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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,, \emph{ Ann. Mat. Pura Appl (4)}, 182 (2003), 247.
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,, \emph{SIAM J. Math. Anal., 31 (2000), 1270.
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,, \emph{J. Math. Anal. Appl.}, 231 (1999), 543.
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,, \emph{Proc. Roy. Soc. Edinburgh. Sect. A}, 104 (1986), 1.
doi: 10.1017/S030821050001903X. |
[5] |
L. Boccardo, M. Escobedo and I. Peral, A Dirichlet problem involving critical exponents,, \emph{Nonlinear Anal., 24 (1995), 1639.
doi: 10.1016/0362-546X(94)E0054-K. |
[6] |
L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures,, \emph{Comm. Partial Differential Equations, 17 (1992), 641.
doi: 10.1080/03605309208820857. |
[7] |
L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations,, \emph{Nonlinear Analysis, 19 (1992), 581.
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,, \emph{Adv. Math.}, 224 (2010), 2151.
doi: 10.1016/j.aim.2010.01.023. |
[9] |
H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$,, \emph{Manuscripta Math.}, 74 (1992), 87.
doi: 10.1007/BF02567660. |
[10] |
E. DiBenedetto, Degenerate Parabolic Equations,, Springer-Verlag, (1993).
doi: 10.1007/978-1-4612-0895-2. |
[11] |
E. DiBenedetto, Intrinsic Harnack type inequalities for solutions of certain degenerate Parabolic equations,, \emph{Arch. Rational Mech. Anal.}, 100 (1988), 129.
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,, \emph{Manuscripta Math.}, 131 (2010), 231.
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, (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,, \emph{Comm. Ac. Scient. Petr. Tom. III}, (1728), 124. Google Scholar |
[15] |
H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+\alpha}$,, \emph{J. Fac. Sci. Univ. Tokyo Sect. I}, 13 (1966), 109.
|
[16] |
H. Fujita and S. Watanabe, On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations,, \emph{Comm. Pure Appl. Math}, 21 (1968), 631.
|
[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,, \emph{Zh. Vychisl. Mat. i Mat. Fiz.}, 23 (1983), 1341.
|
[18] |
V. A. Galaktionov and H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems,, \emph{Nonlinear Anal.}, 34 (1998), 1005.
doi: 10.1016/S0362-546X(97)00716-5. |
[19] |
J. García Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems,, \emph{J. Differential Equations}, 144 (1998), 441.
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, (1998).
|
[21] |
R. G. Iagar, P. Laurençot, Philippe and J. L. Vázquez, Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent,, \emph{Interfaces Free Bound}, 13 (2011), 271.
doi: 10.4171/IFB/258. |
[22] |
A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, \emph{Russian Math Surveys}, 42 (1987), 169.
|
[23] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type,, Translation of Mathematical Monographs, (1968).
|
[24] |
J. L. Lions, Quelques méthodes de resolution des problèmes aux limites non linéaires,, Dunod Gauthier-Villars, (1969).
|
[25] |
J. J. Manfredi and V. Vespri, Large time behavior of solutions to a class of doubly nonlinear parabolic equations,, \emph{Electron. J. Differential Equations}, 2 (1994).
|
[26] |
G. Savaré and V. Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations,, \emph{Nonlinear Anal.}, 22 (1994), 1553.
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,, \emph{Siberian Math. J.}, 45 (2004), 155.
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,, \emph{ Ann. Mat. Pura Appl (4)}, 182 (2003), 247.
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,, \emph{SIAM J. Math. Anal., 31 (2000), 1270.
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,, \emph{J. Math. Anal. Appl.}, 231 (1999), 543.
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,, \emph{Proc. Roy. Soc. Edinburgh. Sect. A}, 104 (1986), 1.
doi: 10.1017/S030821050001903X. |
[5] |
L. Boccardo, M. Escobedo and I. Peral, A Dirichlet problem involving critical exponents,, \emph{Nonlinear Anal., 24 (1995), 1639.
doi: 10.1016/0362-546X(94)E0054-K. |
[6] |
L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right-hand side measures,, \emph{Comm. Partial Differential Equations, 17 (1992), 641.
doi: 10.1080/03605309208820857. |
[7] |
L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations,, \emph{Nonlinear Analysis, 19 (1992), 581.
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,, \emph{Adv. Math.}, 224 (2010), 2151.
doi: 10.1016/j.aim.2010.01.023. |
[9] |
H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$,, \emph{Manuscripta Math.}, 74 (1992), 87.
doi: 10.1007/BF02567660. |
[10] |
E. DiBenedetto, Degenerate Parabolic Equations,, Springer-Verlag, (1993).
doi: 10.1007/978-1-4612-0895-2. |
[11] |
E. DiBenedetto, Intrinsic Harnack type inequalities for solutions of certain degenerate Parabolic equations,, \emph{Arch. Rational Mech. Anal.}, 100 (1988), 129.
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,, \emph{Manuscripta Math.}, 131 (2010), 231.
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, (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,, \emph{Comm. Ac. Scient. Petr. Tom. III}, (1728), 124. Google Scholar |
[15] |
H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+\alpha}$,, \emph{J. Fac. Sci. Univ. Tokyo Sect. I}, 13 (1966), 109.
|
[16] |
H. Fujita and S. Watanabe, On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations,, \emph{Comm. Pure Appl. Math}, 21 (1968), 631.
|
[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,, \emph{Zh. Vychisl. Mat. i Mat. Fiz.}, 23 (1983), 1341.
|
[18] |
V. A. Galaktionov and H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems,, \emph{Nonlinear Anal.}, 34 (1998), 1005.
doi: 10.1016/S0362-546X(97)00716-5. |
[19] |
J. García Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems,, \emph{J. Differential Equations}, 144 (1998), 441.
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, (1998).
|
[21] |
R. G. Iagar, P. Laurençot, Philippe and J. L. Vázquez, Asymptotic behaviour of a nonlinear parabolic equation with gradient absorption and critical exponent,, \emph{Interfaces Free Bound}, 13 (2011), 271.
doi: 10.4171/IFB/258. |
[22] |
A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, \emph{Russian Math Surveys}, 42 (1987), 169.
|
[23] |
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type,, Translation of Mathematical Monographs, (1968).
|
[24] |
J. L. Lions, Quelques méthodes de resolution des problèmes aux limites non linéaires,, Dunod Gauthier-Villars, (1969).
|
[25] |
J. J. Manfredi and V. Vespri, Large time behavior of solutions to a class of doubly nonlinear parabolic equations,, \emph{Electron. J. Differential Equations}, 2 (1994).
|
[26] |
G. Savaré and V. Vespri, The asymptotic profile of solutions of a class of doubly nonlinear equations,, \emph{Nonlinear Anal.}, 22 (1994), 1553.
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,, \emph{Siberian Math. J.}, 45 (2004), 155.
doi: 10.1023/B:SIMJ.0000013021.66528.b6. |
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