# 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 - A, 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.  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.   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.   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.   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.  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.   Google Scholar [8] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425.  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.   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.  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.   Google Scholar [12] S. Kamin, A. Pozio and A. Tesei, Admissible conditions for parabolic equations degenerating at infinity,, Algebra i Analiz, 19 (2007), 105.   Google Scholar [13] S. Kaplan, On the growth of solutions of quasilinear parabolic equations,, Comm. Pure Appl. Math, 16 (1963), 305.  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, (1968).   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.  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.   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.   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.   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.   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.   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.   Google Scholar [22] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (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.   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.   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.   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.   Google Scholar [27] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation,, Israel J. Math, 38 (1981), 29.  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.  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.   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.   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.   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.  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.   Google Scholar [8] A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425.  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.   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.  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.   Google Scholar [12] S. Kamin, A. Pozio and A. Tesei, Admissible conditions for parabolic equations degenerating at infinity,, Algebra i Analiz, 19 (2007), 105.   Google Scholar [13] S. Kaplan, On the growth of solutions of quasilinear parabolic equations,, Comm. Pure Appl. Math, 16 (1963), 305.  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, (1968).   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.  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.   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.   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.   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.   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.   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.   Google Scholar [22] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,", Springer-Verlag, (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.   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.   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.   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.   Google Scholar [27] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation,, Israel J. Math, 38 (1981), 29.  doi: 10.1007/BF02761845.  Google Scholar
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