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

2024/06/25 by Junior, Alexandre Arias, Halit Sevki Aslan, Aslan, Halit Sevki +4
Computer Science · Mathematics · #35A01 #35G25 #35L30 #35L75 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2406.17211

openalex publication_date 2024/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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≤ p≤ q≤ ∞, 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≤ 4, and assuming L¹ regularity on the second datum, we were able to prove global existence for α> max\α_c(n), α_c(n)\ where α_c = 1+4/n and α_c = 2+2/n. However, assuming initial data in H²(ℝⁿ)× L²(ℝⁿ), the presence of the mass term allows us to obtain global in time existence for all 1<α≤ (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]₊.

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