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Nonlinear Heat Equation with a Fractional Laplacian in a Disk

100%

Varlamov V.

Colloquium Mathematicum

1999

tom 81

nr 1

101-122

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.

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

88%

Varlamov V.

Studia Mathematica

2000

tom 142

nr 1

71-99

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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Studia Mathematica

2 Varlamov V.

1 2000

1 1999

Wyniki wyszukiwania

Nonlinear Heat Equation with a Fractional Laplacian in a Disk

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