SOLVABILITY OF DOUBLY NONLINEAR PARABOLIC EQUATION WITH p-LAPLACIAN

In this paper, we consider a doubly nonlinear parabolic equation partial derivative(t)beta(u)-del center dot alpha(x, del u) there exists f with the homogeneous Dirichlet boundary condition in a bounded domain, where beta : R -> 2(R) is a maximal monotone graph satisfying 0 is an element of beta(0) and del center dot alpha(x, del u) stands for a generalized p -Laplacian. Existence of solution to the initial boundary value problem of this equation has been studied in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on beta. However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for 1 < p < 2. Main purpose of this paper is to show the solvability of the initial boundary value problem for any p is an element of (1, infinity) without any conditions for beta except 0 is an element of beta (0). We also discuss the uniqueness of solution by using properties of entropy solution.