Stability of the entropy solutions of a kind of degenerate parabolic equation
[Objective]Consider the equation of the utility function for agent's decision under financial risk uxx+uuy-ut=f(x,y,t,u),which belongs to the group of strongly degenerate parabolic equations.How to choose a suitable boundary such that the uniqueness and the stability of weak solutions are true constitutes a challenging problem.[Methods]By choosing a suitable test function,a partial boundary-value condition is found to match up with this strong degenerate parabolic equation.[Results]Such a result improves the result reported in the relevant literature.Under a partial boundary-value condition,by Kruzkov bi-variables method,we discuss the stability of BV entropy solutions of the equation.Moreover,the stability independent of any boundary value condition is discussed,and some explicit examples are given.[Conclusion]The conclusion reflects on the essential characteristic of a nonlinear degenerate parabolic equation,in which its partial boundary value condition secures a close relationship with the geometry of the domain.Hopefully,the theoretical significance of our work can contribute to the math community.
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