# American Institute of Mathematical Sciences

February  2013, 33(2): 643-662. doi: 10.3934/dcds.2013.33.643

## The Cauchy problem for a nonhomogeneous heat equation with reaction

 1 Departamento de Matemática Aplicada, Universidad Carlos III de Madrid, 28911 Leganés, Spain 2 Departamento de Matemáticas, U. Politécnica de Madrid, 28040 Madrid, Spain 3 Departamento de Matemáticas, U. Rey Juan Carlos, 28933 Móstoles, Spain

Received  July 2011 Revised  July 2012 Published  September 2012

We study the behaviour of the solutions to the Cauchy problem $$\left\{\begin{array}{ll} \rho(x)u_t=\Delta u+u^p,&\quad x\in\mathbb{R}^N ,\;t\in(0,T),\\ u(x,\, 0)=u_0(x),&\quad x\in\mathbb{R}^N , \end{array}\right.$$ with $p>0$ and a positive density $\rho$ satisfying $\rho(x)\sim|x|^{-\sigma}$ for large $|x|$, $0<\sigma<2< N$. We consider both the cases of a bounded density and the singular density $\rho(x)=|x|^{-\sigma}$. We are interested in describing sharp decay conditions on the data at infinity that guarantee local/global existence of solutions, which are unique in classes of functions with the same decay. We prove that larger data give rise to instantaneous complete blow-up. We also deal with the occurrence of finite-time blow-up. We prove that the global existence exponent is $p_0=1$, while the Fujita exponent depends on $\sigma$, namely $p_c=1+\frac2{N-\sigma}$.
We show that instantaneous blow-up at space infinity takes place when $p\le1$.
We also briefly discuss the case $2<\sigma< N$: we prove that the Fujita exponent in this case does not depend on $\sigma$, $\tilde{p}_c=1+\frac2{N-2}$, and for initial values not too small at infinity a phenomenon of instantaneous complete blow-up occurs in the range $1< p < \tilde{p}_c$
Citation: Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643
##### References:
 [1] J. Aguirre and M. Escobedo, On the blow-up of solutions of a convective reaction diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 433-460. doi: 10.1017/S0308210500025828.  Google Scholar [2] J. Aguirre and M. Escobedo, A Cauchy problem for $u_t=\Delta u+u^p$ with $0< p <1$. Asymptotic behaviour of solutions,, Ann. Fac. Sci. Toulouse Math, 8 (): 175.   Google Scholar [3] P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.  Google Scholar [4] P. Baras and R. Kersner, Local and global solvability of a class of semilinear parabolic equations, J. of Differential Equations, 68 (1987), 238-252.  Google Scholar [5] C. Bandle and H. A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc., 316 (1989), 595-622.  Google Scholar [6] K. Deng and H. A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl., 243 (2000), 85-126. doi: 10.1006/jmaa.1999.6663.  Google Scholar [7] S. Eidelman, S. Kamin and F. Porper, Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity, Asymptotic Analysis, 22 (2000), 349-358.  Google Scholar [8] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.  Google Scholar [9] 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 Sec. IA Math, 16 (1966), 105-113.  Google Scholar [10] V. A. Galaktionov and H. A. Levine, On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary, Israel J. Math, 94 (1996), 125-146. doi: 10.1007/BF02762700.  Google Scholar [11] S. Kamin, R. Kersner and A. Tesei, On the Cauchy problem for a class of parabolic equations with variable density, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 9 (1998), 279-298.  Google Scholar [12] S. Kamin, A. Pozio and A. Tesei, Admissible conditions for parabolic equations degenerating at infinity, Algebra i Analiz, 19 (2007), 105-121. (Russian); translation in St. Petersburg Math. J., 19 (2008), 239-251.  Google Scholar [13] S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math, 16 (1963), 305-330. doi: 10.1002/cpa.3160160307.  Google Scholar [14] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, "Linear and Quasilinear Equation of Parabolic Type," (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968 xi+648 pp.  Google Scholar [15] T. Y. Lee and W. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., 333 (1992), 365-378. doi: 10.1090/S0002-9947-1992-1057781-6.  Google Scholar [16] H. A. Levine, The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), 262-288.  Google Scholar [17] A. V. Martynenko and A. F. Tedeev, The Cauchy problem for a quasilinear parabolic equation with a source and nonhomogeneous density, Zh. Vychisl. Mat. Mat. Fiz., 47 (2007), 245-255. (Russian); translation in Comput. Math. Math. Phys., 47 (2007), 238-248.  Google Scholar [18] A. de Pablo and J. L. Vázquez, The balance between strong reaction and slow diffusion, Comm. Partial Differential Equations, 15 (1990), 159-183.  Google Scholar [19] R. G. Pinsky, Existence and nonexistence of global solutions for $u_t=\Delta u+a(x)u^p$ in $\mathbbR^d$, J. Differential Equations, 133 (1997), 152-177.  Google Scholar [20] S. I. Pohozaev and A. Tesei, Nonexistence of local solutions to semilinear partial differential inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 487-502.  Google Scholar [21] M. A. Pozio, F. Punzo and A. Tesei, Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients, Discrete Contin. Dyn. Syst., 30 (2011), 891-916.  Google Scholar [22] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, New York, 1984.  Google Scholar [23] Y. W. Qi, The critical exponents of parabolic equations and blow-up in $R^n$, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 123-136.  Google Scholar [24] G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal., 8 (2009), 493-508.  Google Scholar [25] G. Reyes and J. L. Vázquez, A weighted symmetrization for nonlinear elliptic and parabolic equations, J. Eur. Math. Soc., 8 (2006), 531-554.  Google Scholar [26] C. Wang and S. Zheng, Critical Fujita exponents of degenerate and singular parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 415-430.  Google Scholar [27] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math, 38 (1981), 29-40. doi: 10.1007/BF02761845.  Google Scholar

