Nonuniqueness of solutions for a class of forward–backward parabolic equations

We study the initial–boundary value problem {ut=[φ(u)]xx+ε[ψ(u)]txxinΩ×(0,∞)φ(u)+ε[ψ(u)]t=0in∂Ω×(0,∞)u=u0≥0inΩ×{0} with measure-valued initial data. Here ΩΩ is a bounded open interval, φ(0)=φ(∞)=0φ(0)=φ(∞)=0, φφ is increasing in (0,α)(0,α) and decreasing in (α,∞)(α,∞), and the regularising term ψψ is increasing but bounded. It is natural to study measure-valued solutions since singularities may appear spontaneously in finite time. Nonnegative Radon measure-valued solutions are known to exist and their construction is based on an approximation procedure. Until now nothing was known about their uniqueness.In this note we construct some nontrivial examples of solutions which do not satisfy all properties of the constructed solutions, whence uniqueness fails. In addition, we classify the steady state solutions.