- Series
- PDE Seminar
- Time
- Tuesday, August 21, 2018 - 3:00pm for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Professor Veli Shakhmurov – Okan University – veli.sahmurov@okan.edu.tr
- Organizer
- Ronghua Pan

The boundary value and mixed value problems for linear and
nonlinear degenerate abstract elliptic and parabolic equations are
studied. Linear problems involve some parameters. The uniform
L_{p}-separability properties of linear problems and the optimal
regularity results for nonlinear problems are obtained. The equations
include linear operators defined in Banach spaces, in which by choosing
the spaces and operators we can obtain numerous classes of problems for
singular degenerate differential equations which occur in a wide variety
of physical systems.
In this talk, the classes of boundary value problems (BVPs) and
mixed value problems (MVPs) for regular and singular degenerate
differential operator equations (DOEs) are considered. The main
objective of the present talk is to discuss the maximal regularity
properties of the BVP for the degenerate abstract elliptic and parabolic
equation
We prove that for f∈L_{p} the elliptic problem has a unique
solution u∈ W_{p,α}² satisfying the uniform coercive estimate
∑_{k=1}ⁿ∑_{i=0}²|λ|^{1-(i/2)}‖((∂^{[i]}u)/(∂x_{k}^{i}))‖_{L_{p}(G;E)}+‖Au‖_{L_{p}(G;E)}≤C‖f‖_{L_{p}(G;E)}
where L_{p}=L_{p}(G;E) denote E-valued Lebesque spaces for p∈(1,∞) and
W_{p,α}² is an E-valued Sobolev-Lions type weighted space that to be
defined later. We also prove that the differential operator generated by
this elliptic problem is R-positive and also is a generator of an
analytic semigroup in L_{p}. Then we show the L_{p}-well-posedness with
p=(p, p₁) and uniform Strichartz type estimate for solution of MVP for
the corresponding degenerate parabolic problem. This fact is used to
obtain the existence and uniqueness of maximal regular solution of the
MVP for the nonlinear parabolic equation.