Bound for the solution of a degenerate parabolic equation
The author studies the initial-boundary value problem for homogeneous Dirichlet data for the partial differential equation \[ Lu=u_ t- \partial_ i(a_{ij}\partial_ ju)+cu=f, \] where the elliptic part of L is allowed to degenerate. More precisely, it is assumed that \(a_{ij}\xi_ i\xi_ j\geq \nu | \xi |^ 2\), where \(\nu\) (x,t)\(\geq 0\) and \(1/\nu (\cdot,t)\in L^ p(\Omega)\) for all \(t>0\), for some \(p\geq 1\). It is stated that any solution to this problem is bounded from above by the solution to the corresponding spherically symmetrized problem.
scientific article