| Abstract: |
| The partial differential equation with cubic nonlinearity proposed by Novikov [1] will be discussed in this talk:
\begin{equation*}
(1-\partial_x^2)u_t=(1+\partial_x)( u^2u_{xx}+ uu_x^2-2u^2u_x).
\end{equation*}
Initial value problems corresponding to the equation either on the line or on the circle will be presented. The recent results obtained both for local existence and finite time blow-up of solutions will be provided referring to the ideas recently proved in [2,3] for the quadratic nonlinear equation as a member of the integrable family of equations proposed by Novikov.
[1] V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., vol. 42, paper 342002, (2009).
[2] N. Duruk Mutlubas and I. L. Freire, Existence and uniqueness of periodic pseudospherical surfaces emanating from Cauchy problems, Proc. R. Soc. A., vol. 480, paper 20230670, (2024).
[3] N. Duruk Mutlubas and I. L. Freire, Global and blow-up solutions for a non-local integrable equation with applications to geometry, https://doi.org/10.48550/arXiv.2505.12232, (2025). |
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