TY  - JOUR
ID  - Lapa2023
LB  - Lapa2023
AU  - Lapa, Eugenio Cabanillas
TI  - Global solutions for a nonlinear Kirchhoff type equation with viscosity
ST  - Global solutions for a nonlinear Kirchhoff type equation with viscosity
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2023
VL  - 43
IS  - 5
SP  - 689
EP  - 701
DO  - https://doi.org/10.7494/OpMath.2023.43.5.689
UR  - https://doi.org/10.7494/OpMath.2023.43.5.689
AB  - 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.
KW  - global solutions
KW  - nonlinear Kirchhoff type problem
KW  - exponential decay
SN  - 1232-9274
ER  -