Global analytic solution to the IVP for the BBM equation posed on $$\mathbb {R}$$ R and $$\mathbb {T}$$ T

Mikaela Baldasso () and Mahendra Panthee ()
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Mikaela Baldasso: University of Campinas (UNICAMP)
Mahendra Panthee: University of Campinas (UNICAMP)

Partial Differential Equations and Applications, 2025, vol. 6, issue 6, 1-26

Abstract: Abstract Initial value problem (IVP) for the BBM equation posed on the real line $$\mathbb {R}$$ R and on the torus $$\mathbb {T}$$ T with initial data that are analytic in a strip of width $$2\sigma _0$$ 2 σ 0 in the complex plane is considered. It is known that the local solution preserves the radius of spatial analyticity during the time of existence; however, this radius may decrease as time evolves. In this work, we study the evolution of the radius of spatial analyticity $$\sigma (t)$$ σ ( t ) of the solution over time. For the BBM equation posed on the real line $$\mathbb {R}$$ R , by introducing appropriate damping terms, we show that the radius of spatial analyticity possesses a fixed positive lower bound uniformly in time. For the BBM equation posed on the periodic domain $$\mathbb {T}$$ T , first we show that the evolution of the radius of spatial analyticity cannot decay faster than $$ct^{-\frac{2}{3}}$$ c t - 2 3 as the time t goes to infinity, improving the results obtained by Himonas and Petronilho (Proc Am Math Soc 148:2953–2967, 2020). Next, as in the real-line case, in this case too, the evolution of the radius of spatial analyticity is shown to be bounded below by a fixed constant by introducing damping terms.

Keywords: BBM equation; Initial value problem; Damping effect; Radius of spatial analyticity; Gevrey spaces; Almost conserved quantity; 35A20; 35B40; 35Q35 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-025-00353-y

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