Asymptotic behavior of solutions of a k-Hessian evolution equation

2018/12/07 by Sánchez, Justino
#35B40 #35C06 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary: 47J35 #Secondary: 35K55

paper · doi:10.48550/arxiv.1812.03207

Abstract

We study the long-time behavior of solutions of the k-Hessian evolution equation u_t=S_k(D² u), posed on a bounded domain of the n-dimensional space with homogeneous boundary conditions. To this end, we construct a separable solution and we show that the long-time behavior of u is precisely described by this special solution. Further, we initiate the study of that dynamic phenomenon on the entire space, providing a new class of explicit and radially symmetric self-similar solutions that we call k-Barenblatt solutions. These solutions present some common properties as those of well-known Barenblatt solutions for the porous media equation and the p-Laplacian equation. It is known that self-similar solutions are important in describing the intermediate asymptotic behavior of general solutions.

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