2012/03/29 by Marina Ghisi, Massimo Gobbino, Ghisi, Marina +1
Computer Science · Engineering · Mathematics · #35B25 #35B40 #35L72 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1203.6581
openalex publication_date 2012/03/29 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
We consider a family of Kirchhoff equations with a small parameter epsilon in\nfront of the second-order time-derivative, and a dissipation term with a\ncoefficient which tends to 0 as t -> +infinity.\n It is well-known that, when the decay of the coefficient is slow enough,\nsolutions behave as solutions of the corresponding parabolic equation, and in\nparticular they decay to 0 as t -> +infinity.\n In this paper we consider the nondegenerate and coercive case, and we prove\noptimal decay estimates for the hyperbolic problem, and optimal decay-error\nestimates for the difference between solutions of the hyperbolic and the\nparabolic problem. These estimates show a quite surprising fact: in the\ncoercive case the analogy between parabolic equations and dissipative\nhyperbolic equations is weaker than in the noncoercive case.\n This is actually a result for the corresponding linear equations with\ntime-dependent coefficients. The nonlinear term comes into play only in the\nlast step of the proof.\n