The Cauchy problem for linear growth functionals
The following Cauchy problem is considered \[ \frac{\partial u}{\partial t}=\text{div }\mathbf a(x,Du)\quad \text{in}\quad Q=(0,\infty)\times\mathbb R^N;\qquad u(0,x)=u_0(x)\quad \text{in}\quad x\in\mathbb R^N,\tag{*} \] where \(u_0\in L_{loc}^1(\mathbb R^N)\), \(\mathbf a(x,\xi)=\nabla_\xi f(x,\xi)\) and \(f:\mathbb R^N\times\mathbb R^N\to\mathbb R\) is a function with linear growth as \(\|\xi\|\to\infty\) satisfying some additional assumptions. An example of a function \(f(x,\xi)\) covered by these assumptions is \(f(x,\xi)=\sqrt{1+\|\xi\|^2}\) but the function \(f(x,\xi)=\|\xi\|\) does not satisfy these assumptions. Existence and uniqueness results are stated for the initial data in \(L_{loc}^1(\mathbb R^N)\).
scientific article; zbMATH DE number 1921149