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July  2014, 13(4): 1465-1480. doi: 10.3934/cpaa.2014.13.1465

Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation

1. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China, China

Received  May 2013 Revised  December 2013 Published  February 2014

In this paper, we study the existence of local/global solutions to the Cauchy problem \begin{eqnarray} \rho(x)u_t=\Delta u+q(x)u^p, (x,t)\in R^N \times (0,T),\\ u(x,0)=u_{0}(x)\ge 0, x \in R^N \end{eqnarray} with $p > 0$ and $N\ge 3$. We describe the sharp decay conditions on $\rho, q$ and the data $u_0$ at infinity that guarantee the local/global existence of nonnegative solutions.
Citation: Xie Li, Zhaoyin Xiang. Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1465-1480. doi: 10.3934/cpaa.2014.13.1465
References:
[1]

P. Baras and R. Kersner, Local and global solvability of a class of semilinear parabolic equations,, \emph{J. Differential Equations}, 68 (1987), 238. Google Scholar

[2]

K. Deng and H.A. Levine, The role of critical exponents in blow-up theorems: The sequel,, \emph{J. Math. Anal. Appl.}, 243 (2000), 85. Google Scholar

[3]

S. Eidelman, S. Kamin and F. Porper, Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity,, \emph{Asympotic Analysis}, 22 (2000), 349. Google Scholar

[4]

D. Eidus, The Cauchy problem for the non-linear filtration equation in an inhomogeneous medium,, \emph{J. Differential Equations}, 84 (1990), 309. Google Scholar

[5]

R. Ferreira, A. de Pablo, G. Reyes and A. Sánchez, The interfaces of an inhomogeneous porous medium equation with convection,, \emph{Comm. Partial Differential Equations}, 31 (2006), 497. doi: 10.1080/03605300500481343. Google Scholar

[6]

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

[7]

S. Kamin, R. Kersner and A. Tessi, On the Cauchy problem for a class of parabolic equations with variable density,, \emph{Atti Accad. Naz. Lincei Rend.Cl. Sci. Fis. Mat. Natur.}, 9 (1998), 279. Google Scholar

[8]

S. Kamin, A. Pozio and A. Tessi, Admissible conditions for parabolic equations degenerating at infinity,, \emph{Algebra i Analiz}, 19 (2007), 105. doi: 10.1090/S1061-0022-08-00996-5. Google Scholar

[9]

S. Kaplan, On the growth of solutions of quasi-linear parabolic equations,, \emph{Comm. Pure Appl. Math.}, 16 (1963), 305. Google Scholar

[10]

H. A. Levine, The role of critical exponents in blowup theorems,, \emph{SIAM Rev.}, 32 (1990), 262. doi: 10.1137/1032046. Google Scholar

[11]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996). Google Scholar

[12]

A. V. Martynenko, A. F. Tedeev and V. N. Shramenko, The Cauchy problem for a degenerate parabolic equation with inhomogeneous density and source in the class of slowly decaying initial data,, \emph{Izvestiya: Mathematics}, 76 (2012), 563. doi: 10.1070/IM2012v076n03ABEH002595. Google Scholar

[13]

A. de Pablo, G. Reyes and A. Sánchez, The Cauchy problem for a nonhomogeneous heat equation with reaction,, \emph{Discrete and Continuous Dynamical Systems}, 33 (2013), 643. Google Scholar

[14]

R. G. Pinsky, Existence and nonexistence of global solutions for $u_t=\Delta u+a(x)u^p$ in $\mathbbR^d$},, \emph{J. Differential Equations}, 133 (1997), 152. doi: 10.1006/jdeq.1996.3196. Google Scholar

[15]

M. A. Pozio, F. Punzo and A. Tesei, Uniqueness and nonuniqueness of solutions to parabolic problems with singular coeffcients,, \emph{Discrete and Continuous Dynamical Systems}, 30 (2011), 891. doi: 10.3934/dcds.2011.30.891. Google Scholar

[16]

Y. W. Qi, The critical exponents of parabolic equations and blow-up in $R^n$,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 128 (1998), 123. doi: 10.1017/S0308210500027190. Google Scholar

[17]

G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density,, \emph{Comm. Pure Appl. Anal.}, 8 (2009), 493. doi: 10.3934/cpaa.2009.8.493. Google Scholar

[18]

Y. Wang and Z. Xiang, The interfaces of an inhomogeneous non-Newtonian polytropic filtration equation with convection,, \emph{IMA J. Appl. Math.}, (2013). doi: 10.1093/imamat/hxt043. Google Scholar

[19]

