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Optimal matching problems with costs given by Finsler distances
Optimal reaction exponent for some qualitative properties of solutions to the $p$-heat equation
1. | Dipartimento di Matematica e Informatica, UNICAL, Ponte Pietro Bucci, 31 B, 87036, Arcavacata di Rende, Cosenza, Italy |
2. | Dipartimento di Matematica, UNICAL, Ponte Pietro Bucci 31 B, 87036 Arcavacata di Rende, Cosenza |
3. | Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid |
If $1 < p < 2$ and $0 < q < p-1$, there is no finite extinction time.
If $p > 2$ and $0 < q< 1$, there is no finite speed of propagation. In both cases the result is optimal.
References:
[1] |
B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potencial, Ann. Mat. Pura Appl (4), 182 (2003), no 3, 247-270.
doi: 10.1007/s10231-002-0064-y. |
[2] |
J. A. Aguilar and I. Peral, Global behavior of the Cauchy problem for some critical nonlinear parabolic equations, SIAM J. Math. Anal., 31 (2000), 1270-1294.
doi: 10.1137/S0036141098341137. |
[3] |
D. Andreucci and A. F. Tedeev, A Fujita type result for a degenerate Neumann problem in domains with noncompact boundary, J. Math. Anal. Appl., 231 (1999), 543-567.
doi: 10.1006/jmaa.1998.6253. |
[4] |
F. Bernis, Finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption, Proc. Roy. Soc. Edinburgh. Sect. A, 104 (1986), 1-19.
doi: 10.1017/S030821050001903X. |
[5] |
L. Boccardo, M. Escobedo and I. Peral, A Dirichlet problem involving critical exponents, Nonlinear Anal., 24 (1995), 1639-1648.
doi: 10.1016/0362-546X(94)E0054-K. |
[6] |
L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655.
doi: 10.1080/03605309208820857. |
[7] |
L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Analysis, 19 (1992), 581-597.
doi: 10.1016/0362-546X(92)90023-8. |
[8] |
M. Bonforte, R. G. Iagar and J. L. Vázquez, Local smoothing effects, positivity, and Harnack inequalities for the fast $p$-Laplacian equation, Adv. Math., 224 (2010), 2151-2215.
doi: 10.1016/j.aim.2010.01.023. |
[9] |
H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N2$, Manuscripta Math., 74 (1992), 87-106.
doi: 10.1007/BF02567660. |
[10] |
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[11] |
E. DiBenedetto, Intrinsic Harnack type inequalities for solutions of certain degenerate Parabolic equations, Arch. Rational Mech. Anal., 100 (1988), 129-147.
doi: 10.1007/BF00282201. |
[12] |
E. DiBenedetto, U. Gianazza and V. Vespri, Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations, Manuscripta Math., 131 (2010), 231-245.
doi: 10.1007/s00229-009-0317-9. |
[13] |
E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics. Springer, New York, 2012.
doi: 10.1007/978-1-4614-1584-8. |
[14] |
L. Euler, Nova methodus innumerabiles aequationes differentiales secundi gradus reducendi ad aequationes differentiales primi gradus, Comm. Ac. Scient. Petr. Tom. III, (1728), 124-137. |
[15] |
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. |
[16] |
H. Fujita and S. Watanabe, On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations, Comm. Pure Appl. Math, 21 (1968), 631-652. |
[17] |
V. A. Galaktionov, Conditions for nonexistence in the large and localization of solutions of the Cauchy problem for a class of nonlinear parabolic equations, Zh. Vychisl. Mat. i Mat. Fiz., 23 (1983), 1341-1354 (Russian). |
[18] |
V. A. Galaktionov and H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal., 34 (1998), 1005-1027.
doi: 10.1016/S0362-546X(97)00716-5. |
[19] |
J. García Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476.
doi: 10.1006/jdeq.1997.3375. |
[20] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 Edition, Classics in Mathematics. Springer-Verlag, Berlin, 2001. |
[21] |
R. G. Iagar, P. Laurençot, Philippe and J. L. Vázquez, Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent, Interfaces Free Bound, 13 (2011), 271-295.
doi: 10.4171/IFB/258. |
[22] |
A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Russian Math Surveys, 42 (1987), 169-222. |
[23] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Translation of Mathematical Monographs, 1968, American Mathematical Society. |
[24] |
J. L. Lions, Quelques méthodes de resolution des problèmes aux limites non linéaires, Dunod Gauthier-Villars, Paris, 1969. |
[25] |
J. J. Manfredi and V. Vespri, Large time behavior of solutions to a class of doubly nonlinear parabolic equations, Electron. J. Differential Equations, 2 (1994), 17 pp. |
[26] |
G. Savaré and V. Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations, Nonlinear Anal., 22 (1994), 1553-1565.
doi: 10.1016/0362-546X(94)90188-0. |
[27] |
A. F. Tedeev, Conditions for the time-global existence and nonexistence of a compact support of solutions of the Cauchy problem for quasilinear degenerate parabolic equations, Siberian Math. J., 45 (2004), 155-164.
