Resolvent kernels that constitute an approximation of the identity and linear heat-transfer problems (Q2730503)

The author considers the integral equation NEWLINE\[NEWLINE u_\alpha (t)=F(t)-\int_0^tk_\alpha (t-s)u_\alpha (s) ds,\quad t>0,\tag{1} NEWLINE\]NEWLINE depending on the parameter \(\alpha\to\infty\). Let \(\Gamma_\alpha (t-s)\) be the corresponding resolvent kernel. Conditions on \(k_\alpha\) are found under which \(\Gamma_\alpha\) is a generalized approximation of identity, that is \(\Gamma_\alpha (t)\geq 0\), \(\int_0^\infty \Gamma_\alpha (s) ds\to 1\), and for every \(\delta >0\), \(\int_\delta^\infty \Gamma_\alpha (s) ds\to 0\) as \(\alpha \to \infty\). NEWLINENEWLINENEWLINEIn this case the solution of (1) tends to zero as \(\alpha \to \infty\). \(L^\infty\)-estimates of this convergence are established. The results are applied to obtain estimates of the convergence of solutions of linear heat transfer boundary value problems as the heat transfer coefficient tends to infinity.