Some results on nonlinear heat equations for materials of fading memory type (Q1896029)

We consider a model for the heat conduction for a material covering an \(n\)-dimensional bounded set \(\Omega\) with boundary \(\partial\Omega\), \(n= 1,2,3\). \[ {d\over dt} \Biggl(b_ 0 u(t, x)+ \int^ t_ 0 \beta(t- s) u(s, x)ds\Biggr)= c_ 0 \Delta u(t, x),\;t> 0,\;x\in \Omega,\;u(0, x)= x,\;x\in \Omega, \] where \(u(t, x)\) is the temperature of the point \(x\) at time \(t\) (we assume that the temperature is 0 for \(x\in \partial\Omega\)), \(b_ 0\) is the specific heat and \(c_ 0\) the thermal conductivity. We assume that the specific heat has a term of fading memory type \(\int^ t_ 0 \beta(t- s)u(s, x)ds\), whereas the thermal conductivity is constant. Concerning the kernel \(\beta\) we assume only that it is locally integrable in \([0, \infty[\); this will allow us to consider kernels as \(\beta(t)= e^{-\omega t} t^{\alpha- 1}\), \(\omega\geq 0\), \(\alpha\in ]0, 1[\).