A note on strongly nonlinear parabolic equations of higher order
Let \(\Omega \subset R^ N\) (N\(\geq 1)\) be bounded domain and consider the parabolic initial-boundary value problem over \(\Omega\) : \[ u_ t+A(u)+G(u)=f\text{ in } (0,T)\times \Omega =Q_ T,\quad u(0)=\phi \text{ in } \Omega,\quad D^{\alpha}u|_{(0,T)\times \partial \Omega}=0\text{ for } | \alpha | \leq m-1. \] Here, \(A(u)=\sum_{| \alpha | \leq m}(-1)^{| \alpha |}D^{\alpha}A_{\alpha}(t,x;Du)\). The author discusses sufficient conditions for the perturbation \(G(u)=g(t,x;u)\) in order to establish existence of weak solutions of the problem. Under some technical assumptions as e.g. ellipticity and Carathéodory condition for A, \(\alpha\) sign and Carathéodory condition for G the author proves the theorem: There is a weak solution of this problem with the following properties: (i) \(\frac{\partial}{\partial t}u\in L^ 2((0,T),L^ 2(\Omega))\) and \(u\in C([0,T],L^ 2(\Omega))\) (ii) The solution u itself can be used as a test function: that is to say, we have \[ {1\over2}\int_{\Omega}(u(T))^ 2dx+\int^{T}_{0}(A(u),u)dt+\int^{T}_{0}(G(u),u)dt=f(u)+{1\over2}\i nt_{\Omega}(u(0))^ 2 dx. \]
scientific article; zbMATH DE number 4212601