Abstract
A nonlinear parabolic equation from a two-phase problem is considered in this paper. The existence of weak solutions is proved by the standard parabolically regularized method. Different from the related papers, one of diffusion coefficients in the equation, b(x), is degenerate on the boundary. Then the Dirichlet boundary value condition may be overdetermined. In order to study the stability of weak solution, how to find a suitable partial boundary value condition is the foremost work. Once such a partial boundary value condition is found, the stability of weak solutions will naturally follow.
Highlights
1 Introduction In this paper, we study the following initial-boundary value problem: ut = div |∇u |p(x)–2∇u + b(x)|∇u|q(x)–2∇u), (x, t) ∈ QT = × (0, T), u|t=0 = u0(x), x ∈, u| T = 0, (x, t) ∈ T = ∂ × (0, T), where 1 < p(x), q(x) ∈ C( ), b(x) ∈ C1( ) and satisfies b(x) > 0, x ∈, b(x) = 0, x ∈ ∂
We first noticed that the initial-boundary value problem of the equation ut = div |u|σ (x,t) + d0 |∇u|p(x,t)–2∇u + c(x, t) – b0u(x, t), (x, t) ∈ QT, (1.5)
For a general degenerate parabolic equation, the well-posedness of weak solutions based on a partial boundary value condition has been studied for a long time, relevant literature can be referred to [4, 22, 28, 29, 31–37]
Summary
If f (x, t) = 0, the author of [6] studied the existence of weak solutions to equation (1.6) by the energy functional method. We use the parabolically regularized method to prove the existence of the weak solution to equation (1.1).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
