Further existence results for elliptic–parabolic and forward–backward parabolic equations
Fabio Paronetto · Calculus of Variations and Partial Differential Equations · 2020
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 .