November  2014, 19(9): 3019-3029. doi: 10.3934/dcdsb.2014.19.3019

Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China

Received  May 2013 Revised  September 2013 Published  September 2014

Throughout this paper, we consider the equation \[u_t - \Delta u = e^{|\nabla u|}\] with homogeneous Dirichlet boundary condition. One of our main goals is to show that the existence of global classical solution can derive the existence of classical stationary solution, and the global solution must converge to the stationary solution in $C(\overline{\Omega})$. On the contrary, the existence of the stationary solution also implies the global existence of the classical solution at least in the radial case. The other one is to show that finite time gradient blowup will occur for large initial data or domains with small measure.
Citation: Zhengce Zhang, Yan Li. Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 3019-3029. doi: 10.3934/dcdsb.2014.19.3019
References:
[1]

J. M. Arrieta, A. Rodriguez-Bernal and Ph. Souplet, Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena,, Ann. Scuola. Norm. Super. Pisa Cl. Sci., 3 (2004), 1.   Google Scholar

[2]

M. Fila and G. M. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations,, Differential Integral Equations, 7 (1994), 811.   Google Scholar

[3]

M. Fila, J. Taskinen and M. Winkler, Convergence to a singular steady state of a parabolic equation with gradient blow-up,, Appl. Math. Lett., 20 (2007), 578.  doi: 10.1016/j.aml.2006.07.004.  Google Scholar

[4]

J.-S. Guo and B. Hu, Blowup rate estimates for the heat equation with a nonlinear gradient source term,, Discrete Contin. Dyn. Syst., 20 (2008), 927.  doi: 10.3934/dcds.2008.20.927.  Google Scholar

[5]

M. Hesaaraki and A. Moameni, Blow-up of positive solutions for a family of nonlinear parabolic equations in general domain in $\mathbbR^N$,, Michigan Math. J., 52 (2004), 375.  doi: 10.1307/mmj/1091112081.  Google Scholar

[6]

M. Kardar, G. Parisi and Y. C. Zhang, Dynamic scaling of growing interfaces,, Phys. Rev. Lett., 56 (1986), 889.   Google Scholar

[7]

J. Krug and H. Spohn, Universality classes for deterministic surface growth,, Phys. Rev. A., 38 (1988), 4271.  doi: 10.1103/PhysRevA.38.4271.  Google Scholar

[8]

Y. X. Li and Ph. Souplet, Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains,, Commun. Math. Phys., 293 (2010), 499.  doi: 10.1007/s00220-009-0936-8.  Google Scholar

[9]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (2005).   Google Scholar

[10]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States,, Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, (2007).   Google Scholar

[11]

W. Rudin, Principles of Mathematical Analysis,, $3^{rd}$ edition, (2007).   Google Scholar

[12]

Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions,, Differential Integral Equations, 15 (2002), 237.   Google Scholar

[13]

Ph. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities,, Electron. J. Differential Equations, 2001 (2001), 1.   Google Scholar

[14]

Ph. Souplet and J. L. Vázquez, Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem,, Discrete Contin. Dyn. Syst., 14 (2006), 221.   Google Scholar

[15]

Ph. Souplet and Q. S. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations,, J. dÁnalyse Math., 99 (2006), 335.  doi: 10.1007/BF02789452.  Google Scholar

[16]

Z. C. Zhang and B. Hu, Gradient blowup rate for a semilinear parabolic equation,, Discrete Contin. Dyn. Syst., 26 (2010), 767.  doi: 10.3934/dcds.2010.26.767.  Google Scholar

[17]

Z. C. Zhang and B. Hu, Rate estimates of gradient blowup for a heat equation with exponential nonlinearity,, Nonlinear Anal., 72 (2010), 4594.  doi: 10.1016/j.na.2010.02.036.  Google Scholar

[18]

Z. C. Zhang and Y. Y. Li, Gradient blowup solutions of a semilinear parabolic equation with exponential source,, Comm. Pure Appl. Anal., 12 (2013), 269.  doi: 10.3934/cpaa.2013.12.269.  Google Scholar

[19]

Z. C. Zhang and Y. Y. Li, Boundedness of global solutions for a heat equation with exponential gradient source,, Abstr. Appl. Anal., 2012 (2012), 1.  doi: doi:10.1155/2012/398049.  Google Scholar

[20]

Z. C. Zhang and Z. J. Li, A note on gradient blowup rate of the inhomogeneous Hamilton-Jacobi equations,, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 678.  doi: 10.1016/S0252-9602(13)60029-6.  Google Scholar

