# Lie Symmetries of Nonlinear Parabolic-Elliptic Systems and Their Application to a Tumour Growth Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Form-Preserving Transformations for the Class of Systems (3)

**Definition**

**1.**

**Theorem**

**1.**

**The**

**proof**

**of**

**Theorem**

**1**

**Consequence**

**1.**

## 3. Lie Symmetries of a Class of Parabolic-Elliptic Systems

**Theorem**

**2.**

**Remark**

**1.**

**Remark**

**2.**

**The**

**sketch**

**of**

**the**

**proof**

**of**

**Theorem**

**2**

**Consequence**

**2.**

## 4. Boundary Value Problems for a One-Dimensional Tumour Growth Model with Negligible Cell Viscosity

**Theorem**

**3.**

**Remark**

**3.**

## 5. Higher-Dimensional Tumour Growth Model without Cell Viscosity

**Theorem**

**4.**

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Surfaces representing the concentrations $\alpha $ (

**left**) and c (

**right**) of the form (49) for the parameter values $m=1$, ${\alpha}_{*}=0.5,\phantom{\rule{4pt}{0ex}}{c}_{\infty}=2,\phantom{\rule{4pt}{0ex}}{\omega}_{0}=1,\phantom{\rule{4pt}{0ex}}{q}_{0}=0.5$.

Case | RD System | Basic Operators of MAI |
---|---|---|

1. | ${U}_{t}={\left(D\left(U\right){U}_{x}\right)}_{x}+{e}^{V}f\left(U\right)$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-2{\partial}_{V}$ |

$0={V}_{xx}+{e}^{V}g\left(U\right)$ | ||

2. | ${U}_{t}={\left(D\left(U\right){U}_{x}\right)}_{x}+{V}^{\beta}f\left(U\right)$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2\beta t{\partial}_{t}+\beta x{\partial}_{x}-2V{\partial}_{V}$ |

$0={V}_{xx}+{V}^{\beta +1}g\left(U\right)$ | ||

3. | ${U}_{t}={\left({U}^{k}{U}_{x}\right)}_{x}+{U}^{\gamma +1}f\left({e}^{V}{U}^{\alpha}\right)$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},$ |

$0={V}_{xx}+{U}^{\gamma -k}g\left({e}^{V}{U}^{\alpha}\right)$ | $2\gamma t{\partial}_{t}+(\gamma -k)x{\partial}_{x}-2U{\partial}_{U}+2\alpha {\partial}_{V}$ | |

4. | ${U}_{t}={\left({U}^{k}{U}_{x}\right)}_{x}+{U}^{\gamma +1}f\left(V{U}^{\beta}\right)$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},$ |

$0={V}_{xx}+V{U}^{\gamma -k}g\left(V{U}^{\beta}\right)$ | $2\gamma t{\partial}_{t}+(\gamma -k)x{\partial}_{x}-2U{\partial}_{U}+2\beta V{\partial}_{V}$ | |

5. | ${U}_{t}={\left({U}^{k}{U}_{x}\right)}_{x}+{\alpha}_{1}{e}^{V}{U}^{\gamma +1}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-2{\partial}_{V},$ |

$0={V}_{xx}+{\alpha}_{2}{e}^{V}{U}^{\gamma -k}$ | $kt{\partial}_{t}-U{\partial}_{U}+(\gamma -k){\partial}_{V}$ | |

6. | ${U}_{t}={\left({U}^{k}{U}_{x}\right)}_{x}+{\alpha}_{1}{V}^{\beta}{U}^{\gamma +1}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2\beta t{\partial}_{t}+\beta x{\partial}_{x}-2V{\partial}_{V},$ |

$0={V}_{xx}+{\alpha}_{2}{V}^{\beta +1}{U}^{\gamma -k}$ | $k\beta t{\partial}_{t}-\beta U{\partial}_{U}+(\gamma -k)V{\partial}_{V}$ | |

7. | ${U}_{t}={\left({U}^{k}{U}_{x}\right)}_{x}+{\alpha}_{1}{U}^{k+1}+UV$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}-k\phi \left(t\right){\partial}_{t}+\dot{\phi}\left(t\right)U{\partial}_{U}+$ |

