• Previous Article
    Global strong solution to the two-dimensional density-dependent magnetohydrodynamic equations with vaccum
  • CPAA Home
  • This Issue
  • Next Article
    Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces
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

[1]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[2]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[3]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[4]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[5]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[6]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[7]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[8]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[9]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[10]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[11]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[12]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[13]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[14]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[15]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[16]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116

[17]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344

[18]

Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082

[19]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[20]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]