On a class of forward-backward parabolic equations: Formation of singularities

Abstract

We study the formation of singularities for the problem{ut=[φ(u)]xx+ε[ψ(u)]txxinΩ×(0,T)φ(u)+ε[ψ(u)]t=0in∂Ω×(0,T)u=u0≥0inΩ×{0}, where ϵ and T are positive constants, Ω a bounded interval, u0 a nonnegative Radon measure on Ω, φ a nonmonotone and nonnegative function with φ(0)=φ(∞)=0, and ψ an increasing bounded function. We show that if u0 is a bounded or continuous function, singularities may appear spontaneously. The class of singularities which can arise in finite time is remarkably large, and includes infinitely many Dirac masses and singular continuous measures.