Problem with chemotaxis equations

I am trying to solve a a pair of coupled partial differential equations but it doesn't seem to work. I am getting the error 'Length of the derivative order is not u1[t,x]the same as the number of arguments',

diff1 = 1.;
diff2 = 1.;
L = 100;
{sol1, sol2} =NDSolveValue[{D[u1[t, x], t] == diff1*D[u1[t, x], {x, 2}] + D[(u1[t, x]*D[u2[t, x], x]), x],
D[u2[t, x], t] == diff2*D[u2[t, x], {x, 2}] - u2[x, t],
u1[0, x] == Exp[-x^2], u2[0, x] == PDF[NormalDistribution[20, 1], x], u1[t, -L] == 0, u1[t, L] == 0, u2[t, -L] == -1, u2[t, L] == 1}, {u1, u2}, {t, 0,
20}, {x, -L, L}, {y, -L, L}]

The equations are, $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial t^2}-\frac{\partial}{\partial x}\Big(v\frac{\partial u}{\partial x}\Big)$$ $$\frac{\partial v}{\partial x}=\frac{\partial^2 v}{\partial t^2}$$

• Two issues: 1) there's an extraneous {y, -L, L} indicating a 2D system, 2) - u2[x, t] should be - u2[t, x]. After fixing those, it seems to work. – Chris K Mar 13 at 17:25
• @ChrisK Do you not get a warning about boundary and initial conditions being inconsistent, after applying the fixes you mentioned? – MarcoB Mar 13 at 17:44
• @MarcoB Yes I get that warning, which is correct because the initial conditions don't match the boundary conditions. If OP's interested in the long-term behavior, then it will soon work itself out (try Plot[sol2[t, L], {t, 0, 20}, PlotRange -> All]). If OP's interested in the short term behavior, then the ICs need to be fixed. – Chris K Mar 13 at 17:56
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• BTW, I just noticed that the second equation doesn't match between the code and the text. – Chris K Mar 13 at 17:59

Here is your tuneup code,

PDE1 = D[u1[t, x], t] == diff1*D[u1[t, x], {x, 2}] + D[(u2[t, x]*D[u1[t, x], x]), x];

PDE2 = D[u2[t, x], t] == diff2*D[u2[t, x], {x, 2}] -u2[t, x];

diff1 = 1.; diff2 = 1.; L = 100;

{sol1, sol2} = NDSolveValue[{PDE1, PDE2, u1[0, x] == Exp[-x^2],
u2[0, x] == PDF[NormalDistribution[20, 1], x], u1[t, -L] == 0,
u1[t, L] == 0, u2[t, -L] == -1, u2[t, L] == 1}, {u1, u2}, {t, 0, 20}, {x, -L, L}]

GraphicsRow[{
Plot3D[sol1[t, x], {t, 0, 20}, {x, -L, L}, PlotRange -> All, AxesLabel -> {t, x, u1}],
Plot3D[sol2[t, x], {t, 0, 20}, {x, -L, L}, PlotRange -> All, AxesLabel -> {t, x, u2}]
}, ImageSize -> Large] 