Opuscula Mathematica
Opuscula Math. 43, no. 5 (2023), 689-701

Global solutions for a nonlinear Kirchhoff type equation with viscosity

Eugenio Cabanillas Lapa
Abstract. In this paper we consider the existence and asymptotic behavior of solutions of the following nonlinear Kirchhoff type problem \[u_{tt}- M\left(\,\displaystyle \int_{\Omega}|\nabla u|^{2}\, dx\right)\triangle u - \delta\triangle u_{t}= \mu|u|^{\rho-2}u\quad \text{in } \Omega \times ]0,\infty[,\] where \[M(s)=\begin{cases}a-bs &\text{for } s \in [0,\frac{a}{b}[,\\ 0, &\text{for } s \in [\frac{a}{b}, +\infty[.\end{cases}\] If the initial energy is appropriately small, we derive the global existence theorem and its exponential decay.
Keywords: global solutions, nonlinear Kirchhoff type problem, exponential decay.
Mathematics Subject Classification: 35L80, 35L70, 35B33, 35J75.
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About the authors ▼
Eugenio Cabanillas Lapa
https://orcid.org/0000-0002-8941-4394
Instituto de Investigación, FCM-UNMSM, Av. Venezuela S/N, Lima, Perú
About the article ▼
Communicated by Giovany Figueiredo.
Received: 2023-02-17.
Revised: 2023-05-07.
Accepted: 2023-05-10.
Published online: 2023-06-24.
Cite this article as ▼
Eugenio Cabanillas Lapa, Global solutions for a nonlinear Kirchhoff type equation with viscosity, Opuscula Math. 43, no. 5 (2023), 689-701, https://doi.org/10.7494/OpMath.2023.43.5.689.
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