Browsing by Subject "a priori estimates"

Now showing 1 - 2 of 2

Access status: Open Access ,
Classical solutions of mixed problems for quasilinear first order PFDEs on a cylindrical domain
(2014) Czernous, Wojciech
We abandon the setting of the domain as a Cartesian product of real intervals, customary for first order PFDEs (partial functional differential equations) with initial boundary conditions. We give a new set of conditions on the possibly unbounded domain $\Omega$ with Lipschitz differentiable boundary. Well-posedness is then reliant on a variant of the normal vector condition. There is a neighbourhood of $\partial\Omega$ with the property that if a characteristic trajectory has a point therein, then its every earlier point lies there as well. With local assumptions on coefficients and on the free term, we prove existence and Lipschitz dependence on data of classical solutions on $(0,c)\times\Omega$ to the initial boundary value problem, for small $c$. Regularity of solutions matches this domain, and the proof uses the Banach fixed-point theorem. Our general model of functional dependence covers problems with deviating arguments and integro-differential equations.
Access status: Open Access ,
Existence results and a priori estimates for solutions of quasilinear problems with gradient terms
(Wydawnictwa AGH, 2019) Filippucci, Roberta; Lini, Chiara
In this paper we establish a priori estimates and then an existence theorem of positive solutions for a Dirichlet problem on a bounded smooth domain in $\mathbb{R}^N$ with a nonlinearity involving gradient terms. The existence result is proved with no use of a Liouville theorem for the limit problem obtained via the usual blow up method, in particular we refer to the modified version by Ruiz. In particular our existence theorem extends a result by Lorca and Ubilla in two directions, namely by considering a nonlinearity which includes in the gradient term a power of u and by removing the growth condition for the nonlinearity $f$ at $u=0$.