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On a mixed problem for a coupled nonlinear system. (English) Zbl 0889.35068

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)

MSC:

35M10 PDEs of mixed type
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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