doi: 10.1023/B:SIMJ.0000013021.66528.b6. |
show all references
References:
[1] |
B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potencial, Ann. Mat. Pura Appl (4), 182 (2003), no 3, 247-270.
doi: 10.1007/s10231-002-0064-y. |
[2] |
J. A. Aguilar and I. Peral, Global behavior of the Cauchy problem for some critical nonlinear parabolic equations, SIAM J. Math. Anal., 31 (2000), 1270-1294.
doi: 10.1137/S0036141098341137. |
[3] |
D. Andreucci and A. F. Tedeev, A Fujita type result for a degenerate Neumann problem in domains with noncompact boundary, J. Math. Anal. Appl., 231 (1999), 543-567.
doi: 10.1006/jmaa.1998.6253. |
[4] |
F. Bernis, Finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption, Proc. Roy. Soc. Edinburgh. Sect. A, 104 (1986), 1-19.
doi: 10.1017/S030821050001903X. |
[5] |
L. Boccardo, M. Escobedo and I. Peral, A Dirichlet problem involving critical exponents, Nonlinear Anal., 24 (1995), 1639-1648.
doi: 10.1016/0362-546X(94)E0054-K. |
[6] |
L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655.
doi: 10.1080/03605309208820857. |
[7] |
L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Analysis, 19 (1992), 581-597.
doi: 10.1016/0362-546X(92)90023-8. |
[8] |
M. Bonforte, R. G. Iagar and J. L. Vázquez, Local smoothing effects, positivity, and Harnack inequalities for the fast $p$-Laplacian equation, Adv. Math., 224 (2010), 2151-2215.
doi: 10.1016/j.aim.2010.01.023. |
[9] |
H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbb{R}^N2$, Manuscripta Math., 74 (1992), 87-106.
doi: 10.1007/BF02567660. |
[10] |
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[11] |
E. DiBenedetto, Intrinsic Harnack type inequalities for solutions of certain degenerate Parabolic equations, Arch. Rational Mech. Anal., 100 (1988), 129-147.
doi: 10.1007/BF00282201. |
[12] |
E. DiBenedetto, U. Gianazza and V. Vespri, Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations, Manuscripta Math., 131 (2010), 231-245.
doi: 10.1007/s00229-009-0317-9. |
[13] |
E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics. Springer, New York, 2012.
doi: 10.1007/978-1-4614-1584-8. |
[14] |
L. Euler, Nova methodus innumerabiles aequationes differentiales secundi gradus reducendi ad aequationes differentiales primi gradus, Comm. Ac. Scient. Petr. Tom. III, (1728), 124-137. |
[15] |
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. |
[16] |
H. Fujita and S. Watanabe, On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations, Comm. Pure Appl. Math, 21 (1968), 631-652. |
[17] |
V. A. Galaktionov, Conditions for nonexistence in the large and localization of solutions of the Cauchy problem for a class of nonlinear parabolic equations, Zh. Vychisl. Mat. i Mat. Fiz., 23 (1983), 1341-1354 (Russian). |
[18] |
V. A. Galaktionov and H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal., 34 (1998), 1005-1027.
doi: 10.1016/S0362-546X(97)00716-5. |
[19] |
J. García Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476.
doi: 10.1006/jdeq.1997.3375. |
[20] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 Edition, Classics in Mathematics. Springer-Verlag, Berlin, 2001. |
[21] |
R. G. Iagar, P. Laurençot, Philippe and J. L. Vázquez, Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent, Interfaces Free Bound, 13 (2011), 271-295.
doi: 10.4171/IFB/258. |
[22] |
A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Russian Math Surveys, 42 (1987), 169-222. |
[23] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Translation of Mathematical Monographs, 1968, American Mathematical Society. |
[24] |
J. L. Lions, Quelques méthodes de resolution des problèmes aux limites non linéaires, Dunod Gauthier-Villars, Paris, 1969. |
[25] |
J. J. Manfredi and V. Vespri, Large time behavior of solutions to a class of doubly nonlinear parabolic equations, Electron. J. Differential Equations, 2 (1994), 17 pp. |
[26] |
G. Savaré and V. Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations, Nonlinear Anal., 22 (1994), 1553-1565.
doi: 10.1016/0362-546X(94)90188-0. |
[27] |
A. F. Tedeev, Conditions for the time-global existence and nonexistence of a compact support of solutions of the Cauchy problem for quasilinear degenerate parabolic equations, Siberian Math. J., 45 (2004), 155-164.
doi: 10.1023/B:SIMJ.0000013021.66528.b6. |
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