• The Strongly Solutions of Nonlinear Parabolic Partial Differential Equations Problems in Sobolev Spaces

    Omar Mohammed Abdullah Al-haj, Mohammed Al-Hawmi

    Department of Mathematics, Faculty of Education and Science, University of Saba Region, Marib, Yemen.

    Abstract:

    In this paper, we study the existence of a strong solutions for the initial-boundary value problems of the nonlinear degenerated parabolic equation

    ∂u/∂t + A(u) + g(x, t, u, ∇u) = f in Q

    where A(u) = -div a(x, t, u, ∇u), is a Leary lions operator acted from the weighted Sobolev Space LP(0, T, W01, p(Ω, ω)) in to its dual LP’(0, T, W0-1, p’(Ω, ω*)) and g(x, t, u, ∇u) is a nonlinear term with critical growth condition with respect to u. The source term f is assumed to belong to Lp’(0, T, W0-1, p’(Ω, ω*)).

    Keywords: Weighted Sobolev Spaces, Boundary Value problems, parabolic problems, nonlinear equation, Compactness, a time mollification sequence, measurable, Compact set, Continuous, a symmetric bilinear, weak convergence, Holders inequality, strongly solution, integral, continuity, weak solution, converges strongly, positive constant.

    Pages: 119 – 151 | Full PDF Paper