Advances in Differential Equations

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

Marina Ghisi

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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.

Article information

Source
Adv. Differential Equations Volume 17, Number 1/2 (2012), 1-36.

Dates
First available in Project Euclid: 17 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355703096

Mathematical Reviews number (MathSciNet)
MR2906728

Zentralblatt MATH identifier
1255.35027

Subjects
Primary: 35B25: Singular perturbations 35L70: Nonlinear second-order hyperbolic equations 35B40: Asymptotic behavior of solutions

Citation

Ghisi, Marina. Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations with weak dissipation. Adv. Differential Equations 17 (2012), no. 1/2, 1--36. https://projecteuclid.org/euclid.ade/1355703096.


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