Abstract
This paper studies the global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with appropriate initial and mixed boundary conditions. Under some practicable regularity criteria on diffusion item and nonlinearity, we establish the local existence and uniqueness of classical solutions based on a contraction mapping. This local solution can be continued for all positive time by employing the methods of energy estimates, Lp-theory, and Schauder estimate of linear parabolic equations. A straightforward application of global existence result of classical solutions to a density-dependent diffusion model of in vitro glioblastoma growth is also presented.
Highlights
In this paper, we consider the existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation of the form ut − div [A(x, t, u, ∇u)] = F (x, t, u, ∇u) in Ω × (0, ∞) (1)with initial and mixed boundary conditions u(x, 0) = u0(x) in Ω, ∂u (2)= σ(x, t, u) on ∂Ω × (0, ∞), ∂ν2010 Mathematics Subject Classification
In order to illustrate the main feature of the this study, we will explore the global existence and uniqueness of solution to some models, including an application to a data-based density-dependent diffusion model of in vitro glioblastoma growth
This paper is devoted to the study of the global-in-time solutions for a generalized quasi-linear parabolic equation with applications in biology and medicine
Summary
We consider the existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation of the form ut − div [A(x, t, u, ∇u)] = F (x, t, u, ∇u) in Ω × (0, ∞) There are practical needs for additional studies for establishing both local and global existence of classical solutions to the generalized quasilinear parabolic equations under weaker regularity or fewer growth restrictions.
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