Local and Global Well-Posedness for Fractional Porous Medium Equation in Critical Fourier-Besov Spaces

Ahmed El Idrissi; Brahim El Boukari; Jalila El Ghordaf

doi:10.5269/bspm.67664

Ahmed El Idrissi faculté des sciences et techniques
Brahim El Boukari Sultan Moulay Slimane University
Jalila El Ghordaf Sultan Moulay Slimane University

Resumo

In this paper, we study the Cauchy problem for the Fractional Porous Medium Equation in Rⁿ for n ≥ 2. By using the contraction mapping method, Littlewood-Paley theory and Fourier analysis, we get, when 1 < β ≤ 2, the local solution v ∈ X_T := L_T ^∞(FB_p,r ^{(2 − 2m −β + n/p' )}(Rⁿ))∩ L_T^ρ₁(FB_p,r ^s₁(Rⁿ))∩ L_T^ρ₂(FB_p,r ^s₂ ( Rⁿ)) with 1 ≤ p < ∞, 1 ≤ r ≤ ∞, and the solution becomes global when the initial data is small in critical Fourier-Besov spaces FB_p,r ^{(2 − 2m −β + n/p' )}(Rⁿ) . In addition, We establish a blowup criterion for the solutions. Furthermore, the global existence of solutions with small initial data in FB_∞,1 ^{(2 − 2m −β + n )}(Rⁿ) is also established. In the limit case β = 1, we prove global well-posedness for small initial data in critical Fourier-Besov spaces FB_p,1 ^{(2 − 2m + n/p' )}(Rⁿ) with 1 ≤ p < ∞ and FB_∞,1 ^{(2 − 2m + n )}(Rⁿ), respectively.

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Biografia do Autor

Ahmed El Idrissi, faculté des sciences et techniques

Laboratory LMACS

Brahim El Boukari, Sultan Moulay Slimane University

Laboratory LMACS

Jalila El Ghordaf, Sultan Moulay Slimane University

Laboratory LMACS

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