On the initial boundary-value problem for \(u_ t-\text{div}(| \nabla u| ^{p-2}\nabla u)=0\)
Let \(\Omega \subset \mathbb{R}^N\), \(N \geq 2\), be a bounded domain with Lipschitz boundary \(\partial \Omega\) and let \(T \in (0, \infty)\). The author considers the problem \( {\partial u / \partial t} - \text{div} (|\nabla u |^{p - 2} \nabla u) = 0\) in \(\Omega \times (0,T)\), \(u = 0\) on \(\partial \Omega \times (0,T)\), \(u(x,0) = u_0(x) \in L^1 (\Omega)\), \(1 < p < 2\). To define a suitable notion of a weak solution, he employs the idea of renormalization, first introduced by \textit{R. Di Perna} and \textit{P. L. Lions} [Commun. Pure Appl. Math. 42, No. 6, 729-757 (1989; Zbl 0698.35128)]. The author proves that there exists a unique renormalized solution.
scientific article; zbMATH DE number 754019