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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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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