The existence of the solution of $u_t+u_0u_x+u_{0x}u+u_{xxx}=0,u(x,0)=0$

Prove the existence of the solution of the Cauchy Problem: $$u_t+u_0u_x+u_{0x}u+u_{xxx}=0,u(x,0)=0$$ where$u_0\in C^{\infty}(R).u_0,u_{0x}\to 0,when |x|\to \infty$

AS Robert Bryant's comment,$u=0$ is a solution.I'm wondering if there is a another solution.I want to use semi-group theory, but I don't know how to do that. Can anyone help me?

edited May 26 2012 at 0:22

asked May 25 2012 at 13:30

89085731
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This does not seem to be the right level. – Igor Rivin May 25 2012 at 13:55

Maybe this is too simple, but the existence of a solution is obvious: Just take $u(x,t) = 0$. It doesn't matter what $u_0$ is. Maybe you want to prove uniqueness? – Robert Bryant May 25 2012 at 15:19

@Robert Bryant:Is the uniqueness true? – 89085731 May 25 2012 at 15:54

To understand where this equation comes from just check mathoverflow.net/questions/96405/…. – Jon May 26 2012 at 19:25

Am I interpreting your question correctly that $u_0$ has no connection to u? You are taking initial condition u(0)=0 and an arbitrary smooth, decaying function $u_0$? This seems to be linear in u and you have an operator $A(\phi) := \partial_x^3 + \phi \partial_x + \phi_x$. Now you just want to show that $A(\phi)$ generates a $C_0$ semigroup in some function space setting (which you need to determine... $L_p$? $C^{\alpha}$?...)? I have some ideas, but I want to see if I understand the question correctly first.. – Jeremy LeCrone Jun 21 2012 at 17:03