$0={V}_{xx}+{\alpha}_{2}{U}^{k}$ | $\left(k\dot{\phi}\left(t\right)V+(k+1)\ddot{\phi}\left(t\right)\right){\partial}_{V}$ | |

8. | $\phantom{\rule{4pt}{0ex}}{U}_{t}={\left({U}^{-4/3}{U}_{x}\right)}_{x}+Uf\left(U{V}^{3}\right)$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2x{\partial}_{x}-3U{\partial}_{U}+V{\partial}_{V},$ |

$0={V}_{xx}+Ug\left(U{V}^{3}\right)$ | ${x}^{2}{\partial}_{x}-3xU{\partial}_{U}+xV{\partial}_{V}$ | |

9. | $\phantom{\rule{4pt}{0ex}}{U}_{t}={\left({U}^{-4/3}{U}_{x}\right)}_{x}+{\alpha}_{1}{U}^{1+\gamma}{V}^{3\gamma}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2x{\partial}_{x}-3U{\partial}_{U}+V{\partial}_{V},$ |

$0={V}_{xx}+{\alpha}_{2}{U}^{4/3+\gamma}{V}^{3\gamma +1}$ | ${x}^{2}{\partial}_{x}-3xU{\partial}_{U}+xV{\partial}_{V},$ | |

$4\gamma t{\partial}_{t}+3\gamma U{\partial}_{U}-(4/3+\gamma )V{\partial}_{V}$ | ||

10. | ${U}_{t}={\left({e}^{U}{U}_{x}\right)}_{x}+{e}^{(\gamma +1)U}f\left(V+\alpha U\right)$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},$ |

$0={V}_{xx}+{e}^{\gamma U}g\left(V+\alpha U\right)$ | $2(\gamma +1)t{\partial}_{t}+\gamma x{\partial}_{x}-2{\partial}_{U}+2\alpha {\partial}_{V}$ | |

11. | ${U}_{t}={\left({e}^{U}{U}_{x}\right)}_{x}+{e}^{(\gamma +1)U}f\left(V{e}^{\beta U}\right)$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},$ |

$0={V}_{xx}+V{e}^{\gamma U}g\left(V{e}^{\beta U}\right)$ | $2(\gamma +1)t{\partial}_{t}+\gamma x{\partial}_{x}-2{\partial}_{U}+2\beta V{\partial}_{V}$ | |

12. | ${U}_{t}={\left({e}^{U}{U}_{x}\right)}_{x}+{\alpha}_{1}exp\left((\gamma +1)U+V\right)$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}-2{\partial}_{V},$ |

$0={V}_{xx}+{\alpha}_{2}exp\left(\gamma U+V\right)$ | $t{\partial}_{t}-{\partial}_{U}+\gamma {\partial}_{V}$ | |

13. | ${U}_{t}={\left({e}^{U}{U}_{x}\right)}_{x}+{\alpha}_{1}{e}^{(\gamma +1)U}{V}^{\beta}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2\beta t{\partial}_{t}+\beta x{\partial}_{x}-2V{\partial}_{V},$ |

$0={V}_{xx}+{\alpha}_{2}{e}^{\gamma U}{V}^{\beta +1}$ | $\beta t{\partial}_{t}-\beta {\partial}_{U}+\gamma V{\partial}_{V}$ | |

14. | ${U}_{t}={\left({e}^{U}{U}_{x}\right)}_{x}+{\alpha}_{1}{e}^{U}+V$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}\phi \left(t\right){\partial}_{t}-\dot{\phi}\left(t\right){\partial}_{U}-$ |

$0={V}_{xx}+{\alpha}_{2}{e}^{U}$ | $\left(\dot{\phi}\left(t\right)V+\ddot{\phi}\left(t\right)\right){\partial}_{V}$ |

**Table 2.**Lie symmetries of system (4) in the case ${F}_{V}{G}_{U}=0$ and ${\left({F}_{V}\right)}^{2}+{\left({G}_{U}\right)}^{2}\ne 0$.

