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**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.

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.

Reviewer: J.L.Vazquez (Madrid)

### MSC:

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

35K55 | Nonlinear parabolic equations |

35K65 | Degenerate parabolic equations |

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\textit{V. A. Galaktionov} and \textit{J. L. Vazquez}, Math. Ann. 303, No. 4, 741--769 (1995; Zbl 0842.35006)

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