Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions (Q2758115)

The authors study blow-up and global existence for the system of porous medium equations NEWLINE\[NEWLINEu_t=(u^m)_{xx}, \quad v_t=(v^n)_{xx}, \qquad x>0,\;t>0, NEWLINE\]NEWLINE coupled through the nonlinear boundary conditions NEWLINE\[NEWLINE -(u^m)_x(0,t)=v^p(0,t), \quad -(v^n)_x(0,t)=u^q(0,t), \qquad t>0.NEWLINE\]NEWLINE The initial data \(u(\cdot,0)\), \(v(\cdot,0)\) are assumed to be continuous nonnegative and compactly supported, \(m,n>1\), \(p,q>0\). It is shown that all solutions of this problem exist globally if and only if \(pq\leq(m+1)(n+1)/4\). In the blow-up case, the authors find necessary and sufficient conditions in terms of \(p,q,m,n\) for blow-up of all nontrivial solutions. They also establish the blow-up rate of blowing up solutions which are increasing in time, and they characterize the blow-up sets \(B(u)\), \(B(v)\) of solutions satisfying the blow-up rate mentioned above. Each of \(B(u)\), \(B(v)\) is either \(\{0\}\), a bounded interval containing \(0\) or the interval \([0,\infty)\) and any combination of these alternatives is possible.