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May  2018, 17(3): 1161-1178. doi: 10.3934/cpaa.2018056

Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities

Departamento de Matemáticas; Universidad Carlos Ⅲ de Madrid, Avda. de la Universidad 30, 28911 Leganés, Spain

* Corresponding author

Received  January 2017 Revised  February 2017 Published  January 2018

Fund Project: Work partially supported by Spanish project MTM2011-25287.

We study the short and large time behaviour of solutions of nonlocal heat equations of the form $\partial_tu+\mathcal{L} u = 0$. Here $\mathcal{L}$ is an integral operator given by a symmetric nonnegative kernel of Lévy type, that includes bounded and unbounded transition probability densities. We characterize when a regularizing effect occurs for small times and obtain $L^q$-$L^p$ decay estimates, $1≤ q < p < ∞$ when the time is large. These properties turn out to depend only on the behaviour of the kernel at the origin or at infinity, respectively, without need of any information at the other end. An equivalence between the decay and a restricted Nash inequality is shown. Finally we deal with the decay of nonlinear nonlocal equations of porous medium type $\partial_tu+\mathcal{L}Φ(u) = 0$.

Citation: Cristina Brändle, Arturo De Pablo. Nonlocal heat equations: Regularizing effect, decay estimates and Nash inequalities. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1161-1178. doi: 10.3934/cpaa.2018056
References:
[1]

M. T. BarlowR. F. BassZ.-Q. Chen and M. Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc., 361 (2009), 1963-1999.  doi: 10.1090/S0002-9947-08-04544-3.

[2]

B. BarriosI. PeralF. Soria and E. Valdinoci, A Widder's type theorem for the heat equation with nonlocal diffusion, Arch. Ration. Mech. Anal., 213 (2014), 629-650.  doi: 10.1007/s00205-014-0733-1.

[3]

R. F. Bass and D. A. Levin, Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc., 354 (2002), 2933-2953 (electronic).  doi: 10.1090/S0002-9947-02-02998-7.

[4]

J. Bertoin, Lévy Processes, vol. 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996.

[5]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, vol. 27 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1989.

[6]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[7]

E. A. CarlenS. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Probab. Statist., 23 (1987), 245-287. 

[8]

E. ChasseigneP. FelmerJ. D. Rossi and E. Topp, Fractional decay bounds for nonlocal zero order heat equations, Bull. Lond. Math. Soc., 46 (2014), 943-952.  doi: 10.1112/blms/bdu042.

[9]

E. ChasseigneM. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.  doi: 10.1016/j.matpur.2006.04.005.

[10]

T. Coulhon, Ultracontractivity and Nash type inequalities, J. Funct. Anal., 141 (1996), 510-539.  doi: 10.1006/jfan.1996.0140.

[11]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284.  doi: 10.1002/cpa.21408.

[12]

M. Fukushima, Dirichlet Forms and Markov Processes, vol. 23 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, 1980.

[13]

L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, J. Math. Pures Appl., 92 (2009), 163-187.  doi: 10.1016/j.matpur.2009.04.009.

[14]

M. Kassmann and A. Mimica, Intrinsic scaling properties for nonlocal operators, J. Eur. Math. Soc., 19 (2017), 983-1011.  doi: 10.4171/JEMS/686.

[15]

M. Kassmann and R. W. Schwab, Regularity results for nonlocal parabolic equations, Riv. Math. Univ. Parma (N.S.), 5 (2014), 183-212. 

[16]

T. Komatsu, Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms, Osaka J. Math., 25 (1988), 697-728. 

[17]

A. Mimica, Heat kernel upper estimates for symmetric jump processes with small jumps of high intensity, Potential Anal., 36 (2012), 203-222.  doi: 10.1007/s11118-011-9225-1.

[18]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.  doi: 10.2307/2372841.

[19]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[20]

E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, Ⅲ.

[21]

D. W. Stroock, An Introduction to the Theory of Large Deviations, Universitext, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4613-8514-1.

[22]

P. Sztonyk, Transition density estimates for jump Lévy processes, Stochastic Process. Appl., 121 (2011), 1245-1265.  doi: 10.1016/j.spa.2011.03.002.

[23]

N. T. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal., 63 (1985), 240-260.  doi: 10.1016/0022-1236(85)90087-4.

show all references

References:
[1]

M. T. BarlowR. F. BassZ.-Q. Chen and M. Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc., 361 (2009), 1963-1999.  doi: 10.1090/S0002-9947-08-04544-3.

[2]

B. BarriosI. PeralF. Soria and E. Valdinoci, A Widder's type theorem for the heat equation with nonlocal diffusion, Arch. Ration. Mech. Anal., 213 (2014), 629-650.  doi: 10.1007/s00205-014-0733-1.

[3]

R. F. Bass and D. A. Levin, Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc., 354 (2002), 2933-2953 (electronic).  doi: 10.1090/S0002-9947-02-02998-7.

[4]

J. Bertoin, Lévy Processes, vol. 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996.

[5]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, vol. 27 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1989.

[6]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[7]

E. A. CarlenS. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Probab. Statist., 23 (1987), 245-287. 

[8]

E. ChasseigneP. FelmerJ. D. Rossi and E. Topp, Fractional decay bounds for nonlocal zero order heat equations, Bull. Lond. Math. Soc., 46 (2014), 943-952.  doi: 10.1112/blms/bdu042.

[9]

E. ChasseigneM. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.  doi: 10.1016/j.matpur.2006.04.005.

[10]

T. Coulhon, Ultracontractivity and Nash type inequalities, J. Funct. Anal., 141 (1996), 510-539.  doi: 10.1006/jfan.1996.0140.

[11]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284.  doi: 10.1002/cpa.21408.

[12]

M. Fukushima, Dirichlet Forms and Markov Processes, vol. 23 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, 1980.

[13]

L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, J. Math. Pures Appl., 92 (2009), 163-187.  doi: 10.1016/j.matpur.2009.04.009.

[14]

M. Kassmann and A. Mimica, Intrinsic scaling properties for nonlocal operators, J. Eur. Math. Soc., 19 (2017), 983-1011.  doi: 10.4171/JEMS/686.

[15]

M. Kassmann and R. W. Schwab, Regularity results for nonlocal parabolic equations, Riv. Math. Univ. Parma (N.S.), 5 (2014), 183-212. 

[16]

T. Komatsu, Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms, Osaka J. Math., 25 (1988), 697-728. 

[17]

A. Mimica, Heat kernel upper estimates for symmetric jump processes with small jumps of high intensity, Potential Anal., 36 (2012), 203-222.  doi: 10.1007/s11118-011-9225-1.

[18]

J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954.  doi: 10.2307/2372841.

[19]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.

[20]

E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, Ⅲ.

[21]

D. W. Stroock, An Introduction to the Theory of Large Deviations, Universitext, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4613-8514-1.

[22]

P. Sztonyk, Transition density estimates for jump Lévy processes, Stochastic Process. Appl., 121 (2011), 1245-1265.  doi: 10.1016/j.spa.2011.03.002.

[23]

N. T. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal., 63 (1985), 240-260.  doi: 10.1016/0022-1236(85)90087-4.

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