Abstract
Abstract In this article, a new type of comparison theorem for some second-order nonlinear parabolic systems with nonlinear boundary conditions is given, which can cover classical linear boundary conditions, such as the homogeneous Dirichlet or Neumann boundary condition. The advantage of our comparison theorem over the classical ones lies in the fact that it enables us to compare two solutions satisfying different types of boundary conditions. As an application of our comparison theorem, we can give some new results on the existence of blow-up solutions of some parabolic equations and systems with nonlinear boundary conditions.
Highlights
Mathematical models for various types of phenomena arising from physics, chemistry, biology and so on are often described as reaction diffusion equations which give typical examples of second-order nonlinear parabolic equations
It is widely known that comparison theorems yield very powerful tools for analyzing the second-order parabolic equations, e.g., for constructing super-solutions or sub-solutions, and for examining the asymptotic behavior of solutions
On the other hand, when one fixes right boundary conditions for the heat equations, it should be noted that if no artificial control of flux is given on the boundary, it is natural to consider the nonlinear boundary conditions from a physical point of view
Summary
Mathematical models for various types of phenomena arising from physics, chemistry, biology and so on are often described as reaction diffusion equations which give typical examples of second-order nonlinear parabolic equations. Most of the existing results on comparison theorems for nonlinear diffusion equations deal with the standard linear boundary conditions such as Dirichlet or Neumann boundary conditions (see [1]). It is well known that the subdifferential operator becomes a maximal monotone operator on H (see [3–5]) The problem with this type of boundary conditions appears in models describing diffusion phenomena taking into consideration some nonlinear radiation law on the boundary (see Brézis [5] and Barbu [3]) and the solvability for (P) is examined in detail under various settings (see [3–6]). Comparison theorem with NBC 1167 arises naturally in our setting, by which we are motivated to deal with multivalued mappings in the boundary conditions of (P). We do not apparently need this assumption to derive our comparison theorem, since the existence of solutions satisfying (1.3) is always assumed in our setting
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