Partial integro-differential equation I don't know if there is a method to solve this following integro - differential equation:
$$\partial_t{u(x,t)}\int_{0}^{x}u(\zeta,t)d\zeta=\partial_{xx}u(x,t)$$
Can someone give me some hint?
Thanks.
 A: Let $v(x,t)=\int_0^xu(\zeta,t)\,d\zeta$. Then $0=v_x$ and the original equation is equivalent to
$$
\tag{1}v\,v_{tx}=v_{xxx}.
$$
This is a non-linear equation for which I doubt that a general solution can be found. However, you can find spcial solutions by reducing it to an ODE.
Solutions in separated variables. Looking for solutions of the form $v(x,t)=X(x)\,T(t)$ leads to
$$
\frac{X'''}{X\,X'}=T'=a,\quad\text{$a$ constant.}
$$
The solution of the equation $T'=\lambda$ is $T(t)=a\,t+b$. The equation for $X$ is
$$
X'''=a\,X\,X'=\frac{a}{2}\,(X^2)'\implies X''=\frac{a}{2}\,X^2+b.
$$
Multiplying the last equation by $X'$ and integrating we get the first order equation
$$
(X')^2=3\,a\,X^3+2\,b\,X+c,
$$
whose solution is
$$
\int\frac{dx}{\sqrt{3\,a\,X^3+2\,b\,X+c}}\,dx=\pm\,x+d.
$$
Taking $a=0$ gives $u(x,t)=A\,x+B$, $A,B\in\mathbb{R}$. Taking $a\ne0$ and $b=c=0$ gives the family of solutions
$$
u(x,t)=\frac{a\,(a\,t+B)}{12\,(C\pm\,x)^{3}}.
$$
Self-similar solutions. For any $\beta\in\mathbb{R}$, the function $v(x,t)=t^{1+2\beta}w(x\,t^\beta)$ is a solution of (1) if $w(\xi)$ is a solution of the ODE
$$
(1+2\,\beta)\,w\,w'+\beta\,\xi\,w''=w'''.
$$
Travelling wave solutions. $v(x,t)=\phi(x+c\,t)$ leads to the ODE
$$
c\,\phi\,\phi''=\phi'''.
$$
I have not tried to solve the last two ODE's, and I do not think it will be easy, if at all feasible. But may be some properties of their solutions can be found.
