January  2015, 14(1): 245-268. doi: 10.3934/cpaa.2015.14.245

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

Received  January 2014 Revised  February 2014 Published  September 2014

This article deals with the following quasilinear parabolic problem \begin{eqnarray} u_t-\Delta_p u=h(x)u^{q}, u\geq 0 & \text{in} \,\, \Omega\times (0,\infty),\\ u(x,t)=0 & \text{on}\,\, \partial \Omega\times (0,\infty), \\ u(x,0)=f(x), \,\, f\geq 0 & \text{in} \,\, \Omega, \end{eqnarray} where $-\Delta_p u=-div(|\nabla u|^{p-2}\nabla u)$, $p>1$, $q>0$, $h(x)>0$ and $f(x)\geq 0$ are non negative functions satisfying suitable hypotheses. We assume the domain $\Omega$ is either a bounded regular domain or the whole $\mathbb{R}^N$. The main contribution of this work is to prove that the optimal exponent in the reaction term in order to prove existence of a global positive solution is $q_0=\min\{1,(p-1)\}$. More precisely, we obtain the following conclusions
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.
Citation: Susana Merchán, Luigi Montoro, I. Peral. Optimal reaction exponent for some qualitative properties of solutions to the $p$-heat equation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 245-268. doi: 10.3934/cpaa.2015.14.245
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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$,, \emph{Manuscripta Math.}, 74 (1992), 87. doi: 10.1007/BF02567660. Google Scholar

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E. DiBenedetto, Degenerate Parabolic Equations,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-0895-2. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 Edition, (1998). Google Scholar

[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. Google Scholar

[22]

A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, \emph{Russian Math Surveys}, 42 (1987), 169. Google Scholar

[23]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type,, Translation of Mathematical Monographs, (1968). Google Scholar

[24]

J. L. Lions, Quelques méthodes de resolution des problèmes aux limites non linéaires,, Dunod Gauthier-Villars, (1969). Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$,, \emph{Manuscripta Math.}, 74 (1992), 87. doi: 10.1007/BF02567660. Google Scholar

[10]

E. DiBenedetto, Degenerate Parabolic Equations,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-0895-2. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 Edition, (1998). Google Scholar

[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. Google Scholar

[22]

A. S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,, \emph{Russian Math Surveys}, 42 (1987), 169. Google Scholar

[23]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type,, Translation of Mathematical Monographs, (1968). Google Scholar

[24]

J. L. Lions, Quelques méthodes de resolution des problèmes aux limites non linéaires,, Dunod Gauthier-Villars, (1969). Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

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