Local Hadamard well-posedness, global existence, finite time blow-up, and vacuum isolating phenomena for a generalized Lamé system

Abstract

In this paper, we examine a class of semilinear Lamé systems. Initially, we establish the local existence of solutions using monotone operator theory. Under appropriate assumptions, we then demonstrate that the weak solution is unique and continuously depends on the initial data. Subsequently, by constructing a family of potential wells, we analyze the global existence, asymptotic behavior, and blow-up of solutions for both subcritical and critical initial energy levels. This method is advantageous as our stabilization estimate does not generate lower-order terms, resulting in a more concise proof of the asymptotic behavior. Finally, we establish conditions under which solutions blow up in finite time for any positive initial energy.