Abstract
We give an existence result for first order evolution equation of the type mathcal {R}u' + mathcal {A}u = f where mathcal {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) for a.e. t) and mathcal {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 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}.
Highlights
In this paper we consider differential equations of mixed type in abstract form whose concrete model example is r (x, t)ut − div (|Du|p−2 Du) = f, p ≥ 2, (1)where r is a function which may assume positive, null and negative values and this equation may be of elliptic–parabolic type, parabolic both forward and backward.Equations of mixed type have been considered since at least one century ago, since, as far as many authors say, they are mentioned in [7]
Where r is a function which may assume positive, null and negative values and this equation may be of elliptic–parabolic type, parabolic both forward and backward
In the last section we show with some examples why this setting can be interesting
Summary
In this paper we consider differential equations of mixed type in abstract form whose concrete model example is r (x, t)ut − div (|Du|p−2 Du) = f , p ≥ 2,. Where r is a function which may assume positive, null and negative values and this equation may be of elliptic–parabolic type, parabolic both forward and backward. Equations of mixed type have been considered since at least one century ago, since, as far as many authors say, they are mentioned in [7]. We recall some simple and more known examples: Communicated by C.
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