C. Wang and S. Zheng, Critical Fujita exponents of degenerate and singular parabolic equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 136 (2006), 415. doi: 10.1017/S0308210500004637. Google Scholar

[20]

Z. Xiang, C. Mu and X. Hu, Support properties of solutions to a degenerate equation with absorption and variable density,, \emph{Nonlinear Anal.}, 68 (2008), 1940. doi: 10.1016/j.na.2007.01.021. Google Scholar

show all references

References:
[1]

P. Baras and R. Kersner, Local and global solvability of a class of semilinear parabolic equations,, \emph{J. Differential Equations}, 68 (1987), 238. Google Scholar

[2]

K. Deng and H.A. Levine, The role of critical exponents in blow-up theorems: The sequel,, \emph{J. Math. Anal. Appl.}, 243 (2000), 85. Google Scholar

[3]

S. Eidelman, S. Kamin and F. Porper, Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity,, \emph{Asympotic Analysis}, 22 (2000), 349. Google Scholar

[4]

D. Eidus, The Cauchy problem for the non-linear filtration equation in an inhomogeneous medium,, \emph{J. Differential Equations}, 84 (1990), 309. Google Scholar

[5]

R. Ferreira, A. de Pablo, G. Reyes and A. Sánchez, The interfaces of an inhomogeneous porous medium equation with convection,, \emph{Comm. Partial Differential Equations}, 31 (2006), 497. doi: 10.1080/03605300500481343. Google Scholar

[6]

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

[7]

S. Kamin, R. Kersner and A. Tessi, On the Cauchy problem for a class of parabolic equations with variable density,, \emph{Atti Accad. Naz. Lincei Rend.Cl. Sci. Fis. Mat. Natur.}, 9 (1998), 279. Google Scholar

[8]

S. Kamin, A. Pozio and A. Tessi, Admissible conditions for parabolic equations degenerating at infinity,, \emph{Algebra i Analiz}, 19 (2007), 105. doi: 10.1090/S1061-0022-08-00996-5. Google Scholar

[9]

S. Kaplan, On the growth of solutions of quasi-linear parabolic equations,, \emph{Comm. Pure Appl. Math.}, 16 (1963), 305. Google Scholar

[10]

H. A. Levine, The role of critical exponents in blowup theorems,, \emph{SIAM Rev.}, 32 (1990), 262. doi: 10.1137/1032046. Google Scholar

[11]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996). Google Scholar

[12]

A. V. Martynenko, A. F. Tedeev and V. N. Shramenko, The Cauchy problem for a degenerate parabolic equation with inhomogeneous density and source in the class of slowly decaying initial data,, \emph{Izvestiya: Mathematics}, 76 (2012), 563. doi: 10.1070/IM2012v076n03ABEH002595. Google Scholar

[13]

A. de Pablo, G. Reyes and A. Sánchez, The Cauchy problem for a nonhomogeneous heat equation with reaction,, \emph{Discrete and Continuous Dynamical Systems}, 33 (2013), 643. Google Scholar

[14]

R. G. Pinsky, Existence and nonexistence of global solutions for $u_t=\Delta u+a(x)u^p$ in $\mathbbR^d$},, \emph{J. Differential Equations}, 133 (1997), 152. doi: 10.1006/jdeq.1996.3196. Google Scholar

[15]

M. A. Pozio, F. Punzo and A. Tesei, Uniqueness and nonuniqueness of solutions to parabolic problems with singular coeffcients,, \emph{Discrete and Continuous Dynamical Systems}, 30 (2011), 891. doi: 10.3934/dcds.2011.30.891. Google Scholar

[16]

Y. W. Qi, The critical exponents of parabolic equations and blow-up in $R^n$,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 128 (1998), 123. doi: 10.1017/S0308210500027190. Google Scholar

[17]

G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density,, \emph{Comm. Pure Appl. Anal.}, 8 (2009), 493. doi: 10.3934/cpaa.2009.8.493. Google Scholar

[18]

Y. Wang and Z. Xiang, The interfaces of an inhomogeneous non-Newtonian polytropic filtration equation with convection,, \emph{IMA J. Appl. Math.}, (2013). doi: 10.1093/imamat/hxt043. Google Scholar

[19]

C. Wang and S. Zheng, Critical Fujita exponents of degenerate and singular parabolic equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 136 (2006), 415. doi: 10.1017/S0308210500004637. Google Scholar

[20]

Z. Xiang, C. Mu and X. Hu, Support properties of solutions to a degenerate equation with absorption and variable density,, \emph{Nonlinear Anal.}, 68 (2008), 1940. doi: 10.1016/j.na.2007.01.021. Google Scholar

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