A global approach to fully nonlinear parabolic problems

Abstract

We consider the general initial-boundary value problem (1) ∂ u ∂ t − F ( x , t , u , D 1 u , D 2 u ) = f ( x , t ) , ( x , t ) ∈ Q T ≡ Ω × ( 0 , T ) , \displaystyle {\frac {\partial u}{\partial t}-F(x,t,u,\mathcal {D}^{1}u, \mathcal {D}^{2}u)=f(x,t),\quad (x,t)\in Q_{T}\equiv \Omega \times (0,T),} (2) G ( x , t , u , D 1 u ) = g ( x , t ) , ( x , t ) ∈ S T ≡ ∂ Ω × ( 0 , T ) , \displaystyle {G(x,t,u,\mathcal {D}^{1}u)=g(x,t),\quad (x,t)\in S_{T}\equiv \partial \Omega \times (0,T),} (3) u ( x , 0 ) = h ( x ) , x ∈ Ω , \displaystyle {u(x,0)=h(x),\quad x\in \Omega ,} where Ω \Omega is a bounded open set in R n \mathcal {R}^{n} with sufficiently smooth boundary. The problem (1)-(3) is first reduced to the analogous problem in the space W p ( 4 ) , 0 ( Q T ) W^{(4),0}_{p}(Q_{T}) with zero initial condition and \[ f ∈ W p ( 2 ) , 0 ( Q T ) , g ∈ W p ( 3 − 1 p ) , 0 ( S T ) . f\in W^{(2),0}_{p}(Q_{T}),~g \in W^{(3-\frac {1}{p}),0}_{p}(S_{T}). \] The resulting problem is then reduced to the problem A u = 0 , Au=0, where the operator A : W p ( 4 ) , 0 ( Q T ) → [ W p ( 4 ) , 0 ( Q T ) ] ∗ A:W^{(4),0}_{p}(Q_{T})\to \left [W^{(4),0}_{p}(Q_{T})\right ]^{*} satisfies Condition ( S ) + . (S)_{+}. This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces. The local and global solvability of the operator equation A u = 0 Au=0 are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations.