Initial value problems for nonlinear degenerate systems of filtration type (Q791039)

In this work for the nonlinear degenerate parabolic system of filtration type \((1)\quad u_ t=(\text{grad} \phi(u))_{xx},\) the periodic boundary problem \((2)\quad u(x+2D,t)=u(x,t);u(x,0)=u_ 0(x)\) and the initial value problem \((3)\quad u(x,0)=u_ 0(x)\) are considered, where \(u(x,t)=(u_ 1(x,t),...,u_ N(x,t)\) and \(u_ 0(x)\) are N-dimensional vector functions, \(\phi\) (u) is a scalar function of vector variable \(u\in {\mathbb{R}}^ N\) and ''grad'' denotes gradient operator with respect to u. Suppose that the \(N\times N\) Hessian matrix \(H(u)=\partial^ 2\phi(u)/\partial u_ i\partial u_ j\) of \(\phi\) (u) with respect to u is positively definite for \(| u| \neq 0\) and H(0) is zero definite. The system (1) is degenerate. When \(N=1\), the system is the so- called equation of the filtration type. For the periodic boundary problem (1), (2), there exists a unique global weak N-dimensional vector solution u(x,t), such that \(u(x,t)\in L_{\infty}(0,T;C^{(0,s)}(-D,D))\) and \(\text{grad} \phi(u(x,t))\in L_{\infty}(0,T;H^ 1(-D,D)),\) where \(0<s<1\) is a constant. For the initial value problem (1), (3), there is a unique global weak vector solution u(x,t), such that \(u(x,t)\in L_{\infty}(0,T;C^{(0,s)}({\mathbb{R}}))\) and \(\text{grad} \phi(u(x,t))\in L_{\infty}(0,T;H^ 1({\mathbb{R}})).\) When H(u) is positively definite for \(u\in {\mathbb{R}}^ N\), the system (1) is non-degenerate and the global generalized and classical vector solution are obtained.