The Cauchy problem for the nonlinear integro-partial differential equation in quantitative sociodynamics.
The authors derive asymptotic estimates for solutions to the Cauchy problem for the generic (``master'') partial integro-differential equation \[ \partial v(t, x)/\partial t= -w(t, x)v(t,x)+ \int_{v\in D} W(t; x| y) v(t,y)\,dy, \] where \(D\subset\mathbb{R}^2\) is bounded and Lebesgue measurable; the kernel \(W(t;x| y)\) describes the transition rate, at time \(t\geq 0\), from the point \(y\in D\) to \(x\in D\). While such functional equations have their origin in statistical physics, the present paper is motivated by similar ones arising in quantitative sociodynamics, for example in the mathematical modelling of interregional migration due to regional economic disparity.
nonlinear partial integro-differential equation
scientific article; zbMATH DE number 1861228