Case | RD System | Basic Operators of MAI |
---|---|---|

1. | ${U}_{t}={\left(D\left(U\right){U}_{x}\right)}_{x}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}+x{\partial}_{x}+2V{\partial}_{V},$ |

$0={V}_{xx}+g\left(U\right)$ | $\phi \left(t\right)x{\partial}_{V},\phantom{\rule{4pt}{0ex}}\psi \left(t\right){\partial}_{V}$ | |

2. | ${U}_{t}={\left(D\left(U\right){U}_{x}\right)}_{x}+f\left(U\right)$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}\psi \left(t\right)V{\partial}_{V}$ |

$0={V}_{xx}+Vg\left(U\right)$ | ||

3. | ${U}_{t}={\left(D\left(U\right){U}_{x}\right)}_{x}+f\left(U\right)$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}h(t,x){\partial}_{V}$ |

$0={V}_{xx}+{\alpha}_{2}V+g\left(U\right)$ | ||

4. | ${U}_{t}={\left({U}^{k}{U}_{x}\right)}_{x}+{\alpha}_{0}{U}^{\beta +1}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2\beta t{\partial}_{t}+(\beta -k)x{\partial}_{x}-2U{\partial}_{U},$ |

$0={V}_{xx}+\alpha V{U}^{\beta -k},\phantom{\rule{4pt}{0ex}}\alpha \ne 0$ | $\psi \left(t\right)V{\partial}_{V}$ | |

5. | ${U}_{t}={\left({U}^{k}{U}_{x}\right)}_{x}+UV$ | ${\partial}_{x},\phantom{\rule{4pt}{0ex}}kx{\partial}_{x}+2U{\partial}_{U},\phantom{\rule{4pt}{0ex}}-k\phi \left(t\right){\partial}_{t}+$ |

$0={V}_{xx}$ | $\dot{\phi}\left(t\right)U{\partial}_{U}+(k\dot{\phi}\left(t\right)V+(k+1)\ddot{\phi}\left(t\right)){\partial}_{V}$ | |

6. | ${U}_{t}={\left({U}^{k}{U}_{x}\right)}_{x}+{\alpha}_{1}{U}^{k+1}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}-kt{\partial}_{t}+U{\partial}_{U}+{h}^{0}\left(x\right){\partial}_{V},\phantom{\rule{4pt}{0ex}}h(t,x){\partial}_{V},$ |

$0={V}_{xx}+lnU+{\alpha}_{2}V$ | ||

7. | ${U}_{t}={\left({U}^{k}{U}_{x}\right)}_{x}+{\alpha}_{1}{U}^{k+1}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}-kt{\partial}_{t}+U{\partial}_{U}+\beta V{\partial}_{V},$ |

$0={V}_{xx}+{U}^{\beta}+{\alpha}_{2}V$ | $h(t,x){\partial}_{V}$ | |

8. | ${U}_{t}={\left({U}^{k}{U}_{x}\right)}_{x}+\alpha {U}^{\gamma +1},\phantom{\rule{4pt}{0ex}}\alpha \gamma \ne 0$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2\gamma t{\partial}_{t}+(\gamma -k)x{\partial}_{x}-2U{\partial}_{U}+$ |

$0={V}_{xx}+lnU$ | $\left(2(\gamma -k)V+{x}^{2}\right){\partial}_{V},\phantom{\rule{4pt}{0ex}}\phi \left(t\right)x{\partial}_{V},\phantom{\rule{4pt}{0ex}}\psi \left(t\right){\partial}_{V}$ | |

9. | ${U}_{t}={\left({U}^{k}{U}_{x}\right)}_{x}+\alpha {U}^{\gamma +1},\phantom{\rule{4pt}{0ex}}\alpha \gamma \ne 0$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2\gamma t{\partial}_{t}+(\gamma -k)x{\partial}_{x}-2U{\partial}_{U}+$ |

$0={V}_{xx}+{U}^{\beta}$ | $2(\gamma -\beta -k)V{\partial}_{V},\phantom{\rule{4pt}{0ex}}\phi \left(t\right)x{\partial}_{V},\phantom{\rule{4pt}{0ex}}\psi \left(t\right){\partial}_{V}$ | |

