May be separation of variables is siutable. U(x,t)=f(x)g(t) and then two independent ode for time and space will be derived.

@Farzan: That method is ineffective because of the nonlinear term. Independent ODEs are not obtained.

I don't understand the notation. What is u_xxxx? u_t? partial derivatives?

@Robert: Yes, subscripts denote partial derivatives.

@Robin: The equation arises in the study of the dynamics of solid surfaces with conserved mass. If the nonlinear term were absent, the equation would reduce to the Mullins equation describing the smoothing of a surface due to surface diffusion.

@Aref: Thanks for your suggestion. Unfortunately I could not find anything of help in the pdf you attached.

@Aref: The term means the second partial of u squared with respect to x. There are traveling wave solutions that can be obtained by solving a fourth order nonlinear ODE, but these will not match the initial condition. A better approach is to seek a similarity solution. This gives a nasty fourth order nonlinear ODE as well.

Note there is a conservation law: the integral of u(x,t) over x is constant, independent of time. This shows that a simple similarity solution cannot work. Any such solution would have to be of the form:

u = t^{-p} U(z), with similarity variable z = x/t^p .

The construction above, and only this construction, ensures that \int u d x is constant (independent of time): change variables from x to z and all the t's cancel out.

But the nonlinear part of the equation (u^2)_xx implies that p = 1/3 -- put n=1 in the example in article 51 Landau and Lifshitz, Fluid Mechanics. On the other hand, the linear term u_xxxx implies that p=1/4. So no joy here.

It is interesting that this argument shows that v_t = (v^3)_xx - v_xxxx might have a similarity solution with p =1/4. (This is the case n=2 in L&L). Any chance I can persuade you to modify the nonlinear term as above?

Any body know the CG Rossby(1937) eq.(28)27-? which happen to be the solution of the linearized version of the above problem, u_t+u_xxxx=0