\(L^{2}\)-norm blow-up of solutions to a fourth order parabolic PDE involving the Hessian
The author studies the initial-boundary value problem for the equation \(u_t+\Delta^2u=\text{det}\left(D^2u\right)\) describing the epitaxial growth of a crystal. An answer is given to an open question in [\textit{C. Escudero} et al., J. Math. Pures Appl. 103, 924--957 (2015; Zbl 1406.35114)] concerning possible blowups of solutions. The main result is an ubounded growth of the \(L^2\) norm of a solution emanating from data satisfying a condition on the energy functional associated with the stationary problem. These are also generalizations of results in [\textit{G. Xu} and \textit{J. Zhou}, NoDEA, Nonlinear Differ. Equ. Appl. 24, No. 4, Paper No. 41, 12 p. (2017; Zbl 1375.35240)].
scientific article; zbMATH DE number 6933424
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