# 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 & 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 [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

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
 [1] Lihua Min, Xiaoping Yang. Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 543-566. doi: 10.3934/cpaa.2014.13.543 [2] Jean-Daniel Djida, Juan J. Nieto, Iván Area. Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4031-4053. doi: 10.3934/dcdsb.2019049 [3] S. Bonafede, G. R. Cirmi, A.F. Tedeev. Finite speed of propagation for the porous media equation with lower order terms. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 305-314. doi: 10.3934/dcds.2000.6.305 [4] Mattia Turra. Existence and extinction in finite time for Stratonovich gradient noise porous media equations. Evolution Equations & Control Theory, 2019, 8 (4) : 867-882. doi: 10.3934/eect.2019042 [5] Antoine Benoit. Finite speed of propagation for mixed problems in the $WR$ class. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2351-2358. doi: 10.3934/cpaa.2014.13.2351 [6] Alfonso C. Casal, Jesús Ildefonso Díaz, José M. Vegas. Finite extinction time property for a delayed linear problem on a manifold without boundary. Conference Publications, 2011, 2011 (Special) : 265-271. doi: 10.3934/proc.2011.2011.265 [7] Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715 [8] Yong Zhou, Zhengguang Guo. Blow up and propagation speed of solutions to the DGH equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 657-670. doi: 10.3934/dcdsb.2009.12.657 [9] Rafael Monteiro. Horizontal patterns from finite speed directional quenching. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3503-3534. doi: 10.3934/dcdsb.2018285 [10] Thomas Gauthier, Gabriel Vigny. Distribution of postcritically finite polynomials Ⅱ: Speed of convergence. Journal of Modern Dynamics, 2017, 11: 57-98. doi: 10.3934/jmd.2017004 [11] Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1769-1781. doi: 10.3934/dcdsb.2014.19.1769 [12] Young-Pil Choi, Seung-Yeal Ha, Jeongho Kim. Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication. Networks & Heterogeneous Media, 2018, 13 (3) : 379-407. doi: 10.3934/nhm.2018017 [13] Arno Berger. On finite-time hyperbolicity. Communications on Pure & Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963 [14] Thierry Cazenave, Yvan Martel, Lifeng Zhao. Finite-time blowup for a Schrödinger equation with nonlinear source term. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1171-1183. doi: 10.3934/dcds.2019050 [15] Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems & Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019 [16] David Henry. Infinite propagation speed for a two component Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 597-606. doi: 10.3934/dcdsb.2009.12.597 [17] Hassan Belhadj, Mohamed Fihri, Samir Khallouq, Nabila Nagid. Optimal number of Schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 21-34. doi: 10.3934/dcdss.2018002 [18] Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463 [19] Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929 [20] Baojun Bian, Nan Wu, Harry Zheng. Optimal liquidation in a finite time regime switching model with permanent and temporary pricing impact. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1401-1420. doi: 10.3934/dcdsb.2016002

2018 Impact Factor: 0.925