Abstract
In this paper, we deal with entire solutions to the generalized parabolic k-Hessian equation of the form \(u_t = \mu (F_k(D^2 u)^{1/k})\) in \(\mathbb {R}^n \times (-\infty ,0]\). We prove that for \(1 \le k \le n\), any strictly convex-monotone solution \(u=u(x,t) \in C^{4,2}(\mathbb {R}^n \times (-\infty , 0])\) to \(u_t = \mu ( F_k(D^2 u)^{1/k})\) in \(\mathbb {R}^n \times (-\infty , 0]\) must be a linear function of t plus a quadratic polynomial of x, under some assumptions on \(\mu : (0,\infty ) \rightarrow \mathbb {R}\) and some growth conditions on u.
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Acknowledgments
This research was supported by the Grant-in-Aid for Scientific Research No. 25400169 from Japan Society for the Promotion of Science. The authors would like to thank the anonymous reviewers for careful reading and useful suggestions.
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Nakamori, S., Takimoto, K. (2016). Entire Solutions to Generalized Parabolic k-Hessian Equations. In: Gazzola, F., Ishige, K., Nitsch, C., Salani, P. (eds) Geometric Properties for Parabolic and Elliptic PDE's. GPPEPDEs 2015. Springer Proceedings in Mathematics & Statistics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-41538-3_11
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