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

TL;DR

This paper characterizes blowup and global solutions for a free boundary system with nonlinear inhomogeneous gradient and source terms, revealing how coefficients influence solution behavior and classifying thresholds for blowup or global existence.

Contribution

It introduces a novel analysis of inhomogeneous coefficients' effects on blowup and global solutions in free boundary systems with nonlinear gradients, extending prior constant-coefficient results.

Findings

Established well-posedness of local solutions.

Identified thresholds for blowup and global solutions based on initial data.

Showed inhomogeneous coefficients significantly affect solution dynamics.

Abstract

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.