Concavity and B ‐Concavity of Solutions of Quasilinear Filtration Equations

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′⩾0, ψ⩾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 −λ t + a ). It is established that the global structure of a so‐called proper set B = { 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 ⩽( t +τ) −1 θ″(ξ), due to the B ‐concavity of the solution to the subset B 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 B of particular solutions. Also studied are other aspects of the B ‐concavity/convexity with respect to different subsets of explicit solutions.