The author considers both locally and globally in time stability, decay estimates and blow-up of solutions to the following Cauchy problem for nonlinear dispersive wave equations arising in elasto-plastic flow: \[ u_{tt}+u_{xxxx}+ \lambda u=\sigma(u_x)_x,\;t>0,\;u(x,0)=u_0(x),\;u_t(0)=u_1(t), \] where \(\lambda >0\) and \(\sigma\) is of polynomial growth order. Then the main properties of this paper are described as follows: If \(\sigma(s)\in C^2(\mathbb{R})\), \(|\sigma(s)|\leq C( |s|^\alpha+ |s|)\), \(|\sigma'(s)|\leq C(|s|^{\alpha-1}+1)\) for any \(s\in\mathbb R\) with \(\alpha >1\), \(u_0\in H^2\) and \(u_1\in L^2\), then there exists \(T^0> 0\) such that the Cauchy problem has a local unique solution \(u\in C([0, T^0);H^2)\). In particular, if \(\sup_{t\in(0,T^0)}\{\|u(t)\|_{L^{2,2}}+ \|u_t(t)\|_{L^2}\}< \infty,\) then \(T^0=\infty\), where \(L^{s,p}=(I-\Delta)^{-s/2}L^p\). Moreover, the following stability of the solutions and decay property are proved: \[ \sup_{t\in (0,T)}\bigl\|u(t)-v(t)\bigr\|_{L^{2,2}}\leq C\bigl(\|u_0-v_0\|_{L^{2,2}}+\|u_1-v_1\|_{L^2} \bigr) \] for the corresponding initial data \(u_0,u_1\), and \(v_0,v_1\), and \(v\), respectively, and \[ \bigl\|u(t)\bigr\|_{L^{1,p}}\leq(1+t)^{-(p-2)/ (2p)} \] for \(2