I Couldn't understand what to do with Integration of partial differential kinds equation till now. I wonder if there is a way to find a solution to below equation:
$\partial_{t}(e^{|k|^{a} }p(k,t)) = -c(t) *b* k[e^{t(|k|^{a}-|k+2 \pi/l|^{a})} +e^{t(|k|^{a}-|k-2 \pi/l|^{a})}+ 0.5(e^{t(|k|^{a}-|k+4 \pi/l|^{a})} + e^{t(|k|^{a}-|k+4 \pi/l|^{a})})]$
$c(t)= \frac{1}{\int_{-\infty}^{+\infty} e^{-t|k|^{a}}dk}$ b is an imaginary constant. I will be appreciated if someone knows the answer to this equation and help me. I have been thinking about it for 2 days and after a lot of searches, but I still don't really know what to do.