Global existence and blowup of solutions for nonlinear Love equation

Abstract

In this paper, we are concerned with the nonlinear Love equation, which describes some sort of “propagation process” in physics. Firstly, we establish the local existence and uniqueness of the solution by using the contraction mapping principle and the regularity theory for the elliptic problem. Secondly, we construct a family of potential wells to obtain a threshold for the existence of global solutions and blowup in a finite time of the solution in both cases with subcritical and critical initial energy. Additionally, we provide the decay rate of the global solution and estimate the lifespan of a blow-up solution. Moreover, at the supercritical initial energy level, we provide a sufficient condition for initial data leading to blow-up results. This paper, along with the previous work of A. N. Carvalho, J. Cholewa, Local well-posedness, asymptotic behavior, and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time, Trans. Amer. Math. Soc. 361 (5) (2009) 2567-2586, provides us with a comprehensive and systematic study of the dynamic behavior of the solution to the nonlinear Love equation.