Geometrical properties of the solutions of one-dimensional nonlinear parabolic equations. (English) Zbl 0842.35006

It is well known that the nonnegative solutions \(u(x,t)\) of certain quasilinear parabolic equations, like the heat equation \(u_t = u_{xx}\) and the porous medium equation \(u_t = (u^m)_{xx}\), \(m > 1\), have certain concavity or convexity properties as functions of \(x \in \mathbb{R}\). We show that such properties can be derived from the existence of a certain set \({\mathcal B}\) of explicit solutions with suitable properties of completeness, continuity and monotonicity. The analysis is based on the technique of intersection comparison.
Among the results, we prove that in terms of a suitable dependent variable (the pressure) the initial concavity is preserved in time. This uses comparison with a set \({\mathcal B}\) of solutions which are piecewise-linear in \(x\). We also prove eventual concavity for \(t \gg 1\) for arbitrary compactly supported solutions by a similar method. An important point is that such properties hold for a wide class of equations, like the general filtration equation \(u_t = (\varphi (u))_{xx}\), with increasing \(\varphi\); and also for the equation with gradient-dependent diffusivity \(u_t = (|u_x |^\sigma u_x)_x\), \(\sigma > 0\), and some equations with lower-order terms accounting for absorption, reaction or convection effects. The present technique gives a unified approach to a host of such results, emphasizing their common geometric content.
Another direction is the generalization of the technique to solution sets \({\mathcal B}\) with a more complicated space structure. Then the simple geometrical interpretation is lost but we still obtain so-called \(B\)-concavity \((B\)-convexity), concepts defined with respect to the given functional set \({\mathcal B}\). This property can be conveniently phrased as an estimate of the second derivative \(v_{xx}\) in terms of \(v_x\), \(v\) and \(t\). Estimates of this kind have been useful e.g. in the study of blow-up problems.


