Non-local Hamilton-Jacobi equations arising in dislocation dynamics
The authors consider the following nonlocal Hamilton-Jacobi equation arising in dislocation dynamics: \[ u_t=[c(x,t)+M_p[u(\cdot,t)](x)]|p+Du(x,t)|,\tag{1} \] where \(u:\mathbb R^N\times [0,\infty)\to\mathbb R\), \(p\in\mathbb R^N\), \(Du=(\frac{\partial u}{\partial x_1},\dots,\frac{\partial u}{\partial x_N})\), \[ M_p[\phi](x)=\int_{\mathbb R^N}J(z)[E(\varphi(x+z)-\varphi(x)+p\cdot z)-p\cdot z]\,dz. \] Here, \(J\) is a measurable function on \(\mathbb R^N\), and \(E(r)\) denotes the greatest integer less than or equal to \(r\in\mathbb R\). Firstly, they establish some stability properties of viscosity solutions, comparison theorems between subsolutions and supsolutions. By using the above mentioned results, an existence and unique theorem for the initial value problem for (1) is proved. Furthermore, for \(N=1\), they show the well-posedness for the initial value problem for (1) by weakening the conditions.
scientific article; zbMATH DE number 5770970