  We investigate avascular tumour growth as a two-phase process consisting of
cells and liquid. Based on the one-dimensional continuum moving-boundary model
formulated by (Byrne, King, McElwain, Preziosi, Applied Mathematics Letters,
2003, 16, 567-573), we defined boundary conditions for the analogous model of
tumour growth in two dimensions. We investigate linear stability of one
dimensional time-dependent solution profiles in the moving-boundary formulation
of a limit case (with negligible nutrient consumption and cell drag). For this,
we obtain an asymptotic limit of the two-dimensional perturbations for large
time (in the case where the tumour is growing) by using the method of matched
asymptotic approximations. Having characterised an asymptotic limit of the
perturbations, we compare it to the time-dependent solution profile in order to
analytically obtain a condition for instability. Numerical simulations are
mentioned.
