Equivalence Relation Between Initial Values and Solutions for Evolution $p$-Laplacian Equation in Unbounded Space

Abstract

In this paper, an equivalence relation between the $ω$-limit set of initial values and the $ω$-limit set of solutions is established for the Cauchy problem of evolution $p$-Laplacian equation in the unbounded space $\mathcal{Y}$_$σ$($ℝ$^$N$). To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions, we establish the propagation estimate and the growth estimate for the solutions. It will be demonstrated that the equivalence relation can be used to study the asymptotic behavior of solutions.