Blow-up solution for the degenerate quasilinear parabolic system with nonlinear boundary condition
The authors consider the following system \[ u_t= u^{\alpha_1} u_{xx},\;v_t= v^{\alpha_2} v_{xx}\quad\text{for }x\in (0,\ell),\;t> 0,\tag{1} \] \[ -u_x(0,t)= av^p(0,t),\;-v_x(0,t)= bu^q(0,t),\;u(\ell,t)= v(\ell, t)= 0,\tag{2} \] \[ u(x,0)= u_0(x),\;v(x,0)= v_0(x)\quad\text{for }x\in (0,\ell),\tag{3} \] where \(0< \alpha_1\), \(\alpha_2< 1\), \(p,q,a,b> 0\), and the initial values \(u_0(x)\), \(v_0(x)> 0\) for \(x\in (0,\ell)\) are sufficiently smooth. Under compatibility conditions for (1)--(3) the authors prove the local existence and the uniqueness for (1)--(3). Moreover, they discuss the blow-up solution of (1)--(3) as well as the blow-up point set and the blow-up rate. The existence of the self-similar blow-up solution is discussed.
scientific article; zbMATH DE number 5006973
scientific article