[21]

L. P. Zhu and Z. C. Zhang, Rate of approach to the steady state for a diffusion-convection equation on annular domains,, Electron. J. Qual. Theory Differ. Equ., 39 (2012), 1.   Google Scholar

show all references

References:
[1]

J. M. Arrieta, A. Rodriguez-Bernal and Ph. Souplet, Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena,, Ann. Scuola. Norm. Super. Pisa Cl. Sci., 3 (2004), 1.   Google Scholar

[2]

M. Fila and G. M. Lieberman, Derivative blow-up and beyond for quasilinear parabolic equations,, Differential Integral Equations, 7 (1994), 811.   Google Scholar

[3]

M. Fila, J. Taskinen and M. Winkler, Convergence to a singular steady state of a parabolic equation with gradient blow-up,, Appl. Math. Lett., 20 (2007), 578.  doi: 10.1016/j.aml.2006.07.004.  Google Scholar

[4]

J.-S. Guo and B. Hu, Blowup rate estimates for the heat equation with a nonlinear gradient source term,, Discrete Contin. Dyn. Syst., 20 (2008), 927.  doi: 10.3934/dcds.2008.20.927.  Google Scholar

[5]

M. Hesaaraki and A. Moameni, Blow-up of positive solutions for a family of nonlinear parabolic equations in general domain in $\mathbbR^N$,, Michigan Math. J., 52 (2004), 375.  doi: 10.1307/mmj/1091112081.  Google Scholar

[6]

M. Kardar, G. Parisi and Y. C. Zhang, Dynamic scaling of growing interfaces,, Phys. Rev. Lett., 56 (1986), 889.   Google Scholar

[7]

J. Krug and H. Spohn, Universality classes for deterministic surface growth,, Phys. Rev. A., 38 (1988), 4271.  doi: 10.1103/PhysRevA.38.4271.  Google Scholar

[8]

Y. X. Li and Ph. Souplet, Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains,, Commun. Math. Phys., 293 (2010), 499.  doi: 10.1007/s00220-009-0936-8.  Google Scholar

[9]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific, (2005).   Google Scholar

[10]

P. Quittner and Ph. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States,, Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, (2007).   Google Scholar

[11]

W. Rudin, Principles of Mathematical Analysis,, $3^{rd}$ edition, (2007).   Google Scholar

[12]

Ph. Souplet, Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions,, Differential Integral Equations, 15 (2002), 237.   Google Scholar

[13]

Ph. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities,, Electron. J. Differential Equations, 2001 (2001), 1.   Google Scholar

[14]

Ph. Souplet and J. L. Vázquez, Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem,, Discrete Contin. Dyn. Syst., 14 (2006), 221.   Google Scholar

[15]

Ph. Souplet and Q. S. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations,, J. dÁnalyse Math., 99 (2006), 335.  doi: 10.1007/BF02789452.  Google Scholar

[16]

Z. C. Zhang and B. Hu, Gradient blowup rate for a semilinear parabolic equation,, Discrete Contin. Dyn. Syst., 26 (2010), 767.  doi: 10.3934/dcds.2010.26.767.  Google Scholar

[17]

Z. C. Zhang and B. Hu, Rate estimates of gradient blowup for a heat equation with exponential nonlinearity,, Nonlinear Anal., 72 (2010), 4594.  doi: 10.1016/j.na.2010.02.036.  Google Scholar

[18]

Z. C. Zhang and Y. Y. Li, Gradient blowup solutions of a semilinear parabolic equation with exponential source,, Comm. Pure Appl. Anal., 12 (2013), 269.  doi: 10.3934/cpaa.2013.12.269.  Google Scholar

[19]

Z. C. Zhang and Y. Y. Li, Boundedness of global solutions for a heat equation with exponential gradient source,, Abstr. Appl. Anal., 2012 (2012), 1.  doi: doi:10.1155/2012/398049.  Google Scholar

[20]

Z. C. Zhang and Z. J. Li, A note on gradient blowup rate of the inhomogeneous Hamilton-Jacobi equations,, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 678.  doi: 10.1016/S0252-9602(13)60029-6.  Google Scholar

[21]

L. P. Zhu and Z. C. Zhang, Rate of approach to the steady state for a diffusion-convection equation on annular domains,, Electron. J. Qual. Theory Differ. Equ., 39 (2012), 1.   Google Scholar

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