Admissible \(L_p\) norms for local existence and for continuation in semilinear parabolic systems are not the same (Q2784207)

The authors consider in a bounded smooth domain \(\Omega\subset \mathbb{R}^n\) systems of second-order semilinear parabolic equations NEWLINE\[NEWLINEu_t-\Delta u = f_1(x,t,v, \nabla v),\qquad v_t -\Delta v = f_2 (x,t,u,\nabla u)NEWLINE\]NEWLINE with zero Dirichlet boundary conditions. They impose suitable growth restrictions on \(f_1\), \(f_2\) and discuss admissibility of spaces \(L^{r_1}(\Omega)\times L^{r_2}(\Omega)\) from the point of view of both local well posedness result and continuation property. It is shown that some spaces \(L^{r_1}(\Omega)\times L^{r_2}(\Omega)\) may be admissible for the purpose of continuation, although simultaneously local nonexistence results may occur for certain of their elements \((u_0,v_0)\in L^{r_1}(\Omega)\times L^{r_2}(\Omega)\). This happens e.g. for \(L^r(\Omega)\times L^r(\Omega)\) in the case when \(f_1(v)= v |v|^{p_1-1}\), \(f_2(u)= u |u|^{p_2-1}\) and \({1\over 2}(p_1+ p_2) < 1+{2r\over N} <\max(p_1,p_2)\).