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A nonexistence result to a Cauchy problem in nonlinear one-dimensional thermoelasticity. (English) Zbl 0972.35010

This paper is concerned with the finite-time blow-up of solutions to the Cauchy problem for a one-dimensional semilinear thermoelastic model: \(u_{tt}=a u_{xx} +b\theta_x +d u_x -m u_t+ f(t,u)\), \(c\theta_t=K\theta_{xx} +b u_{xt} +p u_x +q\theta_x\), where \(a,b,c,K>0\) and \(d,m,p,q\geq 0\) are constants. The blow-up of solutions in the case \(d=m=p=q=0\) has been studied recently; this paper deals with the case: \(d,m,p,q\geq 0\). Under some conditions on \(f(t,u)\) and the initial data, the authors prove that the \(L^2\)-norm of \(u\) blows up in finite time. This work is distinguished from a previous work by relaxing the requirements on the initial data and allowing for a slightly more general and nonautonomous forcing term. The main ingredients in the proof are the careful energy estimates and the choice of an appropriate functional. And the crucial step is to apply a lemma given in [H. A. Levine, Trans. Am. Math. Soc. 192, 1-21 (1974; Zbl 0288.35003)] which is in fact a compact version of the concavity method. It should be pointed out that no physical background for the nonlinear term \(f(t,u)\) is given in the the paper.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
74F05 Thermal effects in solid mechanics
74F20 Mixture effects in solid mechanics
35Q72 Other PDE from mechanics (MSC2000)

Citations:

Zbl 0288.35003
Full Text: DOI

References:

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