On a mixed problem for a linear coupled system with variable coefficients. (English) Zbl 0886.35043

Clark, H. R.; San Gil Jutuca, L. P.; Milla Miranda, M.

Electron. J. Differ. Equ. 1998, Paper 4, 20 p. (1998).

Summary: We prove existence, uniqueness, and exponential decay of solutions to the mixed problem \[ u''(x,t)-\mu(t)\Delta u(x,t)+\sum_{i=1}^n{\frac{\partial \theta}{\partial x_i}(x,t)}=0, \quad \theta'(x,t)-\Delta \theta(x,t) +\sum_{i=1}^n{\frac{\partial u'}{\partial x_i}(x,t)}=0, \] with a suitable boundary damping and a positive real-valued function \(\mu\).

MSC:

35F15	Boundary value problems for linear first-order PDEs
35N10	Overdetermined systems of PDEs with variable coefficients
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

hyperbolic-parabolic system; boundary damping; exponential stability; mixed boundary conditions; initial boundary value problem