Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates
Equations of Kirchhoff type have been intensively studied in the recent years. Here, decay estimates, as \(t\to\infty\), are obtained for solutions to the abstract setting of the first order problem \[ Bu=u^{\prime}+m(| A^{\frac{1}{2}}u| ^2)Au=0,\quad\quad u(0)=u_o\tag{1} \] and to the singularly perturbed second order problem \[ \epsilon u^{\prime\prime}_{\epsilon}+Bu_{\epsilon}=0,\quad\quad u_{\epsilon}(0)=u_o, \quad\quad u^{\prime}_{\epsilon}(0)=u_1\tag{2} \] where \(A\) is a self-adjoint non negative operator on a Hilbert space \(H\), with dense domain \(D(A)\), \(m:[0,\infty)\to[0,\infty)\) is a \(C^{1}\)-function, and \((u_o,u_1)\in D(A)\times D(A^{\frac{1}{2}}).\) Conditions of nondegenerescence are imposed on \((u_0,u_1)\). Most of the constants which appear in decay estimates relative to problem (2) are independent on \(\epsilon\), if \(\epsilon\) is small enough. Clear tables summarize the comparison with decay estimates stated, previously, in the literature, when \(A\) is coercive or noncoercive.
scientific article; zbMATH DE number 5365159
scientific article