$L^p-L^q$ estimates for solutions to the plate equation with mass term

TL;DR

This paper derives optimal $L^p-L^q$ estimates for solutions to the linear plate equation with mass term and explores the global existence of solutions to associated semilinear models, revealing critical exponents and their optimality.

Contribution

It provides the first comprehensive $L^p-L^q$ estimates for the linear plate equation with mass term and analyzes the global existence of solutions for semilinear models with sharp critical exponents.

Findings

Established optimal $L^p-L^q$ estimates for the linear problem.

Identified critical exponents for global existence in semilinear models.

Proved the optimality of the upper bound for the nonlinearity exponent.

Abstract

In this paper, we study the Cauchy problem for the linear plate equation with mass term and its applications to semilinear models. For the linear problem we obtain $L^p-L^q$ estimates for the solutions in the full range $1\leq p\leq q\leq \infty$, and we show that such estimates are optimal. In the sequel, we discuss the global in time existence of solutions to the associated semilinear problem with power nonlinearity $|u|^\alpha$. For low dimension space $n\leq 4$, and assuming $L^1$ regularity on the second datum, we were able to prove global existence for $\alpha> \max\{\alpha_c(n), \tilde\alpha_c(n)\}$ where $\alpha_c = 1+4/n$ and $\tilde \alpha_c = 2+2/n$. However, assuming initial data in $H^2(\mathbb{R}^n)\times L^2(\mathbb{R}^n)$, the presence of the mass term allows us to obtain global in time existence for all $1<\alpha \leq (n+4)/[n-4]_+$. We also show that the latter upper bound is optimal, since we prove that there exist data such that a non-existence result for local weak solutions holds when $\alpha > (n+4)/[n-4]_+$.