Global existence of small solutions to the initial value problem for nonlinear thermoelasticity.

*(English)*Zbl 0725.35065The authors consider the initial value problem in three-dimensional nonlinear thermoelasticity (the coupling of a hyperbolic system to a parabolic system) described by
\[
(*)\quad U_{tt}-D'SDU+\gamma \nabla \theta =f^ 1(\nabla U,\nabla^ 2U,\theta,\nabla \theta);\quad \theta_ t-\kappa \Delta \theta +\gamma \nabla 'U_ t=f^ 2(\nabla U,\nabla U_ t,\nabla^ 2U,\theta,\nabla \theta,\nabla^ 2\theta)\text{ in } {\mathbb{R}}^ 3\times (0,\infty)
\]
\[
U(x,0)=U^ 0(x),\quad \theta (x,0)=\theta^ 0(x)\text{ on } {\mathbb{R}}^ 3
\]
where \(\kappa\) \((>0)\), \(\gamma\) (\(\neq 0)\) are constants, \(U=(U_ 1,U_ 2,U_ 3)\), \(\theta\) are the displacement and the temperature respectively, and where D is a differential (6\(\times 3)\) matrix operator and S is a positive definite (6\(\times 6)\) marix (consisting of Lamé constants). Further \(f^ 1=(f^ 1_ 1,f^ 1_ 2,f^ 1_ 3)\) and \(f^ 2\) are the difference from the general state to the initially isotropic one expressed in terms of the nonlinearity. They assume that \(f^ 1,f^ 2\) are smooth and have the nonlinearity of order 2 satisfying \(f^ 1(\nabla U,\nabla^ 2U,0,0)=0\) and \(f^ 2(\nabla U,\nabla U_ t,\nabla^ 2U,0,0,0)=0.\)

Then they prove the existence and uniqueness of global smooth solutions to the problem (*), transforming it to an integral equation, using the \(L^ p\)-L\({}^ q\) estimate of the local solutions in some Sobolev spaces and applying the usual continuation arguments, in case that the initial data are smooth and small. The asymptotic behaviour of solution as \(t\to \infty\) is also described.

Then they prove the existence and uniqueness of global smooth solutions to the problem (*), transforming it to an integral equation, using the \(L^ p\)-L\({}^ q\) estimate of the local solutions in some Sobolev spaces and applying the usual continuation arguments, in case that the initial data are smooth and small. The asymptotic behaviour of solution as \(t\to \infty\) is also described.

Reviewer: T.Kakita (Tokyo)

##### MSC:

35M20 | PDE of composite type (MSC2000) |

35B40 | Asymptotic behavior of solutions to PDEs |

35K45 | Initial value problems for second-order parabolic systems |

35L55 | Higher-order hyperbolic systems |

74B99 | Elastic materials |

##### Keywords:

small initial data; nonlinear thermoelasticity; existence; uniqueness; global smooth solutions; \(L^ p\)-L\({}^ q\) estimate
PDF
BibTeX
XML
Cite

\textit{G. Ponce} and \textit{R. Racke}, J. Differ. Equations 87, No. 1, 70--83 (1990; Zbl 0725.35065)

Full Text:
DOI

##### References:

[1] | Christodoulou, D, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. pure appl. math., 39, 267-282, (1986) · Zbl 0612.35090 |

[2] | Dafermos, C.M; Hsiao, L, Development of singularities in solutions of the equations of nonlinear thermoelasticity, Quart. appl. math., 44, 463-474, (1986) · Zbl 0661.35009 |

[3] | Fujita, H, On the blowing up of solutions of the Cauchy problem for ut = δu + u1 + α, J. fac. sci. univ. Tokyo sect. I, 13, 109-124, (1966) · Zbl 0163.34002 |

[4] | \scS. Jiang, “Global Existence of Smooth Solutions in One-Dimensional Nonlinear Thermoelasticity,” to appear. · Zbl 0723.35044 |

[5] | John, F, Finite amplitude waves in a homogeneous isotropic elastic solid, Comm. pure appl. math., 30, 421-446, (1977) · Zbl 0404.73023 |

[6] | John, F, Blow-up for quasilinear wave equations in three dimensions, Comm. pure appl. math., 34, 29-51, (1981) · Zbl 0453.35060 |

[7] | John, F, Formation of singularities in elastic waves, (), 194-210 |

[8] | John, F, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. pure appl. math., 41, 615-666, (1988) · Zbl 0635.35066 |

[9] | Kawashima, S, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Thesis, (1983), Kyoto |

[10] | Klainerman, S, Global existence for nonlinear wave equations, Comm. pure appl. math., 33, 43-101, (1980) · Zbl 0405.35056 |

[11] | Klainerman, S, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. pure appl. math., 38, 321-332, (1985) · Zbl 0635.35059 |

[12] | Klainerman, S, The null condition and global existence to nonlinear wave equations, (), 293-325 · Zbl 0522.35063 |

[13] | Matsumura, A; Nishida, T, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, (), 337-341 · Zbl 0447.76053 |

[14] | Moser, J, A rapidly convergent iteration method and non-linear partial differential equations, I, Ann. scuola norm. sup. Pisa cl. sci. (4), 20, 265-315, (1966) · Zbl 0144.18202 |

[15] | Ponce, G, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear anal., 9, 399-418, (1985) · Zbl 0576.35023 |

[16] | Racke, R, Initial boundary value problems in thermoelasticity, (), 341-358 |

[17] | Racke, R, On the Cauchy problem in nonlinear 3-d-thermoelasticity, Math. Z., 203, 649-682, (1990) · Zbl 0701.73002 |

[18] | Racke, R, Blow-up in nonlinear three-dimensional thermoelasticity, Math. meth. appl. sci., 12, 267-273, (1990) · Zbl 0705.35081 |

[19] | Slemrod, M, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoelasticity, Arch. rational mech. anal., 76, 97-134, (1981) · Zbl 0481.73009 |

[20] | Weissler, F, Existence and non-existence of global solutions for a semilinear heat equation, Israel J. math., 38, 29-40, (1981) · Zbl 0476.35043 |

[21] | Zheng, S; Shen, W, Global solutions to the Cauchy problem of a class of quasilinear hyperbolic parabolic coupled systems, Sci. sinica ser. A, 30, 11, 1133-1149, (1987) · Zbl 0649.35013 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.