Article Contents

2024, Volume 13, Issue 4: 1101-1125. Doi: 10.3934/eect.2024018

This issue Previous Article Robustness of output feedback control for a wave-ODE cascaded system to the input delay mismatch Next Article Asymptotic behavior and life-span estimates for the damped inhomogeneous nonlinear Schrödinger equation

Global existence and asymptotic behavior for semilinear damped wave equations on measure spaces

1.
Center for Advanced Intelligence Project, RIKEN, Japan
2.
Faculty of Science and Technology, Keio University, Yokohama, 223-8522, Japan
3.
Advanced Institute for Materials Research, Tohoku University, Sendai, 980-8577, Japan
4.
Laboratory of Mathematics, Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

^*Corresponding author: Koichi Taniguchi
^*Corresponding author: Koichi Taniguchi

Received: June 2023

Revised: December 2023

Early access: March 2024

Published: August 2024

This work was supported by JSPS KAKENHI Grant Numbers JP16K17625, JP18H01132, JP19K14581, JP19J00206, JP20K14346, and JST CREST Grant Number JPMJCR1913

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Abstract

The purpose of this paper is to prove the small data global existence of solutions to the semilinear damped wave equation

$ \partial_t^2 u + Au + \partial_t u = |u|^{p-1}u $

on a measure space $ X $ with a self-adjoint operator $ A $ on $ L^2(X) $. Under a certain decay estimate on the corresponding heat semigroup, we establish the linear estimates which generalize the so-called Matsumura estimates. Our approach is based on a direct spectral analysis analogous to the Fourier analysis. The self-adjoint operators treated in this paper include some important examples such as the Laplace operators on Euclidean spaces, the Dirichlet Laplacian on an arbitrary open set, the Robin Laplacian on an exterior domain, the Schrödinger operator, the elliptic operator, the Laplacian on Sierpinski gasket, and the fractional Laplacian.

Keywords:

Mathematics Subject Classification: Primary: 35L15, 35L70, 35L90; Secondary: 35A01, 35B40.

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