10. | $\phantom{\rule{4pt}{0ex}}{U}_{t}={\left({U}^{k}{U}_{x}\right)}_{x}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}-2kt{\partial}_{t}+2U{\partial}_{U}-{x}^{2}{\partial}_{V},$ |

$0={V}_{xx}+lnU$ | $2t{\partial}_{t}+x{\partial}_{x}+2V{\partial}_{V},\phantom{\rule{4pt}{0ex}}\phi \left(t\right)x{\partial}_{V},\phantom{\rule{4pt}{0ex}}\psi \left(t\right){\partial}_{V}$ | |

11. | $\phantom{\rule{4pt}{0ex}}{U}_{t}={\left({U}^{k}{U}_{x}\right)}_{x}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}-kt{\partial}_{t}+U{\partial}_{U}+\beta V{\partial}_{V},$ |

$0={V}_{xx}+{U}^{\beta}$ | $2t{\partial}_{t}+x{\partial}_{x}+2V{\partial}_{V},\phantom{\rule{4pt}{0ex}}\phi \left(t\right)x{\partial}_{V},\phantom{\rule{4pt}{0ex}}\psi \left(t\right){\partial}_{V}$ | |

12. | $\phantom{\rule{4pt}{0ex}}{U}_{t}={\left({U}^{-4/3}{U}_{x}\right)}_{x}+\alpha U,\phantom{\rule{4pt}{0ex}}\alpha \ne 0$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2x{\partial}_{x}-3U{\partial}_{U}+V{\partial}_{V},\phantom{\rule{4pt}{0ex}}\phi \left(t\right)x{\partial}_{V}$ |

$0={V}_{xx}+U$ | ${x}^{2}{\partial}_{x}-3xU{\partial}_{U}+xV{\partial}_{V},\phantom{\rule{4pt}{0ex}}\psi \left(t\right){\partial}_{V}$ | |

13. | $\phantom{\rule{4pt}{0ex}}{U}_{t}={\left({U}^{-4/3}{U}_{x}\right)}_{x}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2x{\partial}_{x}-3U{\partial}_{U}+V{\partial}_{V},$ |

$0={V}_{xx}+U$ | ${x}^{2}{\partial}_{x}-3xU{\partial}_{U}+xV{\partial}_{V},$ | |

$4t{\partial}_{t}+3\left(U{\partial}_{U}+V{\partial}_{V}\right),\phantom{\rule{4pt}{0ex}}\phi \left(t\right)x{\partial}_{V},\phantom{\rule{4pt}{0ex}}\psi \left(t\right){\partial}_{V}$ | ||

14. | ${U}_{t}={\left({e}^{U}{U}_{x}\right)}_{x}+{\alpha}_{0}{e}^{(\beta +1)U}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2(\beta +1)t{\partial}_{t}+\beta x{\partial}_{x}-2{\partial}_{U},$ |

$0={V}_{xx}+\alpha V{e}^{\beta U},\phantom{\rule{4pt}{0ex}}\alpha \ne 0$ | $\psi \left(t\right)V{\partial}_{V}$ | |

15. | ${U}_{t}={\left({e}^{U}{U}_{x}\right)}_{x}+V$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}x{\partial}_{x}+2{\partial}_{U},$ |

$0={V}_{xx}$ | $\phi \left(t\right){\partial}_{t}-\dot{\phi}\left(t\right){\partial}_{U}-\left(\dot{\phi}\left(t\right)V+\ddot{\phi}\left(t\right)\right){\partial}_{V}$ | |

16. | ${U}_{t}={\left({e}^{U}{U}_{x}\right)}_{x}+{\alpha}_{1}{e}^{U}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}-t{\partial}_{t}+{\partial}_{U}+{h}^{0}\left(x\right){\partial}_{V},\phantom{\rule{4pt}{0ex}}h(t,x){\partial}_{V}$ |

$0={V}_{xx}+U+{\alpha}_{2}V$ | ||

17. | ${U}_{t}={\left({e}^{U}{U}_{x}\right)}_{x}+{\alpha}_{1}{e}^{U}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}-t{\partial}_{t}+{\partial}_{U}+\beta V{\partial}_{V},\phantom{\rule{4pt}{0ex}}h(t,x){\partial}_{V}$ |

