) ∈ Rn : x1 > 0}, ν = (−1, 0, 0, . . . , 0) is the outer unit normal on the boundary ∂Rn+ = {x ∈ Rn : x1 = 0} and q > 1. In addition, we also provide an application of our Liouville theorem. The nonexistence of positive classical stationary solutions of (1) is known for q < q S , where +∞ if n ≤ 2, q S := n if n > 2, n−2 and the condition q < q S is optimal for the nonexistence, see [5,7] and the references therein. Our main result is the following Liouville theorem. Theorem 1 Let 1 < q < q S . Then problem (1) does not possess positive classical bounded solutions. The nonexistence result in Theorem 1 follows from the Fujita-type results in [2,3] if q ≤ (n + 1)/n. It has also been proved for n = 1, q > 1 (for solutions with bounded spatial derivatives, see [15]), and for n ≥ 1 and q < qsg or q = qsg (see [12] or [13], respectively), where +∞ if n ≤ 2, qsg := n−1 if n > 2. n−2 Assuming on the contrary that a so
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