Removable singularities of semilinear parabolic equations (Q5891300)

The author studies the removability property for solutions of the semilinear parabolic equation NEWLINE\[NEWLINE u_t-\Delta u=F(x,t,u,\nabla u) \quad\text{ in }(\Omega\setminus\{0\})\times (0,T), NEWLINE\]NEWLINE where \(\Omega\) is a domain of \({\mathbb R}^n\), (\(n\geq 3\)) containing the origin and \(T>0\). The first result of the paper establishes that if \(F\) satisfies NEWLINE\[NEWLINE |F(x,t,u,\nabla u)|\leq C(1+|u|^p) NEWLINE\]NEWLINE for some \(0\leq p<n/(n-2)\), then a solution \(u\) has removable singularities on \(\{0\}\times (0,T)\) if and only if for any \(0<t_1<t_2<T\) and any \(0<\delta \leq 1\) there exists \(r>0\) such that NEWLINE\[NEWLINE |u(x,t)|\leq \delta |x|^{2-n}\quad\text{ for any } 0<|x|<r, t\in [t_1,t_2]. NEWLINE\]NEWLINENEWLINENEWLINEIn the second result of the paper the author obtains that if \(F\) satisfies \(|F(x,t,u,\nabla u)|\leq C(1+|u|^p)\) for some \(p>1\) and if there exists \(q<2/(p-1)\), \(r>0\) and \(\delta>0\) such that NEWLINE\[NEWLINE |u(x,t)\leq |x|^{-q}\quad\text{ for any }|x|<r,t\in (T-\delta,T)NEWLINE\]NEWLINE then the solution \(u\) does not blow up at the point \((x,t)=(0,T)\).NEWLINENEWLINEThe proofs rely on parabolic potential theory argument and iterations techniques.