Is the following eq. elliptic PDE:

u_{tt} + u_{xx} = 0

t here is the time;

i think this pde is hyperbolic pde.

with the + sign your equation is elliptic ( and so the idea that t is time is meaningless; i.e. the initial value problem is ill-posed).

u_{tt} - u_{xx} =0 on the other hand is hyperbolic and so the initial value problem is well-posed.

Tom.

Thanks Tom; this is what I want to understand, -ve & +ve. How the cases you stated hold and at what condition one can decide is depend on the application ?

I appreciate your reply

To understand this it's best to look at a book on pdes such as

Garabedian, P.R., Partial differential equations.

You want to look at sections on the Cauchy problem and Hadamard instability; this final topic demonstrates the ill-posedness of the initial value problem for your original equation.

Tom.

I've looked to the book but felt like am lost;

You should look for a more elementary introduction.

I was taught from the book "Partial differential equations" by W. E. Williams

you could also try

Applied Partial Differential Equations by Ockendon et al.

or any elementary introduction (just make sure it covers the Cauchy problem and Hadamard instability).

You may also want to look at volume 2 of Courant and Hilbert.