zbMATH — the first resource for mathematics

Global solutions and applications to a class of quasilinear hyperbolic- parabolic coupled systems. (English) Zbl 0581.35056
The author investigates the I-BV problem for the following quasilinear hyperbolic-parabolic coupled system in two variables \[ \left( \begin{matrix} 1\\ 0\end{matrix} \begin{matrix} 0\\ a(u,v)\end{matrix} \right)\frac{\partial}{\partial t}\left( \begin{matrix} u_ 1\\ u_ 2\end{matrix} \right)+\left( \begin{matrix} u_ 2b_ 1\\ b_ 2\end{matrix} \begin{matrix} b_ 2\\ u_ 2b_ 3\end{matrix} \right)\frac{\partial}{\partial x}\left( \begin{matrix} u_ 1\\ u_ 2\end{matrix} \right)+\left( \begin{matrix} 0\\ \beta \partial v/\partial x\end{matrix} \right)=0 \] \[ C_ 0(u_ 1,v)\partial v/\partial t-\frac{\partial}{\partial x}[A(u_ 1,v)\partial v/\partial x]+u_ 2C(u,v)\partial v/\partial x+ \] \[ \beta (u_ 1,v)\partial u_ 2/\partial x+D(u,v,\partial u/\partial x,\partial v/\partial x)=0 \] t\(=0:\) \(u=u_ 0(x)\), \(v=v_ 0(x)\); \(x=0,1:\) \(u_ 2=0\), \(\partial v/\partial x=0.\)
Using the energy method and the continuation argument, he proves the global existence and uniqueness of smooth solutions and the exponential decay of solutions as \(t\to \infty\) for sufficiently small initial data. The results thus obtained are applied to the equations of radiation hydrodynamics and of viscoelasticity.
Reviewer: P.Bassanini

35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
35G30 Boundary value problems for nonlinear higher-order PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs