Existence results for a class of evolution equations of mixed type

Abstract

We give an existence result for the evolution equation (Ru)' + Au = f in the space W = {u is an element of V\ (Ru)' is an element of V'}where V is a Banach space and R is a non-invertible operator (the equation may be partially elliptic and partially parabolic, both forward and backward) and we study the "Cauchy-Dirichlet" problem associated to this equation (indeed also for the inclusion (Ru)' + Au There Exists f). We also investigate continuous and compact embeddings of W and regularity in time of the solution. At the end we give some examples of different R

Asymptotic behaviour of a class of degenerate elliptic-parabolic operators: a unitary approach

“…In this paper we consider strongly degenerate parabolic, or elliptic-parabolic, operators like P u = ∂ ∂t (ru) − div(a · Du) w i t hr = r(x, t) 0 ( 4 ) and study the limit behaviour of the sequence of Cauchy-Dirichlet problems (for the existence result we refer to [8], but see also [11])…”

“…Elliptic-parabolic operators like those in (4) were already studied, as regards the existence of the solution, probably first by Showalter (see, for instance, [10] for one of the first papers and [11] for a recent book) and recently in [8] for a more general class of operators (nonlinear and possibly forward, backward and stationary).…”

An existence result for a class of second order evolution equations of mixed type

“…As regards evolution equations of first order (partially elliptic, partially parabolic) we recall some classical books (see [2,14]) where many degenerate problems are presented in which the coefficients in front of temporal derivatives may be non-negative. Among the situations in which the coefficient in front of temporal derivative may be also negative we recall [1,7,8].…”

“…To do this, we reduce by a change of variable to a first order evolution equation and use an existence result for mixed equation of first order (see [8]) which follows by a classical result for perturbation of monotone operators (Theorem 2.5). Although in [8] the operator A in (1) may be non-linear, here we manage to consider A only linear (this is due to Proposition 2.3, needed when we reduce to a first order equation).…”

“…Although in [8] the operator A in (1) may be non-linear, here we manage to consider A only linear (this is due to Proposition 2.3, needed when we reduce to a first order equation).…”