I wan to now whetehr this equqation is of hyperbolic type: $$(I-\partial_{xx})y_{tt}+y_{xxxx}=0$$ with boundary conditions $$y(t,0)=y(t,1)=y_x(t,0)=y_x(t,1)=0.$$ I would say that the answer is yes. By taking the pricipal symbols we get $$-\tau^2+\frac{\xi^4}{1+\xi^2}=0$$ which has the same behavior as $$-\tau^2+\xi^2=0$$ which is clearly the wave equations for high frequency. I'm write?. Does the boundary conditions affects the type of PDEs?. Thank you.

I want to now whether this equation is of hyperbolic type: $$(1-\partial_{xx})y_{tt}+y_{xxxx}=0$$ with boundary conditions $$y(t,0)=y(t,1)=y_x(t,0)=y_x(t,1)=0.$$ I would say that the answer is yes. By taking the principal symbols we get $$-\tau^2+\frac{\xi^4}{1+\xi^2}=0$$ which has the same behavior as $$-\tau^2+\xi^2=0$$ which is clearly the wave equations for high frequency. Am I right ? Does the boundary conditions affects the type of PDEs  ? Thank you.