×

Local and global estimates of the solutions of the Cauchy problem for quasilinear parabolic equations with a nonlinear operator of Baouendi-Grushin type. (English. Russian original) Zbl 1182.35146

Math. Notes 85, No. 3, 385-396 (2009); translation from Mat. Zametki 85, No. 3, 395-407 (2009).
The authors study the solution of the Cauchy problem for a quasilinear degenerate parabolic equation of the following form:
\[ \begin{cases}\frac{\partial u}{\partial t}=L_{\lambda ,\alpha}[u]= \text{div}_L(|D_Lu|^{\lambda -1} D_Lu), &(x,y,t)\in \mathbb{R}^{N+M}\times(0,T), \\ u(x,y,0)= u_0(x,y), &x\in \mathbb{R}^N,\;y\in \mathbb{R} ^M,\\ u_0(x,y)\geq 0&\text{a.e. }x\in\mathbb{R}^N,\;y\in\mathbb{R}^M. \end{cases} \]
Here \(\lambda >1,N\geq 1,M\geq 1.\)
\[ \begin{aligned} D_Lu &= \left( \frac{\partial u}{\partial x_1}, \frac{\partial u}{\partial x_2} ,\dots,\frac{\partial u}{\partial x_N},|x|^\alpha \frac{\partial u }{\partial y_1},|x|^\alpha \frac{\partial u}{\partial y_2} ,\dots,|x|^\alpha \frac{\partial u}{\partial y_M}\right),\\ |D_Lu| &= \sqrt{\sum_{i=1}^N\left( \frac{\partial u}{ \partial x_i}\right) ^2+|x|^{2\alpha }\sum_{j=1}^M\left( \frac{\partial u}{\partial y_j}\right) ^2},\\ \text{div}_L\overrightarrow{F}(x,y) &= \sum_{i=1}^N\frac{\partial F_i}{\partial x_i}+|x|^\alpha \sum_{j=1}^M \frac{\partial F_{j+N}}{\partial y_j}. \end{aligned} \]
The authors study the qualitative properties of the solution of \((1)\) where the operator is of Baouendi-Grushin type. They derive sharp local and global (with respect to the spatial and temporal variables) estimates of the solutions of \((1)\). The authors establish also the property of the finiteness of the support of the solution.

MSC:

35K59 Quasilinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35K65 Degenerate parabolic equations
35B45 A priori estimates in context of PDEs
Full Text: DOI

References:

[1] A. S. Kalashnikov, ”Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations,” UspekhiMat. Nauk 42(2), 135–176 (1987) [RussianMath. Surveys 42 (2), 169–222 (1987)].
[2] V. V. Grushin, ”A certain class of hypoelliptic operators,” Mat. Sb. 83(3), 456–473 (1970).
[3] M. S. Baouendi, ”Sur une classe d’opérateurs elliptiques dégénérés,” Bull. Soc. Math. France 95, 45–87 (1967). · Zbl 0179.19501 · doi:10.24033/bsmf.1647
[4] B. Franchi and E. Lanconelli, ”Une métrique associée à une classe d’opérateurs elliptiques dégénérés,” in Conference on Linear Partial and Pseudodifferential Operators, Rend. Sem.Mat. Univ. Politec. Torino, Torino, 1982 (Univ. Politec. Torino, Torino, 1984), Vol. 1983, pp. 105–114.
[5] B. Franchi and E. Lanconelli, ”Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10(4), 523–541 (1983). · Zbl 0552.35032
[6] N. Garofalo and D.-M. Nhieu, ”Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces,” Comm. Pure Appl. Math. 49(10), 1081–1144 (1996). · Zbl 0880.35032 · doi:10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A
[7] E. Di Benedetto and M. A. Herrero, ”On the Cauchy problem and initial traces for a degenerate parabolic equation,” Trans. Amer. Math. Soc. 314(1), 187–224 (1989). · Zbl 0691.35047 · doi:10.1090/S0002-9947-1989-0962278-5
[8] S. N. Antontsev, ”On the localization of solutions of nonlinear degenerate elliptic and parabolic equations,” Dokl. Akad. Nauk SSSR 260(6), 1289–1293 (1981) [SovietMath. Dokl. 24, 420–424 (1981)].
[9] J. I. Díaz and L. Véron, ”Local vanishing properties of solutions of elliptic and parabolical quasilinear equations,” Trans. Amer.Math. Soc. 290(2), 787–814 (1985). · Zbl 0579.35003 · doi:10.2307/2000315
[10] D. Andreucci and A. F. Tedeev, ”Universal bounds at the blow-up time for nonlinear parabolic equations,” Adv. Differential Equations 10(1), 89–120 (2005). · Zbl 1122.35042
[11] D. Andreucci and A. F. Tedeev, ”Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity,” Adv. Differential Equations 5(7–9), 833–860 (2000). · Zbl 0987.35090
[12] D. Andreucci and A. F. Tedeev, ”Finite speed of propagation for thin film equations and other higher order parabolic equations with general nonlinearity,” Interfaces Free Bound. 3(3), 233–264 (2001). · Zbl 1002.35059 · doi:10.4171/IFB/40
[13] O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Fizmatlit, Moscow, 1967) [in Russian].
[14] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires (Dunod, Paris, 1969; MirMoscow, 1972).
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.