CONCAVITY AND B-CONCAVITY OF SOLUTIONS OF QUASILINEAR FILTRATION EQUATIONS

Abstract

Spatial concavity properties of non-negative weak solutions of the filtration equations with absorption u_t = (φ(u))_xx−ψ(u) in Q = R×(0, ∞), φ′[ges ]0, ψ[ges ]0 are studied. Under certain assumptions on the coefficients φ, ψ it is proved that concavity of the pressure function is a consequence of a ‘weak’ convexity of travelling-wave solutions of the form V(x, t) = θ(x−λt+a). It is established that the global structure of a so-called proper set [Bscr ] = {V} of such particular solutions determines a property of B-concavity for more general solutions which is preserved in time. For the filtration equation u_t = (φ(u))_xx a semiconcavity estimate for the pressure, v_xx[les ](t+τ)⁻¹θ″(ξ), due to the B-concavity of the solution to the subset [Bscr ] of the explicit self-similar solutions θ(x/√(t+τ)) is proved.

The analysis is based on the intersection comparison based on the Sturmian argument of the general solution u(x, t) with subsets [Bscr ] of particular solutions. Also studied are other aspects of the B-concavity/convexity with respect to different subsets of explicit solutions.