Evolution of brain tumours backwards in time is studied using well-established brain tumour growth models being semilinear parabolic equations of reaction-diffusion type. To run the models backwards, the tumour cell density data at a fixed (final) time is used, rendering an inverse ill-posed problem. This problem is recast as the minimisation of a cost functional matching the data against the solution at a final time of a forward parabolic model having the initial cell density as a control function. Regularisation is incorporated via penalising terms involving Sobolev norms. Mathematical properties of the semilinear parabolic equations are shown in Sobolev-Bochner spaces including uniqueness of a solution to the inverse problem. Differentiability of the control-to-state map is established rendering a sensitivity problem. The derivative of the cost functional is calculated and the adjoint state is derived via the Lagrange formalism. A non-linear conjugate gradient method (NCG) is presented for the minimisation. Numerical realisation of the minimisation on the BraTS'20 dataset is included using a standard finite difference discretisation of the space and time derivatives, showing that tumour evolution backwards in time can be accomplished and that the initial tumour cell density can be reconstructed. Comparison is done with a non-linear Landweber method.