$0={V}_{xx}+{e}^{\beta U}+{\alpha}_{2}V$ | ||

18. | ${U}_{t}={\left({e}^{U}{U}_{x}\right)}_{x}+\alpha {e}^{\gamma U},\phantom{\rule{4pt}{0ex}}\alpha \gamma \ne 0$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2\gamma t{\partial}_{t}+(\gamma -1)x{\partial}_{x}-2{\partial}_{U}+$ |

$0={V}_{xx}+U$ | $\left(2(\gamma -1)V+{x}^{2}\right){\partial}_{V},\phantom{\rule{4pt}{0ex}}\phi \left(t\right)x{\partial}_{V},\phantom{\rule{4pt}{0ex}}\psi \left(t\right){\partial}_{V}$ | |

19. | ${U}_{t}={\left({e}^{U}{U}_{x}\right)}_{x}+\alpha {e}^{\gamma U},\phantom{\rule{4pt}{0ex}}\alpha \gamma \ne 0$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2\gamma t{\partial}_{t}+(\gamma -1)x{\partial}_{x}-2{\partial}_{U}+$ |

$0={V}_{xx}+{e}^{\beta U}$ | $2(\gamma -\beta -1)V{\partial}_{V},\phantom{\rule{4pt}{0ex}}\phi \left(t\right)x{\partial}_{V},\phantom{\rule{4pt}{0ex}}\psi \left(t\right){\partial}_{V}$ | |

20. | $\phantom{\rule{4pt}{0ex}}{U}_{t}={\left({e}^{U}{U}_{x}\right)}_{x}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}2t{\partial}_{t}-2{\partial}_{U}+{x}^{2}{\partial}_{V},$ |

$0={V}_{xx}+U$ | $2t{\partial}_{t}+x{\partial}_{x}+2V{\partial}_{V},\phantom{\rule{4pt}{0ex}}\phi \left(t\right)x{\partial}_{V},\phantom{\rule{4pt}{0ex}}\psi \left(t\right){\partial}_{V}$ | |

21. | $\phantom{\rule{4pt}{0ex}}{U}_{t}={\left({e}^{U}{U}_{x}\right)}_{x}$ | ${\partial}_{t},\phantom{\rule{4pt}{0ex}}{\partial}_{x},\phantom{\rule{4pt}{0ex}}t{\partial}_{t}-{\partial}_{U}-\beta V{\partial}_{V},$ |

$0={V}_{xx}+{e}^{\beta U}$ |

**Table 3.**RD systems that are reduced to those in Table 1 by the form-preserving transformations.

RD System | Transformation of Variables | Case of Table 1 | |
---|---|---|---|

1. | ${u}_{\tau}={\left({u}^{k}{u}_{y}\right)}_{y}+{u}^{k+1}f\left({e}^{v}{u}^{\alpha}\right)+\lambda u$ | $U={e}^{-\lambda \tau}u,\phantom{\rule{4pt}{0ex}}V=v+\alpha \lambda \tau ,$ | 3 |

$0={v}_{yy}+g\left({e}^{v}{u}^{\alpha}\right)$ | $t=\frac{1}{k\lambda}\phantom{\rule{0.166667em}{0ex}}{e}^{k\lambda \tau}$ | with $\gamma =k$ | |

2. | ${u}_{\tau}={\left({u}^{k}{u}_{y}\right)}_{y}+{u}^{k+1}f\left(v{u}^{\beta}\right)+\lambda u$ | $U={e}^{-\lambda \tau}u,\phantom{\rule{4pt}{0ex}}V={e}^{\beta \lambda \tau}v,$ | 4 |

$0={v}_{yy}+vg\left(v{u}^{\alpha}\right)$ | $t=\frac{1}{k\lambda}\phantom{\rule{0.166667em}{0ex}}{e}^{k\lambda \tau}$ | with $\gamma =k$ | |

