Clark, M. R.; Lima, O. A. On a mixed problem for a coupled nonlinear system. (English) Zbl 0889.35068 Electron. J. Differ. Equ. 1997, Paper 6, 11 p. (1997). This paper studies an initial boundary value problem for the following coupled hyperbolic-parabolic system: \[ u_{tt}-M(\int_\Omega |\nabla u|^2dx)\Delta u +|u|^\rho u +\theta =f,\quad \theta_t -\Delta\theta +u_t =g,\qquad x\in\Omega,\;t >0 , \] where \(\Omega\subset \mathbb{R}^n\) is a domain. Under the conditions that \(M\in C^1[0,\infty)\) and \(M(s)\geq m_0>0\) for \(s\geq 0\), \(\rho\geq 0\) for \(n=1,2,3,4\) and \(0<\rho\leq 2/(n-2)\) for \(n\geq 5\), and \(f,g\in C^0(0,T;H^1_0(\Omega ))\), the authors prove the local existence and uniqueness of global weak solutions. The main ingredients in the proof are the use of the Galerkin method, energy estimates, and a compactness argument. Reviewer: S.Jiang (Beijing) Cited in 4 Documents MSC: 35M10 PDEs of mixed type 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) Keywords:initial boundary value problem; coupled hyperbolic-parabolic system; thermal effect; weak solutions; existence; uniqueness of global weak solutions; Galerkin method; energy estimates; compactness argument PDFBibTeX XMLCite \textit{M. R. Clark} and \textit{O. A. Lima}, Electron. J. Differ. Equ. 1997, Paper 6, 11 p. (1997; Zbl 0889.35068) Full Text: EuDML EMIS