COLÓQUIO MAP
“Lp − Lq decay estimates for Evolution Operators and Applications”
Prof. Marcelo Rempel Ebert (USP – Ribeirão Preto)
Dia: 26 de maio, sexta-feira, Auditório Antonio Gilioli, Sala 247/262, Bloco A, IMEUSP, das 16 às 17h, Café às 15h30, na sala 265 A (Chefia do MAP).
Transmissão online: http://www.ime.usp.br/comunicacao/eventos/cat.listevents/
Resumo da palestra:
In this talk we discuss the critical exponent for global small data solutions to the σ − evolution equation with structural effective damping utt + 2µ(−∆)δ ut + (−∆)σ u = |∂t k u|p , t > 0, x in Rn , (u, ut)(0, x) = (u0, u1)(x), with µ > 0, 2δ in (0, σ), k = 0, 1 and we denote by (−∆)b f = F −1(|ξ|2b F(f)), with b > 0, the fractional Laplacian operator. According to different cases, we fix data regularity and solution spaces which allow us to derive optimal decay estimates for the solution, in the sense that the obtained decay rate is the same of the linear case. In particular, we show which benefits come by assuming smallness of the initial data in spaces on L1 ∩L∞ basis, when the fractional power of the damping term is close to the limit case at which the damping cease to be effective. We prepare several high frequencies estimates for the corresponding linear problem, applying Mikhlin-Hörmander type multiplier theorems and Riesz potential properties. We couple these estimates with low frequencies estimates, which optimality is guaranteed by the diffusion phenomenon, and we deal with the nonlinear term by using a standard contraction argument in a suitable solution space.
References [1]
M. D’Abbicco, M.R. Ebert, A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations. Nonlinear Analysis, 149 (2017),1–40
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1) Partially supported by São Paulo Research Foundation (FAPESP), grant 2015/16038-2
