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A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients. (English) Zbl 1180.35305
The uniqueness question for a nonlinear strongly degenerate parabolic convection-diffusion initial-value problem is discussed. Here \(u(x,t)\) is the solution to the problem
\[ \frac{\partial u}{\partial t} + \frac{\partial}{\partial x}(F(u,x)-\frac{\partial}{\partial x}D(u,x))=0,\quad u(x,0)=u_0(x),\quad x\in R,\;t\in (0,T), \] with the discontinuous flux and diffusion functions
\[ F(u,x)=\begin{cases} f(u),&x>0,\\g(u),&x<0, \end{cases} \qquad D(u,x)=\begin{cases} A(u),&x>0,\\B(u),&x<0.\end{cases} \] The author presents an interface entropy condition at \(x=0\) which implies uniqueness for the problem. The time-independent piecewise smooth solution is constructed for the geometric interpretation of the uniqueness condition.

MSC:
35K65 Degenerate parabolic equations
35L65 Hyperbolic conservation laws
35R05 PDEs with low regular coefficients and/or low regular data
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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