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The existence, uniqueness and regularities of the generalized global solutions and classical global solutions to the equation $$u_t=-A\left(t\right)u_{x^4}+B\left(t\right)u_{x^2}+g{\left(u\right)}_{x^2}+f{\left(u\right)}_x+h{\left(u_x\right)}_x+G\left(u\right)$$ with the initial boundary value conditions $$u(-\ell ,t)=u(\ell ,t)=0,u_{x^2}(-\ell ,t)=u_{x^2}(\ell ,t)=0,u(x,0)=\varphi \left(x\right),$$ or with the initial boundary value conditions $$u_x(-\ell ,t)=u_x(\ell ,t)=0,u_{x^3}(-\ell ,t)=u_{x^3}(\ell ,t)=0,u(x,0)=\varphi \left(x\right),$$ are proved. Moreover, the asymptotic behavior of these solutions is considered under some conditions.

The existence and uniqueness of classical global solution and blow up of non-global solution to the first boundary value problem and the second boundary value problem for the equation $$u_{tt}-\alpha u_{xx}-\beta u_{xxtt}=\varphi {\left(u_x\right)}_x$$ are proved. Finally, the results of the above problem are applied to the equation arising from nonlinear waves in elastic rods $$u_{tt}-\left[a_0+n{a}_1{\left(u_x\right)}^{n-1}\right]u_{xx}-a_2{u}_{xxtt}=0.$$