Time-periodic solutions of quasilinear parabolic differential equations. III: Conormal boundary conditions (Q5945992)

For part I, see J. Math. Anal. Appl. 264, No. 2, 617--638 (2000; Zbl 1002.35037).NEWLINENEWLINE From the introduction: Let \(P\) be a second-order quasilinear parabolic operator, let \(N\) be a first-order nonlinear oblique operator, and let \(\Omega\subset\mathbb{R}^{n+1}\) with lateral surface \(S\Omega\) and initial surface \(\omega\). In this work, we consider the problem NEWLINE\[NEWLINEPu=0\text{ in } \Omega,\quad Nu=0\text{ on }S\Omega, \quad u(\cdot,0)=u(\cdot,T)\text{ in } \omegaNEWLINE\]NEWLINE under the basic assumption that there are scalar-valued function \(B\) and \(\psi\) and a vector-valued function \(A\) such that NEWLINE\[NEWLINEPu=-u_t+\text{div}\,A (X,u,Du)+ B(X,u,Du)NEWLINE\]NEWLINE and NEWLINE\[NEWLINENu=A(X,u,Du)\cdot\gamma+\psi(X,u).NEWLINE\]NEWLINE With this special structure, we can prove existence and regularity.