# PDE with integral

I am trying to solve PDE containing integral. Its form is showed below:

D[Pa[t],t] == Integrate[-k[x]*Pa[t],{x,0,1}]
k[x_] := 2*x


with initial conditions:

Pa[0] == 100


And NDSolve command gives fine result:

sol1 = NDSolve[{D[Pa[t], t] == Integrate[-k[x]*Pa[t], {x, 0, 1}], Pa[0] == 100}, Pa, {t, 0, 5}]


However, is it possible to get rid of integral and solve the equation as PDE:

D[Pa[t,x],t,x] == -k[x]*Pa[t,x]


What kind of Initial conditions are needed in this case?

The first part of your question is a differential equation

D[Pa[t],t] == Integrate[-k[x],{x,0,1}]*Pa[t]


which doesn't depend on x directly and can be easily solved analytical

C=Integrate[k[x],{x,0,1}]
Pa[t]== Pa0 Exp[-C t]


The last part of your question concerning the pde might be solved with the separation of variables

Pa[t,x]=X[x]T[t]


X'[x]T'[t] == -k[x]*X[x]T[t]

T'[t]/T[t] == -((k[x]*X[x])/X'[x]) ==  -const