TY - JOUR

T1 - Asymptotics for fractional nonlinear heat equations

AU - Hayashi, Nakao

AU - Kaikina, Elena I.

AU - Naumkin, Pavel I.

N1 - Funding Information:
Received 16 March 2004; revised 14 October 2004. 2000 Mathematics Subject Classification 35K55 (primary), 35B40 (secondary). The work of the second and third authors was partially supported by CONACYT.

PY - 2005/12

Y1 - 2005/12

N2 - The Cauchy problem is studied for the nonlinear equations with fractional power of the negative Laplacian {ut+(-δ)α\2 u + u1+σ = 0, u(0,x) = u0(x), x ∈ Rn, t> 0, u(0,x) = u0(x), x ∈ Rn , where α ∈ ( 0,2), with critical σ = α/n and sub-critical σ ∈ (0,α/n) powers of the nonlinearity. Let u0∈ L 1,a} ∩L∞∩ C, u0(x)≥ 0 in Rn}, θ =}∫Rnn u0( x) dx>0. The case of not small initial data is of interest. It is proved that the Cauchy problem has a unique global solution u ∈ C([0,∞); L∞∩ L1,a∩ C) and the large time asymptotics are obtained.

AB - The Cauchy problem is studied for the nonlinear equations with fractional power of the negative Laplacian {ut+(-δ)α\2 u + u1+σ = 0, u(0,x) = u0(x), x ∈ Rn, t> 0, u(0,x) = u0(x), x ∈ Rn , where α ∈ ( 0,2), with critical σ = α/n and sub-critical σ ∈ (0,α/n) powers of the nonlinearity. Let u0∈ L 1,a} ∩L∞∩ C, u0(x)≥ 0 in Rn}, θ =}∫Rnn u0( x) dx>0. The case of not small initial data is of interest. It is proved that the Cauchy problem has a unique global solution u ∈ C([0,∞); L∞∩ L1,a∩ C) and the large time asymptotics are obtained.

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U2 - 10.1112/S0024610705006782

DO - 10.1112/S0024610705006782

M3 - Article

AN - SCOPUS:29144528081

SN - 0024-6107

VL - 72

SP - 663

EP - 688

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 3

ER -