Uniqueness of piecewise smooth weak solutions of multidimensional degenerate parabolic equations
The paper deals with uniqueness of piecewise smooth weak solutions to the Cauchy problem for a degenerate parabolic equation of the form \[ u_t -\nabla f=\nabla (Q\nabla u) +g,\quad (x,t)\in \mathbb{R}^N\times \mathbb{R}^+, \] where the flux \(f,\) the viscosity coefficient \(Q,\) and the source term \(g\) depend on \((x,t,u),\) and \(Q=Q(x,t,u)=(Q_{i,j}(x,t,u))^N_{i,j=1}\) is nonnegative definite. In this case, classical solutions as a rule do not exist and, therefore, weak solutions are considered. It is well known that the Cauchy problem has weak solutions. However, due to the possible degeneracy, weak solutions, in general, are not uniquely determined by the initial data. In order to have uniqueness, the author introduces a class of piecewise smooth weak solutions which are \(C^2-\) smooth everywhere apart from a closed nowhere dense collection of smooth manifolds. He proves that the solution operator is \(L^1-\) stable in this class, and as a consequence, the Cauchy problem admits at most one \(L^1(\mathbb{R}^N)\) piecewise smooth weak solution.
Degenerate nonlinear parabolic equations
scientific article; zbMATH DE number 1046948
scientific article