# American Institute of Mathematical Sciences

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 $\mathbb{R}^N2$, 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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