Softcover ISBN:  9783985470204 
Product Code:  EMSMEM/1 
List Price:  $75.00 
AMS Member Price:  $60.00 
Softcover ISBN:  9783985470204 
Product Code:  EMSMEM/1 
List Price:  $75.00 
AMS Member Price:  $60.00 

Book DetailsMemoirs of the European Mathematical SocietyVolume: 1; 2022; 292 ppMSC: Primary 35
A kink is a stationary solution to a cubic onedimensional wave equation \((\partial_{t}{^2}\partial_{x}^{2})\phi=\phi\phi^{3}\) that has different limits when \(x\) goes to \(\infty\) and \(+\infty\), like \(H(x) = \mathrm{tanh}(x/\sqrt{2})\)). Asymptotic stability of this solution under small odd perturbation in the energy space has been studied in a recent work of Kowalczyk, Martel and Muñoz. They have been able to show that the perturbation may be written as the sum \(a(t)Y(x) + \psi (t,x)\), where \(Y\) is a function in Schwartz space, \(a(t)\) a function of time having some decay properties at infinity, and \(\psi(t, x)\) satisfies some local in space dispersive estimate. These results are likely to be optimal when the initial data belong to the energy space. On the other hand, for initial data that are smooth and have some decay at infinity, one may ask if precise dispersive time decay rates for the solution in the whole spacetime, and not just for \(x\) in a compact set, may be obtained. The goal of this work is to attack these questions.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

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A kink is a stationary solution to a cubic onedimensional wave equation \((\partial_{t}{^2}\partial_{x}^{2})\phi=\phi\phi^{3}\) that has different limits when \(x\) goes to \(\infty\) and \(+\infty\), like \(H(x) = \mathrm{tanh}(x/\sqrt{2})\)). Asymptotic stability of this solution under small odd perturbation in the energy space has been studied in a recent work of Kowalczyk, Martel and Muñoz. They have been able to show that the perturbation may be written as the sum \(a(t)Y(x) + \psi (t,x)\), where \(Y\) is a function in Schwartz space, \(a(t)\) a function of time having some decay properties at infinity, and \(\psi(t, x)\) satisfies some local in space dispersive estimate. These results are likely to be optimal when the initial data belong to the energy space. On the other hand, for initial data that are smooth and have some decay at infinity, one may ask if precise dispersive time decay rates for the solution in the whole spacetime, and not just for \(x\) in a compact set, may be obtained. The goal of this work is to attack these questions.
A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.