## Osaka Journal of Mathematics

### Large time behavior of global solutions to nonlinear wave equations with frictional and viscoelastic damping terms

#### Abstract

In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms in ${\mathbb R}^{n}$. As is pointed out by [10], in this combination, the frictional damping term is dominant for the viscoelastic one for the global dynamics of the linear equation. In this note we observe that if the initial data is small, the frictional damping term is again dominant even in the nonlinear equation case. In other words, our main result is diffusion phenomena: the solution is approximated by the heat kernel with a suitable constant. Especially, the result obtained for the $n = 3$ case is essentially new. Our proof is based on several estimates for the corresponding linear equations.

#### Article information

Source
Osaka J. Math., Volume 56, Number 4 (2019), 807-830.

Dates
First available in Project Euclid: 21 October 2019

https://projecteuclid.org/euclid.ojm/1571623223

Mathematical Reviews number (MathSciNet)
MR4020638

Zentralblatt MATH identifier
07144186

#### Citation

Ikehata, Ryo; Takeda, Hiroshi. Large time behavior of global solutions to nonlinear wave equations with frictional and viscoelastic damping terms. Osaka J. Math. 56 (2019), no. 4, 807--830. https://projecteuclid.org/euclid.ojm/1571623223

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