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Description
We examine the tumour growth model proposed in [1]. In the 2D case, the governing equations after some simplifications take the form
$\begin{array} {l}
\alpha_t + \left(\alpha u^1\right)_{x}+\left(\alpha u^2\right)_{y}=
S(\alpha), \
u^1_{x} + u^2_{y} = \nabla\cdot\left(D(\alpha)\nabla p\right), \\
\Big[(2+\lambda)\alpha u^1_{x} + \lambda \alpha u^2_{y}\Big]_x + \Big[\alpha u^1_{y} + \alpha u^2_{x}\Big]_y =
p_x + (\alpha\Sigma(\alpha))_x, \hskip1.2cm (1)\\
\Big[\alpha u^1_{y} + \alpha u^2_{x}\Big]_x +\Big[(2+\lambda)\alpha u^2_{y} +
\lambda \alpha u^1_{x}\Big]_y = p_y + (\alpha\Sigma(\alpha))_y,
\end{array}$
where $D, \ S$ and $\Sigma$ are some functions and their typical forms are listed in [1]. Assuming that the tumour boundary is prescribed by a curve $\Gamma(t,x,y)=0$, where $\Gamma$ is an unknown function, the boundary сonditions have the form
$\begin{array} {l}
u^1\Gamma_{x}+ u^2\Gamma_{y}= -\Gamma_t, \quad p=0, \\
\Big[(2+\lambda) u^1_{x} + \lambda u^2_{y}\Big]\Gamma_x + \Big[ u^1_{y} + u^2_{x}\Big]\Gamma_y = 0, \hskip3.5cm (2)\\
\Big[ u^1_{y} + u^2_{x}\Big]\Gamma_x +\Big[(2+\lambda) u^2_{y} + \lambda u^1_{x}\Big]\Gamma_y = 0.
\end{array}$
So, we have the nonlinear boundary value problem (1)-(2) with the unknown moving boundary $\Gamma(t,x,y)=0$.
Using the definition proposed in [2] and assuming $\Gamma$ to be a closed curve for any $t\geq 0$, we examined the Lie symmetry and constructed the exact solutions of the boundary value problem (1)-(2). For instance, the following statement takes place:
The system of nonlinear PDEs (1) with arbitrary functions $D, \ S$ and $\Sigma$ is invariant with respect to the infinite-dimensional Lie algebra generated by the Lie symmetry operators
$\begin{array} {l}
\partial_t, \quad F(t) \partial_p, \quad
G_g = g(t)\partial_x + \dot g\partial_{u^1}, \quad G_h = h(t)\partial_y + \dot h\partial_{u^2}\\
J_f = f(t) \Big[y\partial_x - x\partial_y +(u^2+\frac{\dot f}{f}y)\partial_{u^1} -
(u^1+\frac{\dot f}{f}x)\partial_{u^2}\Big].
\end{array}$
Here $ F, f, g, $ and $h$ are arbitrary smooth functions and the upper dot means differentiation with respect to time.
[1] H. Byrne, J.R. King, D.L.S. McElwain, L. Preziosi. A two-phase model of solid tumour growth. Appl. Math. Letters. 16 (2003) 567-573.
[2] R. Cherniha, S. Kovalenko. Lie symmetries and reductions of multi-dimensional boundary value problems of the Stefan type. J. Phys. A: Math. Theor. 44 (2011) 485202 (25 pp.)
