June  2020, 40(6): 3057-3073. doi: 10.3934/dcds.2020046

Evolution equations involving nonlinear truncated Laplacian operators

1. 

IMAG, Univ. Montpellier, CNRS, Montpellier, France

2. 

Dipartimento di Matematica Guido Castelnuovo, Sapienza Universitaà di Roma, 00185, Roma, Italia

* Corresponding author

Received  March 2019 Published  October 2019

Fund Project: Matthieu Alfaro is supported by the ANR I-SITE MUSE, project MICHEL 170544IA (n° ANR IDEX-0006) I. Birindelli is supported by INDAM-Gnampa and Ateneo Sapienza

We first study the so-called Heat equation with two families of elliptic operators which are fully nonlinear, and depend on some eigenvalues of the Hessian matrix. The equation with operators including the "large" eigenvalues has strong similarities with a Heat equation in lower dimension whereas, surprisingly, for operators including "small" eigenvalues it shares some properties with some transport equations. In particular, for these operators, the Heat equation (which is nonlinear) not only does not have the property that "disturbances propagate with infinite speed" but may lead to quenching in finite time. Last, based on our analysis of the Heat equations (for which we provide a large variety of special solutions) for these operators, we inquire on the associated Fujita blow-up phenomena.

Citation: Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964

[2]

M. Alfaro, Fujita blow up phenomena and hair trigger effect: The role of dispersal tails, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1309-1327.  doi: 10.1016/j.anihpc.2016.10.005.

[3]

L. Ambrosio and H. M. Soner, Level set approach to mean curvature flow in arbitrary codimension, J. Differential Geom., 43 (1996), 693-737.  doi: 10.4310/jdg/1214458529.

[4]

I. BirindelliG. Galise and H. Ishii, A family of degenerate elliptic operators: Maximum principle and its consequences, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 417-441.  doi: 10.1016/j.anihpc.2017.05.003.

[5]

I. Birindelli, G. Galise and H. Ishii, Towards a reversed Faber-Krahn inequality for the truncated laplacian, preprint, (2018), arXiv: 1803.07362.

[6]

I. BirindelliG. Galise and F. Leoni, Liouville theorems for a family of very degenerate elliptic nonlinear operators, Nonlinear Anal., 161 (2017), 198-211.  doi: 10.1016/j.na.2017.06.002.

[7]

P. Blanc, C. Esteve and J. D. Rossi, The evolution problem associated with eigenvalues of the Hessian, preprint, (2019), arXiv: 1901.01052.

[8]

P. Blanc and J. D. Rossi, Games for eigenvalues of the Hessian and concave/convex envelopes, preprint, (2018), arXiv: 1801.03383. doi: 10.1016/j.matpur.2018.08.007.

[9]

L. CaffarelliY. Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations Ⅲ: Viscosity solutions including parabolic operators, Comm. Pure Appl. Math., 66 (2013), 109-143.  doi: 10.1002/cpa.21412.

[10]

M. G. Crandall and P.-L. Lions, Quadratic growth of solutions of fully nonlinear second order equations in $R^n$, Differential Integral Equations, 3 (1990), 601-616. 

[11]

M. G. CrandallH. Ishii and P.-L. Lions, User guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[12]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681.  doi: 10.4310/jdg/1214446559.

[13]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u + u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124. 

[14]

G. Galise, On positive solutions of fully nonlinear degenerate Lane-Emden type equations, J. Differential Equations, 266 (2019), 1675-1697.  doi: 10.1016/j.jde.2018.08.014.

[15]

M.-H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-Similar Solutions, Progress in Nonlinear Differential Equations and their Applications, 79. Birkhäuser Boston, Inc., Boston, MA, 2010. doi: 10.1007/978-0-8176-4651-6.

[16]

C. F. GuiW.-M. Ni and X. F. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $R^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.

[17]

A. Haraux and F. B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167-189.  doi: 10.1512/iumj.1982.31.31016.

[18]

HarveyLawson and Jr., Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math., 62 (2009), 396-443.  doi: 10.1002/cpa.20265.

[19]

HarveyLawson and Jr., $p$-convexity, $p$-plurisubharmonicity and the Levi problem, Indiana Univ. Math. J., 62 (2013), 149-169.  doi: 10.1512/iumj.2013.62.4886.

[20]

S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330.  doi: 10.1002/cpa.3160160307.

[21]

O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 423-452.  doi: 10.1016/S0294-1449(16)30358-4.

[22]

R. Meneses and A. Quaas, Fujita type exponent for fully nonlinear parabolic equations and existence results, J. Math. Anal. Appl., 376 (2011), 514-527.  doi: 10.1016/j.jmaa.2010.10.049.

[23]

A. M. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc., 363 (2011), 5871-5886.  doi: 10.1090/S0002-9947-2011-05240-2.

