Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball

Vladimir Varlamov

Studia Mathematica (2000)

  • Volume: 142, Issue: 1, page 71-99
  • ISSN: 0039-3223

Abstract

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The nonlinear heat equation with a fractional Laplacian [ u t + ( - Δ ) α / 2 u = u 2 , 0 < α ≤ 2 ] , is considered in a unit ball B . Homogeneous boundary conditions and small initial conditions are examined. For 3/2 + ε₁ ≤ α ≤ 2, where ε₁ > 0 is small, the global-in-time mild solution from the space C ⁰ ( [ 0 , ∞ ) , H ₀ κ ( B ) ) with κ < α - 1/2 is constructed in the form of an eigenfunction expansion series. The uniqueness is proved for 0 < κ < α - 1/2, and the higher-order long-time asymptotics is calculated.

How to cite

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Varlamov, Vladimir. "Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball." Studia Mathematica 142.1 (2000): 71-99. <http://eudml.org/doc/216790>.

@article{Varlamov2000,
abstract = {The nonlinear heat equation with a fractional Laplacian $[u_t+(-Δ)^\{α/2\} u = u^2, 0 < α ≤ 2]$, is considered in a unit ball $B$. Homogeneous boundary conditions and small initial conditions are examined. For 3/2 + ε₁ ≤ α ≤ 2, where ε₁ > 0 is small, the global-in-time mild solution from the space $C⁰([0,∞), H₀^\{κ\}(B))$ with κ < α - 1/2 is constructed in the form of an eigenfunction expansion series. The uniqueness is proved for 0 < κ < α - 1/2, and the higher-order long-time asymptotics is calculated.},
author = {Varlamov, Vladimir},
journal = {Studia Mathematica},
keywords = {first initial-boundary value problem; nonlinear heat equation; construction of solutions; higher-order long-time asymptotics; fractional Laplacian; long time asymptotics},
language = {eng},
number = {1},
pages = {71-99},
title = {Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball},
url = {http://eudml.org/doc/216790},
volume = {142},
year = {2000},
}

TY - JOUR
AU - Varlamov, Vladimir
TI - Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball
JO - Studia Mathematica
PY - 2000
VL - 142
IS - 1
SP - 71
EP - 99
AB - The nonlinear heat equation with a fractional Laplacian $[u_t+(-Δ)^{α/2} u = u^2, 0 < α ≤ 2]$, is considered in a unit ball $B$. Homogeneous boundary conditions and small initial conditions are examined. For 3/2 + ε₁ ≤ α ≤ 2, where ε₁ > 0 is small, the global-in-time mild solution from the space $C⁰([0,∞), H₀^{κ}(B))$ with κ < α - 1/2 is constructed in the form of an eigenfunction expansion series. The uniqueness is proved for 0 < κ < α - 1/2, and the higher-order long-time asymptotics is calculated.
LA - eng
KW - first initial-boundary value problem; nonlinear heat equation; construction of solutions; higher-order long-time asymptotics; fractional Laplacian; long time asymptotics
UR - http://eudml.org/doc/216790
ER -

References

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