3. | ${u}_{\tau}={\left({u}^{-4/3}{u}_{y}\right)}_{y}+uf\left(u{v}^{3}\right)-3{u}^{-1/3}$ | $U=u{cos}^{3}y,\phantom{\rule{4pt}{0ex}}V=v{cos}^{-1}y,$ | 8 |

$0={v}_{yy}+ug\left(u{v}^{3}\right)+v$ | $x=tany$ | ||

4. | ${u}_{\tau}={\left({u}^{-4/3}{u}_{y}\right)}_{y}+uf\left(u{v}^{3}\right)+3{u}^{-1/3}$ | $U={e}^{3y}u,\phantom{\rule{4pt}{0ex}}V={e}^{-y}v,$ | 8 |

$0={v}_{yy}+ug\left(u{v}^{3}\right)-v$ | $x=\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{e}^{-2y}$ | ||

5. | ${u}_{\tau}={\left({u}^{-4/3}{u}_{y}\right)}_{y}+{\alpha}_{1}{u}^{1+\gamma}{v}^{3\gamma}-3{u}^{-1/3}$ | $U=u{cos}^{3}y,\phantom{\rule{4pt}{0ex}}V=v{cos}^{-1}y,$ | 9 |

$0={v}_{yy}+{\alpha}_{2}{u}^{4/3+\gamma}{v}^{3\gamma +1}+v$ | $x=tany$ | ||

6. | ${u}_{\tau}={\left({u}^{-4/3}{u}_{y}\right)}_{y}+{\alpha}_{1}{u}^{1+\gamma}{v}^{3\gamma}+3{u}^{-1/3}$ | $U={e}^{3y}u,\phantom{\rule{4pt}{0ex}}V={e}^{-y}v,$ | 9 |

$0={v}_{yy}+{\alpha}_{2}{u}^{4/3+\gamma}{v}^{3\gamma +1}-v$ | $x=\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{e}^{-2y}$ | ||

7. | ${u}_{\tau}={\left({e}^{u}{u}_{y}\right)}_{y}+{e}^{u}f\left(v+\alpha u\right)+\lambda $ | $U=u-\lambda \tau ,\phantom{\rule{4pt}{0ex}}V=v+\alpha \lambda \tau ,$ | 10 |

$0={v}_{yy}+g\left(v+\alpha u\right)$ | $t=\frac{1}{\lambda}\phantom{\rule{0.166667em}{0ex}}{e}^{\lambda \tau}$ | with $\gamma =0$ | |

8. | ${u}_{\tau}={\left({e}^{u}{u}_{y}\right)}_{y}+{e}^{u}f\left(v{e}^{\beta u}\right)+\lambda $ | $U=u-\lambda \tau ,\phantom{\rule{4pt}{0ex}}V={e}^{\beta \lambda \tau}v,$ | 11 |

$0={v}_{yy}+vg\left(v{e}^{\beta u}\right)$ | $t=\frac{1}{\lambda}\phantom{\rule{0.166667em}{0ex}}{e}^{\lambda \tau}$ | with $\gamma =0$ |

**Table 4.**RD systems that are reduced to those in Table 2 by the form-preserving transformations.

RD System | Transformation of Variables | Case of Table 2 | |
---|---|---|---|

1. | ${u}_{\tau}={\left({u}^{k}{u}_{y}\right)}_{y}+{\alpha}_{1}{u}^{k+1}+\lambda u$ | $U={e}^{-\lambda \tau}u,\phantom{\rule{4pt}{0ex}}t=\frac{1}{k\lambda}\phantom{\rule{0.166667em}{0ex}}{e}^{k\lambda \tau},$ | 6 |

$0={v}_{yy}+lnu+{\alpha}_{2}v$ | $V=\left\{\begin{array}{cc}v+\frac{\lambda}{{\alpha}_{2}}\phantom{\rule{0.166667em}{0ex}}\tau ,\hfill & if\phantom{\rule{4pt}{0ex}}{\alpha}_{2}\ne 0,\hfill \\ v+\frac{\lambda}{2}\phantom{\rule{0.166667em}{0ex}}\tau {y}^{2},\hfill & if\phantom{\rule{4pt}{0ex}}{\alpha}_{2}=0\hfill \end{array}\right.$ | ||

