Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping
Résumé
In this article, we investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation, namely
Under the assumption of sufficiently large and smooth initial data, we establish that the blow-up curve is continuously differentiable (C 1). A key step in our analysis involves the characterization of the blow-up profile of the solution. The proof relies on transforming the equation into a first-order system and adapting the techniques of Sasaki in [23, 24] which have elegantly extended the method of Caffarelli and Friedman [2] to nonlinear wave equations with time derivative nonlinearity, but without the scale-invariant term (µ = 0).
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