Existence and nonexistence of global solutions for the equation of dislocation of crystals. (English) Zbl 0926.35073
Authors’ abstract: We consider the Cauchy problem for the equation of dislocation of crystals
\[
u_{tt}- \Delta u+u= u^2+ u^3.
\]
The necessary and sufficient conditions of the existence of global solutions are obtained for
\[
E(0)= \textstyle{{1\over 2}}\Biggl(\| u_1\|^2_{L^2}+ \| u_0\|^2_{H^1}- \displaystyle{\int_{\mathbb{R}^n} \int^{u_0}_0} f(s)ds dx< d\Biggr)
\]
(\(f(s)= s^2+ s^3\), \(d\) is a given constant). We give the estimation of life span for the nonglobal solution. The existence and the nonexistence of solutions for \(E(0)= d\) are also considered.
Reviewer: C.Y.Chan (Lafayette)
MSC:
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 35L70 | Second-order nonlinear hyperbolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
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