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 & Continuous Dynamical Systems - A, 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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[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.  Google Scholar

[27]

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

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  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[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.  Google Scholar

[27]

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

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