Further existence results for elliptic–parabolic and forward–backward parabolic equations

Abstract

We give an existence result for first order evolution equation of the type $$\mathcal {R}u' + \mathcal {A}u = f$$ R u ′ + A u = f where $$\mathcal {R}$$ R may be a function depending also on time assuming positive, null and negative sign, then the equation may be elliptic–parabolic, both forward and backward. The result is given in an abstract setting with Banach spaces depending on time (the functions u are defined in an interval [0, T] and $$u(t) \in X(t)$$ u ( t ) ∈ X ( t ) for a.e. t) and $$\mathcal {R}$$ R which is in fact a linear operator. We also extend a previous existence result for the equation $$(\mathcal {R}u)' + \mathcal {A}u = f$$ ( R u ) ′ + A u = f to the setting of moving Banach spaces. We also give a time regularity result in a particular case and give many examples of different possible choices of $$\mathcal {R}$$ R .

Two generalized Tricomi equations

“…(see (9) for the definition ofH − ,H 0 ,H + ). The following result follows from Theorem 3.13 in [8].…”

“…Proof Point i) is an immediate consequence of Theorem 3.10 in [8], while points ii) and iii) follow by Theorem 3.13 in [8] since…”

“…the two spaces, without specifying if not necesary. We now recall some results contained in [8] and [9].…”

“…We show here a simple example, but for further details we refer to the analogous example in [9] and [8]. In dimension 1 consider r assuming only two values, 1 and −1.…”

$G$-convergence of elliptic and parabolic operators depending on vector fields

“…In the following proposition we recall some result regarding the space W for which we refer to [27] and [30].…”