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Global solutions to the Cauchy problem of quasilinear hyperbolic parabolic coupled systems. (English) Zbl 0649.35013
We consider the Cauchy problem of a class of quasilinear coupled hyperbolic parabolic systems. First, we derive the $$L^ p$$ decay estimates of solutions for the corresponding linearized problem, and then on the basis of these $$L^ p$$ decay estimates, delicately deriving the uniform a priori estimates of solutions to nonlinear coupled systems, we establish the global existence, uniqueness of smooth solutions and the decay rates as $$t\to +\infty$$. We also apply these results to the Cauchy problem for the systems of thermoelasticity and radiation hydrodynamics.

##### MSC:
 35F25 Initial value problems for nonlinear first-order PDEs 35L60 First-order nonlinear hyperbolic equations 35K45 Initial value problems for second-order parabolic systems 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B65 Smoothness and regularity of solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B45 A priori estimates in context of PDEs