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

Alexandre Arias Junior, Halit Sevki Aslan, Antonio Lagioia, Marcelo Rempel Ebert

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 $|u|^α$. For low dimension space $n\leq 4$, and assuming $L^1$ regularity on the second datum, we were able to prove global existence for $α> \max\{α_c(n), \tildeα_c(n)\}$ where $α_c = 1+4/n$ and $\tilde α_c = 2+2/n$. 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 $1<α\leq (n+4)/[n-4]_+$. 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 $α> (n+4)/[n-4]_+$.

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

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 estimates for the solutions in the full range , 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 . For low dimension space , and assuming regularity on the second datum, we were able to prove global existence for where and . However, assuming initial data in , the presence of the mass term allows us to obtain global in time existence for all . 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 .

Paper Structure

This paper contains 11 sections, 26 theorems, 219 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Estimates for the linear Cauchy problem
  4. Existence results to the semilinear Cauchy problem
  5. Proof of Theorem \ref{['Thm:1']}
  6. Estimates for high frequencies
  7. Estimates for small frequencies
  8. Proof of local and global in time existence results
  9. Non-local existence result
  10. Auxiliary results
  11. Proof of Lemma \ref{['Lemma_optimality']}

Key Result

Theorem \oldthetheorem

Let $u_1\in L^p(\mathbb{R}^n)$. Then, the solution $u$ to the Cauchy problem Eq:MainProblemLinear satisfies the following estimates: for all $1\leq p\leq q\leq \infty$ such that $d_{\text{pl}}(p,q) < 1$ and for all $1< p\leq 2 \leq q< \infty$ such that $d_{\text{pl}}(p,q)=1$, where $d_{\text{pl}}(p,q)$ is given by Eq:d-plate and provided that in the case $(p,q)=(1,3)$ or $(p,q)=(3/2,\infty)$ wi

Figures (1)

  • Figure 1: Description of the results in the $1/p-1/q$ plane with the optimal long-time decay estimates

Theorems & Definitions (51)

  • Theorem \oldthetheorem
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.1
  • Theorem \oldthetheorem
  • Remark 2.4
  • Theorem \oldthetheorem
  • Definition 2.1
  • Theorem \oldthetheorem
  • ...and 41 more