I am dealing with this (partial) integro-differential equation: \begin{align} \frac{\partial v}{\partial t}(t,x) &= c_{3}v(t,x) + \int_{x}^{1} K(x,\xi) v(t,\xi)d\xi,\\ v(0,x)&= v_{0}(x) \end{align}
where $t\in[0,\infty)$, $x\in[0,1]$, $K(x,\xi)$ is a bounded kernel, $c_{3}< 0$, and $v_{0}$ is a given initial condition.
I know that for $x=1$ the solution converges to zero because in this case the equation is reduced to \begin{align} \frac{\partial v}{\partial t}(t,1) &= c_{3}v(t,1),\\ v(0,1)&= v_{0}(1). \end{align}
However, I want to see if the solution is unstable for $x<1$.
Can anyone help me? Any kind of reference would be grateful.
Thanks.