On the characterization of Bow-up and Global Radial solution for free boundary system with nonlinear inhomogeneous gradient and source term

This paper concerns the characterization of blowup and global radial solutions of a two-free boundaries system read by \begin{align}\label{bs_pr} \tag{1.1} \left\{\begin{array}{rl} u_t(t,r)= \Delta u(t,r) - \lambda(t,x)|\nabla u(t,r)|^{\alpha} + a(t,x)v^{p}(t,r),& t>0,\ 0<r<h(t),\\ v_t(t,r) = \Delta v(t,r) - \lambda(t,x) |\nabla v(t,r)|^{\alpha}+ a(t,x)u^{p}(t,r), & t>0,\ 0 < r < g(t), \end{array}\right. \end{align} where $r = |x|,\ x \in \R^N$, $p, \alpha >1$ are given constants and $\lambda(t,x), a(t,x)$ satisfy suitable prescribed growth conditions. First, we show the well-posedness of the local solution to (\ref{bs_pr}). Second, we succeed to classify the blowup and global phenomena by establishing some relations between $\alpha$, $p$ and growth rate of the coefficients, in which proving a comparison principle based on the Stampacchia truncation method plays the central role. In particular, if $1<\alpha < p$ and $(u(0,r);v(0,r)) = A (\phi(r); \varphi(r))$ is an initial data of (\ref{bs_pr}), we find two certain positive thresholds $A^*_G$ and $A^*_B$ such that the global fast solution exists for $0 < A < A_G^{*}$ and the global slow solution exists for a suitable value of $A$ such that $A^*_G \leq A < A^*_{B}$ while blow-up solutions hold for $A \geq A^{*}_{B}$. On the other hand, if $\alpha \geq p$ incorporating with a suitable comparison on $\beta, p$ and $\alpha$, then there exist global solutions with nonnegative initial data of exponential decay. Our approach is being far different from the celebrated works \cite{MF,PS}, where the authors can only handled the equations with constant coefficients. To our knowledge, this is the first work revealing the influence of the inhomogeneous coefficients to the blow-up and global phenomena to the cooperative system with nonlinear gradient and different free boundaries.