On exact multidimensional solutions of a nonlinear system of reaction-diffusion equations
The main goal of this paper is to construct exact solutions to the following Cauchy problem for the parabolic system of two equations \[ \begin{cases} u_t=\nabla\,\cdot \big(u^{\lambda}\,\nabla u\big)+\alpha u^{1-\lambda} v^{\mu},\quad (x,t)\in \mathbb{R}^n\times \mathbb{R}^+,\\ v_t=\nabla\,\cdot \big(v^{\mu}\,\nabla v\big)+\beta u^{\lambda} v^{1-\mu},\quad (x,t)\in \mathbb{R}^n\times \mathbb{R}^+,\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in \mathbb{R}^n. \end{cases}\tag{1} \] The authors prove that the problem (1) has exact solutions of the form \[ \begin{cases} u(x,t)=\psi _1(t)\,\big[2^{-1}\,( A x,x)+(B,x)+C+\varphi_1(t)\big]^{1/\lambda},\\ u(x,t)=\psi_2(t)\,\big[2^{-1}\,(Ax,x)+(B,x)+C+\varphi_2(t)\big]^{1/\mu}, \end{cases} \] where real \(n\times n\) matrix \(A\), real vector \(B\) and real constant \(C\) satisfy the following algebraic system of equations: \(\displaystyle A=2\,\sigma\, A^2, \quad B=2\,\sigma A\,B,\quad C=\sigma\,|B|^2\) with \(\sigma\neq 0\). the functions \(\psi_1,\, \psi_2,\,\varphi_1,\,\varphi_2\) are solutions to the following Cauchy problems: \[ \begin{cases} \psi'_1=\big(\lambda^{-1}\,\text{tr} A+\sigma^{-1}\,\lambda^{-2}\big)\,\psi_1^{1+\lambda}+\alpha\, \psi_1^{1-\lambda}\, \psi_2^{\mu},\\ \psi'_2=\big(\mu^{-1}\,\text{tr} A+\sigma^{-1}\,\mu^{-2}\big)\,\psi_2^{1+\mu}+\beta\, \psi_1^{\lambda} \psi_2\,^{1-\mu},\\ \psi_1(0)=\psi_{1 0}\neq 0,\quad \psi_2(0)=\psi_{2 0}\neq 0,\quad \psi_{1 0}, \, \psi_{2 0}\in \mathbb{R}, \end{cases} \] \[ \begin{cases} \varphi'_1=\big(\text{tr} A\,\psi_1^{\lambda}-\lambda\,\psi_1'\,\psi^{-1}_1\big)\,\,\varphi_1+\alpha\, \lambda\, \psi_1^{-\lambda}\,\psi_2^{\mu}\,\varphi_2,\\ \varphi'_2=\big(\text{tr} A\,\psi_1^{\mu}-\mu\,\psi_2'\,\psi^{-1}_2\big)\,\,\varphi_2+\beta\, \mu\, \psi_1^{\lambda}\,\psi_2^{-\mu}\,\varphi_1,\\ \varphi_1(0)=\varphi_{1 0},\quad \varphi_2(0)=\varphi_{2 0},\quad \varphi_{1 0}, \, \varphi_{2 0}\in \mathbb{R}. \end{cases} \] The authors also provide examples of the problem (1) for specific values of the \(\varphi_{1 0}, \varphi_{2 0}\), \(\psi_{1 0}, \psi_{2 0}\), \(n\), \(\lambda\), \(\mu\), \(\alpha\) and \(\beta\).
quasilinear reaction-diffusion equations
scientific article; zbMATH DE number 6899065
scientific article