Xmax = 5;

Tmax = 5;

eq1 = D[u[x, t], t] == D[u[x, t], x, x] + (x - NIntegrate[xu[x, t], {x, 0, Xmax}]) u[x, t]

iv5 = {u[x, 0] == 2/(Sqrt[Pi]*Exp[x^2])};

bcs = {u[0, t] == 2/Sqrt[Pi], u[Xmax, t] == 0};

s10 = NDSolve[ {eq1, iv5, bcs} , {u[x, t] } , {x, 0, Xmax} , {t, 0, Tmax} ];

y = Table[u[x, t] /. s10, {x, 0, Xmax}, {t, 0, Tmax}]

Plot3D[u[x, t] /. s10, {x, 0, Xmax}, {t, 0, Tmax}, PlotRange -> All]