Nonexistence theorems for systems of quasilinear parabolic inequalities (Q2757093)

Consider the initial-boundary value problem for the system of degenerate parabolic inequalities NEWLINE\[NEWLINE\begin{cases} u_t\geq\bigl(|u|^{m-1} u\bigr)+ |v|^q,\;(x,t)\in \mathbb{R}^N\times (0,\infty),\\ v_t\geq\bigl( |v|^{n-1} v\bigr)+|u|^p,\;(x,t)\in \mathbb{R}^N\times (0,\infty),\\ u(x,0)=u_0(x),\;v(x,0)=v_0(x),\;x\in\mathbb{R}^N, \end{cases}\tag{S}NEWLINE\]NEWLINE where \(m,n>0\), \(p> \max\{m,1\}\), \(q>\max\{n,1\}\) and \(u_0\), \(v_0\in L^1(\mathbb{R}^N)\) satisfy the following positivity condition NEWLINE\[NEWLINE\int_{\mathbb{R}^N} u_0(x)>0, \int_{\mathbb{R}^N} v_0(x) >0.NEWLINE\]NEWLINE In the present paper the author has given explicit conditions on the parameters \(p,q,m,n\) in order that no global solution of (S) exists when \(m>1\) or \(n>1\).