Extinction and non-extinction of solutions for a nonlocal reaction-diffusion problem

We investigate extinction properties of solutions for the homogeneous Dirichlet boundary value problem of the nonlocal reaction-diffusion equation $u_t-d\Delta u+k u^p=\int_\Omega u^q(x,t)\,dx$ with $p, q\in (0, 1)$ and $k, d >0$. We show that $q=p$ is the critical extinction exponent. Moreover, the precise decay estimates of solutions before the occurrence of the extinction are derived

Several error inequalities for a quadrature formula with a parameter and applications

AbstractIn this paper, we will derive several error inequalities for a quadrature formula with a parameter, which will not only provide some generalizations of the known results, but also give some other interesting quadrature formulae as special cases. Furthermore, sharp upper and lower error bounds for the double error inequalities are obtained. Applications in numerical integration are also given