Nonlinear degenerate parabolic equations with changing evolution direction (Q2760727)

Let \(Q=(0,1)\times(0,l)\). Assume further that there are functions \(a=a(x,t,s,p)\) and \(b=b(x,t,s,p)\) such that for \((x,t)\in\overline Q\), \(u\in {\mathbb R}\), \(p\in {\mathbb R}\) the following hold: NEWLINE\[NEWLINE \begin{gathered} 0<\gamma_0\leq{a(x,t,|u|,p)\over|u|^\gamma(1+|p|)^n} \leq\gamma_1<+\infty,\quad 0<\gamma<1,\;n>0; \\ \text{sign} u\cdot b(x,t,|u|,0)\leq\mu_0\cdot|u|\quad \text{for } |u|\geq N_0>1 \end{gathered} NEWLINE\]NEWLINE (with \(\gamma_0\), \(\gamma_1\), \(\mu_0\), and \(N_0\) constants). In the case of \(u\neq 0\), \(a\) and \(b\) are Hölder continuous with first derivatives in all arguments. The authors consider the following boundary-value problem: Find a solution in \(Q\) to the equations NEWLINE\[NEWLINE \begin{gathered} \text{sign} u\cdot u_t=a(x,t,|u|,u_x)u_{xx} +b(x,t,|u|,u_x), \tag{1} \\ u(k,t)=0,\quad (-1)^ku(x,kl)=u_0(x)\geq 0,\quad k=0,1. \tag{2} \end{gathered} NEWLINE\]NEWLINE Note that this is a Dirichlet problem and so the regular solution to this problem (if it exists) must change the sign in the cylinder \(Q\).NEWLINENEWLINENEWLINEThe study of the problem rests on the elliptic regularization of (1), a priori estimates, and passage to the limit. The main result of the article reads: under subordination conditions the passage to the limit in the regularized equation yields the existence of a degenerate solution.