\(p\)-Laplacian with a concentrated nonlinear source (Q1941123)

The authors consider the Cauchy problem for the \(p\)-Laplacian with a concentrated nonlinear source \[ \begin{cases} {{\partial u}\over{\partial t}}-\text{div\,}\big(|\nabla u|^{p-2}\nabla u\big)={{\partial \chi_b(x)}\over{\partial\nu}} f(u), & (x,t)\in Q_T,\\ u(x,0)=u_0(x), & x\in \mathbb R^n, \end{cases} \] where \(p>2,\) \(Q_T=\mathbb R^n\times (0,T)\), \(n\geq2\), \(T\in(0,\infty)\), \(B\) is the \(n\)-dimensional ball \(\{x\in\mathbb R^n: |x|<b\}\), \(\chi_B\) is its characteristic function and \(\nu\) stands for the inward normal to \(\partial B\). Existence of generalized solutions to that problem is obtained on the base of some a~priori estimates.