Uniform blow-up rate for nonlocal diffusion-like equations with nonlocal nonlinear source (Q330478)

Let \(\Omega\) be the unitary ball in \(\mathbb R^N\) and \(J:\mathbb R^N\to\mathbb R\), be a nonnegative, smooth, radially symmetric function, supported in the unit ball, with \(\int_{\mathbb R^N} J(z)dz=1\), \(p>1, r\geq 1,\) \(a,u_o\in C^2(\Omega)\), \(a,u_o>0\) in \(\Omega\), and \(a(x)=u_o(x)=0\) out of \(\Omega\). The paper deals with blow-up results for problem (P) NEWLINE\[NEWLINEu_t=\int_{\mathbb R^N} J(x-y) \bigg(u(y,t)-u(x,t)\bigg)dy+a(x)\bigg(\int_{\Omega}|u(x,t)|^r dx\bigg)^{\frac{p}{r}},\quad x\in\Omega,t>0,NEWLINE\]NEWLINE together with the condition \(u(x,t)=0\) for \(x\notin\Omega,t\geq 0\), and the initial condition \(u_o(x),x\in\Omega\). The author shows that, if \(u(x,t)\) is a solution to (P), and if \(p>1\), then there exists \(0<T<+\infty, \) such that NEWLINE\[NEWLINE\underset{t\to T^{-}} {\lim} ||u(x,t)||_{L^{\infty}(\Omega)}=+\inftyNEWLINE\]NEWLINE (Theorem 1.1). Moreover, if the above blow-up solution \(u(x,t)\) of (P) is non-decreasing in time, then NEWLINE\[NEWLINE\underset{t\to T^{-}}{\lim}(T-t)^{\frac{1}{p-1}}u(x,t)=a(x)(p-1)^{\frac{1}{1-p}}\bigg(\int_{\Omega}a^r(x)dx \bigg)^{\frac{p}{r(1-p)}},NEWLINE\]NEWLINE uniformly in all compact subsets of \(\Omega\) (Theorem 1.2).