[24]

J.-P. Sha, $p$-convex Riemannian manifolds, Invent. Math., 83 (1986), 437-447.  doi: 10.1007/BF01394417.

[25]

G. Szegö, Orthogonal Polynomials, Fourth edition, American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., 1975.

[26]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40.  doi: 10.1007/BF02761845.

[27]

H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J., 36 (1987), 525-548.  doi: 10.1512/iumj.1987.36.36029.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964

[2]

M. Alfaro, Fujita blow up phenomena and hair trigger effect: The role of dispersal tails, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1309-1327.  doi: 10.1016/j.anihpc.2016.10.005.

[3]

L. Ambrosio and H. M. Soner, Level set approach to mean curvature flow in arbitrary codimension, J. Differential Geom., 43 (1996), 693-737.  doi: 10.4310/jdg/1214458529.

[4]

I. BirindelliG. Galise and H. Ishii, A family of degenerate elliptic operators: Maximum principle and its consequences, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 417-441.  doi: 10.1016/j.anihpc.2017.05.003.

[5]

I. Birindelli, G. Galise and H. Ishii, Towards a reversed Faber-Krahn inequality for the truncated laplacian, preprint, (2018), arXiv: 1803.07362.

[6]

I. BirindelliG. Galise and F. Leoni, Liouville theorems for a family of very degenerate elliptic nonlinear operators, Nonlinear Anal., 161 (2017), 198-211.  doi: 10.1016/j.na.2017.06.002.

[7]

P. Blanc, C. Esteve and J. D. Rossi, The evolution problem associated with eigenvalues of the Hessian, preprint, (2019), arXiv: 1901.01052.

[8]

P. Blanc and J. D. Rossi, Games for eigenvalues of the Hessian and concave/convex envelopes, preprint, (2018), arXiv: 1801.03383. doi: 10.1016/j.matpur.2018.08.007.

[9]

L. CaffarelliY. Y. Li and L. Nirenberg, Some remarks on singular solutions of nonlinear elliptic equations Ⅲ: Viscosity solutions including parabolic operators, Comm. Pure Appl. Math., 66 (2013), 109-143.  doi: 10.1002/cpa.21412.

[10]

M. G. Crandall and P.-L. Lions, Quadratic growth of solutions of fully nonlinear second order equations in $R^n$, Differential Integral Equations, 3 (1990), 601-616. 

[11]

M. G. CrandallH. Ishii and P.-L. Lions, User guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.

[12]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681.  doi: 10.4310/jdg/1214446559.

[13]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u + u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109-124. 

[14]

G. Galise, On positive solutions of fully nonlinear degenerate Lane-Emden type equations, J. Differential Equations, 266 (2019), 1675-1697.  doi: 10.1016/j.jde.2018.08.014.

[15]

M.-H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-Similar Solutions, Progress in Nonlinear Differential Equations and their Applications, 79. Birkhäuser Boston, Inc., Boston, MA, 2010. doi: 10.1007/978-0-8176-4651-6.

[16]

C. F. GuiW.-M. Ni and X. F. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $R^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.

[17]

A. Haraux and F. B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J., 31 (1982), 167-189.  doi: 10.1512/iumj.1982.31.31016.

[18]

HarveyLawson and Jr., Dirichlet duality and the nonlinear Dirichlet problem, Comm. Pure Appl. Math., 62 (2009), 396-443.  doi: 10.1002/cpa.20265.

[19]

HarveyLawson and Jr., $p$-convexity, $p$-plurisubharmonicity and the Levi problem, Indiana Univ. Math. J., 62 (2013), 149-169.  doi: 10.1512/iumj.2013.62.4886.

[20]

S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330.  doi: 10.1002/cpa.3160160307.

[21]

O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 423-452.  doi: 10.1016/S0294-1449(16)30358-4.

[22]

R. Meneses and A. Quaas, Fujita type exponent for fully nonlinear parabolic equations and existence results, J. Math. Anal. Appl., 376 (2011), 514-527.  doi: 10.1016/j.jmaa.2010.10.049.

[23]

A. M. Oberman and L. Silvestre, The Dirichlet problem for the convex envelope, Trans. Amer. Math. Soc., 363 (2011), 5871-5886.  doi: 10.1090/S0002-9947-2011-05240-2.

[24]

J.-P. Sha, $p$-convex Riemannian manifolds, Invent. Math., 83 (1986), 437-447.  doi: 10.1007/BF01394417.

[25]

G. Szegö, Orthogonal Polynomials, Fourth edition, American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., 1975.

[26]

F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40.  doi: 10.1007/BF02761845.

[27]

H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J., 36 (1987), 525-548.  doi: 10.1512/iumj.1987.36.36029.

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