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##### References:
 [1] J. Aguirre and M. Escobedo, On the blow-up of solutions of a convective reaction diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 433-460. doi: 10.1017/S0308210500025828.  Google Scholar [2] J. Aguirre and M. Escobedo, A Cauchy problem for $u_t=\Delta u+u^p$ with $0< p <1$. Asymptotic behaviour of solutions,, Ann. Fac. Sci. Toulouse Math, 8 (): 175.   Google Scholar [3] P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.  Google Scholar [4] P. Baras and R. Kersner, Local and global solvability of a class of semilinear parabolic equations, J. of Differential Equations, 68 (1987), 238-252.  Google Scholar [5] C. Bandle and H. A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc., 316 (1989), 595-622.  Google Scholar [6] K. Deng and H. A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl., 243 (2000), 85-126. doi: 10.1006/jmaa.1999.6663.  Google Scholar [7] S. Eidelman, S. Kamin and F. Porper, Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity, Asymptotic Analysis, 22 (2000), 349-358.  Google Scholar [8] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.  Google Scholar [9] 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 Sec. IA Math, 16 (1966), 105-113.  Google Scholar [10] V. A. Galaktionov and H. A. Levine, On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary, Israel J. Math, 94 (1996), 125-146. doi: 10.1007/BF02762700.  Google Scholar [11] S. Kamin, R. Kersner and A. Tesei, On the Cauchy problem for a class of parabolic equations with variable density, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 9 (1998), 279-298.  Google Scholar [12] S. Kamin, A. Pozio and A. Tesei, Admissible conditions for parabolic equations degenerating at infinity, Algebra i Analiz, 19 (2007), 105-121. (Russian); translation in St. Petersburg Math. J., 19 (2008), 239-251.  Google Scholar [13] S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math, 16 (1963), 305-330. doi: 10.1002/cpa.3160160307.  Google Scholar [14] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, "Linear and Quasilinear Equation of Parabolic Type," (Russian) Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968 xi+648 pp.  Google Scholar [15] T. Y. Lee and W. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., 333 (1992), 365-378. doi: 10.1090/S0002-9947-1992-1057781-6.  Google Scholar [16] H. A. Levine, The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), 262-288.  Google Scholar [17] A. V. Martynenko and A. F. Tedeev, The Cauchy problem for a quasilinear parabolic equation with a source and nonhomogeneous density, Zh. Vychisl. Mat. Mat. Fiz., 47 (2007), 245-255. (Russian); translation in Comput. Math. Math. Phys., 47 (2007), 238-248.  Google Scholar [18] A. de Pablo and J. L. Vázquez, The balance between strong reaction and slow diffusion, Comm. Partial Differential Equations, 15 (1990), 159-183.  Google Scholar [19] R. G. Pinsky, Existence and nonexistence of global solutions for $u_t=\Delta u+a(x)u^p$ in $\mathbbR^d$, J. Differential Equations, 133 (1997), 152-177.  Google Scholar [20] S. I. Pohozaev and A. Tesei, Nonexistence of local solutions to semilinear partial differential inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 487-502.  Google Scholar [21] M. A. Pozio, F. Punzo and A. Tesei, Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients, Discrete Contin. Dyn. Syst., 30 (2011), 891-916.  Google Scholar [22] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Springer-Verlag, New York, 1984.  Google Scholar [23] Y. W. Qi, The critical exponents of parabolic equations and blow-up in $R^n$, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 123-136.  Google Scholar [24] G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal., 8 (2009), 493-508.  Google Scholar [25] G. Reyes and J. L. Vázquez, A weighted symmetrization for nonlinear elliptic and parabolic equations, J. Eur. Math. Soc., 8 (2006), 531-554.  Google Scholar [26] C. Wang and S. Zheng, Critical Fujita exponents of degenerate and singular parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 415-430.  Google Scholar [27] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math, 38 (1981), 29-40. doi: 10.1007/BF02761845.  Google Scholar
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