# Convolution and differential equations

Consider the following system of differential equations: \begin{align} x_1'=f_1(x_1,x_2)\\ x_2'=f_2(x_1,x_2) \end{align} Assume that a solution $x(t)$ exists for $t\in (-T,T)$. Let $g:\mathbb{R}^2\to\mathbb{R}$ be a smooth function. Now we consider the following system of differential equations: \begin{align} x_1'=(g*f_1)(x_1,x_2)\\ x_2'=(g*f_2)(x_1,x_2) \end{align} What can we say about the solution of the above system? Can we say anything about the interval of validity? Or the proximity of the solution to the solution of the original equation?

In particular I am interested when $g$ is the Gaussian convolution kernal: \begin{align} g(x)=\frac{1}{4\pi\sigma}exp\Big({-\frac{||x||^2}{4\sigma}}\Big) \end{align}

• One may assume that all functions appearing above are smooth or $C^\infty$. – user42388 Oct 16 '15 at 22:31
• This is a possible approach: math.stackexchange.com/q/22567/8157 (under some assumptions, convolution against $g$ preserves or even decreases the Lipschitz constant of the vector valued function $(f_1, f_2)$). – Giuseppe Negro Oct 16 '15 at 22:48