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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 \[ u_t- u_{txx}+ 3uu_x= \gamma(2u_x u_{xx}+ uu_{xxx}),\quad x\in\mathbb T,\tag{1} \] where \(\gamma\in\mathbb R\) and \(\mathbb T= \mathbb R\) or \(S^1= \mathbb R/\mathbb Z\). The author obtains local well-posedness for (1) with initial datum \(u_0\in H^s(\mathbb R)\), \(s> 3/2\), 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.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
49K10 Optimality conditions for free problems in two or more independent variables
74B20 Nonlinear elasticity
74H20 Existence of solutions of dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
49K40 Sensitivity, stability, well-posedness
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
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