Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations with weak dissipation

Abstract

We consider Kirchhoff equations with a small parameter ${\varepsilon}$ such as $$ \varepsilon {u_{{\varepsilon}}}''(t)+(1+t)^{-p}{u_{{\varepsilon}}}'(t)+ \m{{u_{{\varepsilon}}}(t)}A{u_{{\varepsilon}}}(t)=0. $$ We prove the existence of global solutions when $\varepsilon$ is small with respect to the size of initial data, for all $0\leq p \leq 1$ and $\gamma \geq 1$. Then we provide global-in-time error estimates on ${u_{{\varepsilon}}} - u$ where $u$ is the solution of the parabolic problem obtained setting formally $\varepsilon = 0$ in the previous equation.