35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
Full Text: DOI EuDML


[1] [An] S. Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math.390 (1988), 79-96 · Zbl 0644.35050
[2] [A] D.G. Aronson, Regularity properties of flows through porous media, SIAM J. Appl. Math.17 (1969), 461-467 · Zbl 0187.03401
[3] [AB] D.G. Aronson and Ph. B?nilan, R?gularit? des solutions de l’?quation des milieux poreux dansR N , C.R. Acad. Sci. Paris. Ser. I288 (1979), 103-105
[4] [AV] D.G. Aronson and J.L. Vazquez, EventualC ?-regularity and concavity for flows in one-dimensional porous media, Arch. Ration. Mech. Anal.99 (1987), 329-348 · Zbl 0642.76108
[5] [AP] F.V. Atkinson and L.A. Peletier, Similarity solutions of the nonlinear diffusion equation, Arch. Ration. Mech. Anal.54 (1974), 373-392 · Zbl 0293.35039
[6] [BC] Ph. B?nilan and M.G. Crandall, The continuous dependence on ? of the solutions ofu t =??(u), Indiana Univ. Math. J.30 (1981), 161-177 · Zbl 0482.35012
[7] [BCP] Ph. B?nilan, M.G. Crandall and M. Pierre, Solutions of the porous media equation inR N under optimal conditions on initial values, Indiana Univ. Math. J.33 (1984), 51-87 · Zbl 0552.35045
[8] [BV] Ph. B?nilan and J.L. Vazquez, Concavity of solutions of the porous medium equation, Trans. Amer. Math. Soc.299 (1987), 81-93 · Zbl 0628.76092
[9] [Br] H. Brezis, Op?rateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, North Holland, Amsterdam, 1973
[10] [G] V.A. Galaktionov, Invariant subspaces and new explicit solutions to evolution equations with quadratic nonlinearities, Proc. Roy. Soc. Edinburgh 125A (1995), 225-246 (Report No. AM-91-11, School of Math., Univ. of Bristol, 1991) · Zbl 0824.35128
[11] [GP1] V.A. Galaktionov and S.A. Posashkov, Applications of a new comparison theorem in the investigations of unbounded solutions of nonlinear parabolic equations, Differentsial’nye Uravneniya22 (1986), 1165-1173 (in Russian). English translation: Differ. Equat.22 (1986), 809-815
[12] [GP2] V.A. Galaktionov and S.A. Posashkov, Any large solution of nonlinear heat conduction equation becomes monotone in time, Proc. Roy. Soc. Edinburgh118A (1991), 13-20 · Zbl 0741.35004
[13] [GP3] V.A. Galaktionov and S.A. Posashkov, Single point blow-up forN-dimensional quasilinear equations with gradient diffusion and source, Indiana Univ. Math. J.40 (1991), 1041-1060 · Zbl 0767.35033
[14] [GP4] V.A. Galaktionov and S.A. Posashkov, Explicit solutions and invariant subspaces for nonlinear equations with gradient-dependent diffusivity, Comp. Maths Math. Phys.34 (1994), 313-321
[15] [GV] V.A. Galaktionov and J.L. Vazquez, Extinction for a quasilinear heat equation with absorption I. Technique of intersection comparison, Comm. Partial Diff. Equat.19 (1994), 1075-1106 · Zbl 0831.35073
[16] [K1] A.S. Kalashnikov, The Cauchy problem in a class of growing functions for equations of nonstationary filtration type, Vestnik Moscow Univ., Ser. 1, Math., Mech.6 (1973), 17-27 (in Russian)
[17] [K2] A.S. Kalashnikov, On the Cauchy problem in classes of growing initial functions for some quasilinear degenerate parabolic equations, Differentsial’nye Uravneniya9 (1973), 682-691 (in Russian)
[18] [K3] A.S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate parabolic second-order equations, Uspehi Matem. Nauk42 (1987), 135-176 (in Russian). English translation: Russian Math. Surveys42 (1987), 169-222
[19] [KP] K. Kunisch and G. Peichl, On the shape of the solutions of second-order parabolic differential equations, J. Diff. Equat.75 (1988), 329-353 · Zbl 0676.35037
[20] [M] H. Matano, Nonincrease of the lap number of a solution of a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo, Sect. IA29 (1982), 401-441 · Zbl 0496.35011
[21] [N] K. Nickel, Gestaltaussagen ?ber L?sungen parabolischer Differentialgleichungen, J. Reine Angew. Math.211 (1962), 78-94 · Zbl 0127.31801
[22] [OKC] O.A. Ole?nik, A.S. Kalashnikov and Y.-L. Chou, The Cauchy problem and boundary value problems for equations of the type of nonstationary filtration, Izv. Acad. Nauk SSSR, Ser. Mat.22 (1958), 667-704 (in Russian) · Zbl 0093.10302
[23] [PV] A. de Pablo and J.L. Vazquez, The balance between strong reaction and slow diffusion, Comm. Partial Diff. Equat.15 (1990), 159-183 · Zbl 0705.35072
[24] [SGKM] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov and A.P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Nauka, Moscow, 1987 (in Russian). English translation. Walter de Gruyter, Berlin, 1995
[25] [SZKM] A.A. Samarskii, N.V. Zmitrenko, S.P. Kurdyumov and A.P. Mikhailov, Thermal structures and fundamental length in a medium with non-linear heat conduction and volumetric heat sources, Doklady AN SSSR, Ser. Math. Phys.227 (1976), 321-324 (in Russian). English translation: Soviet Phys. Dokl.,21 (1976), 141-143
[26] [S] D.H. Sattinger, On the total variation of solutions of parabolic equations, Math. Ann.183 (1969), 78-92 · Zbl 0176.40501
[27] [St] C. Sturm, M?moire sur une classe d’?quations ? diff?rences partielles, J. Math. Pure Appl.1 (1836), 373-444
[28] [VI] J.L. Vazquez, Convexity properties of the solutions of nonlinear heat equations, in ?Contributions to Nonlinear Partial Differential Equations?, vol. II, J.I. D?az and P.L. Lions eds., Pitman Research Notes in Math., Longman, 1987, 267-275
[29] [V2] J.L. Vazquez, Singular solutions and asymptotic behaviour of nonlinear parabolic equations, in ?Proceedings of Equadiff91?, Vol. 1, World Scientific, Singapore/Hong Kong, 1993, 234-249
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.