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 and 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, 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 $\mathbbR^N$, 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 $\mathbbR^N$, 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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