Existence of solutions for a k-Hessian equation and its connection with self-similar solutions

2023/05/30 by Sánchez, Justino
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2305.19364

Abstract

Let α,β be real parameters and let a>0. We study radially symmetric solutions of S_k(D²v)+αv+βξ⋅∇ v=0, vgt;0 in ℝⁿ, v(0)=a, where S_k(D²v) denotes the k-Hessian operator of v. For α≤(β(n-2k))/(k) and β>0, we prove the existence of a unique solution to this problem, without using the phase plane method. We also prove existence and properties of the solutions of the above equation for other ranges of the parameters α and β. These results are then applied to construct different types of explicit solutions, in self-similar forms, to a related evolution equation. In particular, for the heat equation, we have found a new family of self-similar solutions of type II which blows up in finite time. These solutions are represented as a power series, called the Kummer function.

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