Nonlinear Heat Equation with a Fractional Laplacian in a Disk

Vladimir Varlamov

Colloquium Mathematicae (1999)

  • Volume: 81, Issue: 1, page 101-122
  • ISSN: 0010-1354

Abstract

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For the nonlinear heat equation with a fractional Laplacian u t + ( - Δ ) α / 2 u = u 2 , 1 < α ≤ 2, the first initial-boundary value problem in a disk is considered. Small initial conditions, homogeneous boundary conditions, and periodicity conditions in the angular coordinate are imposed. Existence and uniqueness of a global-in-time solution is proved, and the solution is constructed in the form of a series of eigenfunctions of the Laplace operator in the disk. First-order long-time asymptotics of the solution is obtained.

How to cite

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Varlamov, Vladimir. "Nonlinear Heat Equation with a Fractional Laplacian in a Disk." Colloquium Mathematicae 81.1 (1999): 101-122. <http://eudml.org/doc/210722>.

@article{Varlamov1999,
abstract = {For the nonlinear heat equation with a fractional Laplacian $u_t + (-Δ)^\{α/2\} u = u^2$, 1 < α ≤ 2, the first initial-boundary value problem in a disk is considered. Small initial conditions, homogeneous boundary conditions, and periodicity conditions in the angular coordinate are imposed. Existence and uniqueness of a global-in-time solution is proved, and the solution is constructed in the form of a series of eigenfunctions of the Laplace operator in the disk. First-order long-time asymptotics of the solution is obtained.},
author = {Varlamov, Vladimir},
journal = {Colloquium Mathematicae},
keywords = {nonlinear heat equation; long-time asymptotics; fractional Laplacian; initial-boundary value problem in a disk; global-in-time mild solution; eigenfunction expansions; perturbation theory; long-time asymptotic behaviour},
language = {eng},
number = {1},
pages = {101-122},
title = {Nonlinear Heat Equation with a Fractional Laplacian in a Disk},
url = {http://eudml.org/doc/210722},
volume = {81},
year = {1999},
}

TY - JOUR
AU - Varlamov, Vladimir
TI - Nonlinear Heat Equation with a Fractional Laplacian in a Disk
JO - Colloquium Mathematicae
PY - 1999
VL - 81
IS - 1
SP - 101
EP - 122
AB - For the nonlinear heat equation with a fractional Laplacian $u_t + (-Δ)^{α/2} u = u^2$, 1 < α ≤ 2, the first initial-boundary value problem in a disk is considered. Small initial conditions, homogeneous boundary conditions, and periodicity conditions in the angular coordinate are imposed. Existence and uniqueness of a global-in-time solution is proved, and the solution is constructed in the form of a series of eigenfunctions of the Laplace operator in the disk. First-order long-time asymptotics of the solution is obtained.
LA - eng
KW - nonlinear heat equation; long-time asymptotics; fractional Laplacian; initial-boundary value problem in a disk; global-in-time mild solution; eigenfunction expansions; perturbation theory; long-time asymptotic behaviour
UR - http://eudml.org/doc/210722
ER -

References

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