2. | ${u}_{\tau}={\left({u}^{k}{u}_{y}\right)}_{y}+{\alpha}_{1}{u}^{k+1}+\lambda u$ | $U={e}^{-\lambda \tau}u,\phantom{\rule{4pt}{0ex}}t=\frac{1}{k\lambda}\phantom{\rule{0.166667em}{0ex}}{e}^{k\lambda \tau},$ | 7 |

$0={v}_{yy}+{u}^{\beta}+{\alpha}_{2}v+\gamma $ | $V=\left\{\begin{array}{cc}\left(v+\frac{\gamma}{{\alpha}_{2}}\right){e}^{-\beta \lambda \tau},\hfill & if\phantom{\rule{4pt}{0ex}}{\alpha}_{2}\ne 0,\hfill \\ \left(v+\frac{\gamma}{2}\phantom{\rule{0.166667em}{0ex}}{y}^{2}\right){e}^{-\beta \lambda \tau},\hfill & if\phantom{\rule{4pt}{0ex}}{\alpha}_{2}=0\hfill \end{array}\right.$ | ||

3. | ${u}_{\tau}={\left({u}^{k}{u}_{y}\right)}_{y}+\alpha {u}^{\gamma +1}$ | $V=v+\frac{\lambda}{2}\phantom{\rule{0.166667em}{0ex}}{y}^{2}$ | 9, if $\alpha \ne 0$ |

$0={v}_{yy}+{u}^{\beta}+\lambda $ | 11, if $\alpha =0$ | ||

4. | ${u}_{\tau}={\left({u}^{k}{u}_{y}\right)}_{y}+\lambda u$ | $U={e}^{-\lambda \tau}u,\phantom{\rule{4pt}{0ex}}t=\frac{1}{k\lambda}\phantom{\rule{0.166667em}{0ex}}{e}^{k\lambda \tau},$ | 10 |

$0={v}_{yy}+lnu$ | $V=v+\frac{\lambda}{2}\phantom{\rule{0.166667em}{0ex}}\tau {y}^{2}$ | ||

5. | ${u}_{\tau}={\left({u}^{k}{u}_{y}\right)}_{y}+\lambda u$ | 11 | |

$0={v}_{yy}+{u}^{\beta}+\gamma $ | $V=\left(v+\frac{\gamma}{2}\phantom{\rule{0.166667em}{0ex}}{y}^{2}\right){e}^{-\beta \lambda \tau}$ | ||

6. | ${u}_{\tau}={\left({u}^{-4/3}{u}_{y}\right)}_{y}+\alpha u-3{u}^{-1/3}$ | $U=u{cos}^{3}y,\phantom{\rule{4pt}{0ex}}V=v{cos}^{-1}y,$ | 12 |

$0={v}_{yy}+u+v$ | $x=tany$ | ||

7. | ${u}_{\tau}={\left({u}^{-4/3}{u}_{y}\right)}_{y}+\alpha u+3{u}^{-1/3}$ | $U={e}^{3y}u,\phantom{\rule{4pt}{0ex}}V={e}^{-y}v,$ | 12 |

$0={v}_{yy}+u-v$ | $x=\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{e}^{-2y}$ | ||

8. | ${u}_{\tau}={\left({u}^{-4/3}{u}_{y}\right)}_{y}-3{u}^{-1/3}$ | $U=u{cos}^{3}y,\phantom{\rule{4pt}{0ex}}V=v{cos}^{-1}y,$ | 13 |

$0={v}_{yy}+u+v$ | $x=tany$ | ||

9. | ${u}_{\tau}={\left({u}^{-4/3}{u}_{y}\right)}_{y}+3{u}^{-1/3}$ | $U={e}^{3y}u,\phantom{\rule{4pt}{0ex}}V={e}^{-y}v,$ | 13 |

$0={v}_{yy}+u-v$ | $x=\frac{1}{2}\phantom{\rule{0.166667em}{0ex}}{e}^{-2y}$ | ||

10. | ${u}_{\tau}={\left({e}^{u}{u}_{y}\right)}_{y}+{\alpha}_{1}{e}^{u}+\lambda $ | $U=u-\lambda \tau ,\phantom{\rule{4pt}{0ex}}t=\frac{1}{\lambda}\phantom{\rule{0.166667em}{0ex}}{e}^{\lambda \tau},$ | 16 |

