Ammar, Kaouther Renormalized entropy solutions for degenerate nonlinear evolution problems. (English) Zbl 1182.35147 Electron. J. Differ. Equ. 2009, Paper No. 147, 32 p. (2009). Summary: We study the degenerate differential equation\[ b(v)_t -\operatorname{div} a(v,\nabla g(v))=f \quad \text{on }Q:= (0,T) \times \Omega \]with the initial condition \(b(v(0,\cdot))=b(v_0)\) on \(\Omega\) and boundary condition \(v=u\) on some part of the boundary \(\Sigma:=(0,T) \times \partial \Omega\) with \(g(u)\equiv 0\) a.e. on \(\Sigma\). The vector field \(a\) is assumed to satisfy the Leray-Lions conditions, and the functions \(b,g\) to be continuous, locally Lipschitz, nondecreasing and to satisfy the normalization condition \(b(0)=g(0)=0\) and the range condition \(R(b+g)=\mathbb{R}\). We assume also that \(g\) has a flat region \([A_1,A_2]\) with \(A_1\leq 0\leq A_2\). Using Kruzhkov’s method of doubling variables, we prove an existence and comparison result for renormalized entropy solutions. Cited in 3 Documents MSC: 35K65 Degenerate parabolic equations 35K55 Nonlinear parabolic equations 35J65 Nonlinear boundary value problems for linear elliptic equations 35J70 Degenerate elliptic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B51 Comparison principles in context of PDEs Keywords:homogeneous boundary conditions; continuous flux; Leray-Lions conditions; Kruzhkov’s method PDFBibTeX XMLCite \textit{K. Ammar}, Electron. J. Differ. Equ. 2009, Paper No. 147, 32 p. (2009; Zbl 1182.35147) Full Text: EuDML Link