integro- (partial) differential equation

Question

I'm dealing with a certain kind of integro-differential equation. The equation reads as : \begin{equation} \frac{du(t,x)}{dt} = \int_{\Omega} u(t,y)K(x,y) dy \end{equation}

for some nice kernel function $K(x,y)$.

I want to do a kind of decay estimate of $\|u(t,\cdot)\|_{L^2(\Omega)}$. can anyone help me? Any kind of reference would be grateful.

Thanks.

Answer 1

I don't think the question is detailed enough to have one 'correct' answer. However, using limited analysis knowledge, here is one approach:

$$\sup_{\Omega}|\partial_t||u||^2| = 2\sup_{\Omega}\left |\int_{\Omega} uu_t K dy\right | = 2\sup_{\Omega}\left|\int_{\Omega} u(t,z) \int_{\Omega} u(t,y)K(z,y) dy dz\right| \leq 2\left(\sup_{\Omega}K\right)\left|\int_{\Omega} u(t,z) dz\right|^2.$$

I'm not sure how satisfactory this is. Of course, if $u$ and/or $\Omega$ are bounded, you can do more.