The uniqueness of nonnegative \(C^{1}\)-solution for nonlinear differential equations (Q1826987)

This paper is devoted to a study of the nonlinear degenerate parabolic system \[ \begin{cases} u_t= u^{\alpha_1}(u_{xx}+ av),\quad & x\in(-\ell,\ell),\;0< t< T,\\ v_t= v^{\alpha_2}(v_{xx}+ bu),\quad & x\in(-\ell,\ell),\;0< t< T,\\ u(\pm\ell, t)= v(\pm\ell, t)= 0,\quad & 0< t< T,\end{cases}\tag{1} \] where \(\alpha_1\), \(\alpha_2\), \(a\), \(b> 0\) are constants. For (1) the authors estimate the blow up rate and analyze the asymptotic behaviour of the blow up solution near the blow up time. Uniqueness and some numerical results are presented.