Concavity and B-Concavity of Solutions of Quasilinear Filtration Equations |
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Authors: | Galaktionov Victor A. |
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Affiliation: | Department of Mathematical Sciences, University of Bath Claverton Down, Bath BA2 7AY, vag{at}maths.bath.ac.uk |
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Abstract: | Spatial concavity properties of non-negative weak solutionsof the filtration equations with absorption ut = ((u))xx(u)in Q = Rx(0, ), '0, 0 are studied. Under certain assumptionson the coefficients , it is proved that concavity of the pressurefunction is a consequence of a weak convexityof travelling-wave solutions of the form V(x, t) = (xt+a).It is established that the global structure of a so-called properset B = {V} of such particular solutions determines a propertyof B-concavity for more general solutions which is preservedin time. For the filtration equation ut = ((u))xx a semiconcavityestimate for the pressure, vxx(t+)1'(), due to the B-concavityof the solution to the subset B of the explicit self-similarsolutions (x/t+)) is proved. The analysis is based on the intersection comparison based onthe Sturmian argument of the general solution u(x, t) with subsetsB of particular solutions. Also studied are other aspects ofthe B-concavity/convexity with respect to different subsetsof explicit solutions. |
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