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Local well-posedness and blow-up criteria of solutions for a rod equation. (English) Zbl 1125.35103
The present study deals with the equation $$u_t- u_{txx}+ 3uu_x= \gamma(2u_x u_{xx}+ uu_{xxx}),\quad x\in\Bbb T,\tag1$$ where $\gamma\in\Bbb R$ and $\Bbb T= \Bbb R$ or $S^1= \Bbb R/\Bbb Z$. The author obtains local well-posedness for (1) with initial datum $u_0\in H^s(\Bbb R)$, $s> 3/2$, and the lifespan of the corresponding solution is finite if and only if its first-order derivative blows up. He studies various sufficient condition (which are different for different $\gamma$) of the initial datum to guarantee the finite time blow-up for the periodic case. Blow-up criteria are established for the non-periodic case.

35Q72Other PDE from mechanics (MSC2000)
35B30Dependence of solutions of PDE on initial and boundary data, parameters
35B40Asymptotic behavior of solutions of PDE
49K10Free problems in several independent variables (optimality conditions)
74B20Nonlinear elasticity
74H20Existence of solutions for dynamical problems in solid mechanics
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
49K40Sensitivity, stability, well-posedness of optimal solutions
37L05General theory, nonlinear semigroups, evolution equations
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