Abstract
This paper is concerned with fast diffusion equations for coupling via nonlinear boundary flux. By means of the theory of linear equations and constructing self-similar super-solutions and sub-solutions, we obtain a critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results. In addition, we show that the constant $\varepsilon_{0}$ of the linear system $$A(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{s})^{\mathrm{T}} = (\varepsilon_{0}, \varepsilon_{0}, \ldots, \varepsilon_{0})^{\mathrm{T}} $$ plays an important role in our discussion.
Highlights
In this paper, we investigate the existence and non-existence of global weak solutions to the following porous medium equations:(ui)t = umi i xx, i =, . . . , s, x >, < t < T ( . )coupled via nonlinear boundary flux– umi i x(, t) = upi i (, t)uqi+i (, t), i =, . . . , s, us+ := u, < t < T, with continuous, nonnegative initial data ui(x, ) = u i(x), i =, . . . , s, x >The particular feature of equations ( . ) is their gradient-dependent diffusivity
Such equations can be used to provide a model for nonlinear heat propagation, they appear in several branches of applied mathematics such as plasma physics, population dynamics, Ling Journal of Inequalities and Applications (2015) 2015:175 chemical reactions, and so on
These equations are called the Newtonian filtration equations, which have been intensively studied since the last century
Summary
Local in time existence of weak solution Let T be the maximal existence time of a solution The problem of determining critical Fujita exponents is very interesting for various nonlinear parabolic equations of mathematical physics. In , Quirós and Rossi [ ] considered the following degenerate equations coupled via variational nonlinear boundary flux (s = ): The solutions may blow up in a finite time.
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