$0={v}_{yy}+u+{\alpha}_{2}v$ | $V=\left\{\begin{array}{cc}v+\frac{\lambda}{{\alpha}_{2}}\phantom{\rule{0.166667em}{0ex}}\tau ,\hfill & if\phantom{\rule{4pt}{0ex}}{\alpha}_{2}\ne 0,\hfill \\ v+\frac{\lambda}{2}\phantom{\rule{0.166667em}{0ex}}\tau {y}^{2},\hfill & if\phantom{\rule{4pt}{0ex}}{\alpha}_{2}=0\hfill \end{array}\right.$ | ||

11. | ${u}_{\tau}={\left({e}^{u}{u}_{y}\right)}_{y}+{\alpha}_{1}{e}^{u}+\lambda $ | $U=u-\lambda \tau ,\phantom{\rule{4pt}{0ex}}t=\frac{1}{\lambda}\phantom{\rule{0.166667em}{0ex}}{e}^{\lambda \tau},$ | 17 |

$0={v}_{yy}+{e}^{\beta u}+{\alpha}_{2}v+\gamma $ | $V=\left\{\begin{array}{cc}\left(v+\frac{\gamma}{{\alpha}_{2}}\right){e}^{-\beta \lambda \tau},\hfill & if\phantom{\rule{4pt}{0ex}}{\alpha}_{2}\ne 0,\hfill \\ \left(v+\frac{\gamma}{2}\phantom{\rule{0.166667em}{0ex}}{y}^{2}\right){e}^{-\beta \lambda \tau},\hfill & if\phantom{\rule{4pt}{0ex}}{\alpha}_{2}=0\hfill \end{array}\right.$ | ||

12. | ${u}_{\tau}={\left({e}^{u}{u}_{y}\right)}_{y}+\alpha {e}^{\gamma u}$ | $V=v+\frac{\lambda}{2}\phantom{\rule{0.166667em}{0ex}}{y}^{2}$ | 19, if $\alpha \ne 0$ |

$0={v}_{yy}+{e}^{\beta u}+\lambda $ | 21, if $\alpha =0$ | ||

13. | ${u}_{\tau}={\left({e}^{u}{u}_{y}\right)}_{y}+\lambda $ | $U=u-\lambda \tau ,\phantom{\rule{4pt}{0ex}}t=\frac{1}{\lambda}\phantom{\rule{0.166667em}{0ex}}{e}^{\lambda \tau},$ | 20 |

$0={v}_{yy}+u$ | $V=v+\frac{\lambda}{2}\phantom{\rule{0.166667em}{0ex}}\tau {y}^{2}$ | ||

14. | ${u}_{\tau}={\left({e}^{u}{u}_{y}\right)}_{y}+\lambda $ | 21 | |

$0={v}_{yy}+{e}^{\beta u}+\gamma $ | $V=\left(v+\frac{\gamma}{2}\phantom{\rule{0.166667em}{0ex}}{y}^{2}\right){e}^{-\beta \lambda \tau}$ |

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**MDPI and ACS Style**

Cherniha, R.; Davydovych, V.; King, J.R.
Lie Symmetries of Nonlinear Parabolic-Elliptic Systems and Their Application to a Tumour Growth Model. *Symmetry* **2018**, *10*, 171.
https://doi.org/10.3390/sym10050171

**AMA Style**

Cherniha R, Davydovych V, King JR.
Lie Symmetries of Nonlinear Parabolic-Elliptic Systems and Their Application to a Tumour Growth Model. *Symmetry*. 2018; 10(5):171.
https://doi.org/10.3390/sym10050171

**Chicago/Turabian Style**

Cherniha, Roman, Vasyl’ Davydovych, and John R. King.
2018. "Lie Symmetries of Nonlinear Parabolic-Elliptic Systems and Their Application to a Tumour Growth Model" *Symmetry* 10, no. 5: 171.
https://doi.org/10.3390/sym10050171