January  2011, 10(1): 161-177. doi: 10.3934/cpaa.2011.10.161

Blowing up at zero points of potential for an initial boundary value problem

1. 

Department of Mathematics, Tamkang University, 151, Ying-Chuan Road, Tamsui, Taipei County 25137, Taiwan

2. 

National Center for Theoretical Sciences, Taipei Office, 1, S-4, Roosevelt Road, Taipei 10617, Taiwan

Received  January 2010 Revised  May 2010 Published  November 2010

We study nonnegative radially symmetric solutions for a semilinear heat equation in a ball with spatially dependent coefficient which vanishes at the origin. Our aim is to construct a solution that blows up at the origin where there is no reaction. For this, we first prove that the blow-up is complete, if the origin is not a blow-up point and if there is no blow-up point on the boundary. Then we prove that a threshold solution exists such that it blows up in finite time incompletely and there is no blow-up point on the boundary. On the other hand, we prove that any zero of nonnegative potential is not a blow-up point for a more general problem under the assumption that the solution is monotone in time.
Citation: Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure & Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161
References:
[1]

H. Amann, Dual semigroups and second order linear elliptic boundary value problems,, Israel J. Math., 45 (1983), 225. doi: doi:10.1007/BF02774019. Google Scholar

[2]

P. Baras and L. Cohen, Complete blow-up after $T_{m a x}$ for the solution of a semilinear heat equation,, J. Funct. Anal., 71 (1987), 142. doi: doi:10.1016/0022-1236(87)90020-6. Google Scholar

[3]

X.Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations,, J. Differential Equations, 78 (1989), 160. doi: doi:10.1016/0022-0396(89)90081-8. Google Scholar

[4]

X. Y. Chen and P. Poláčik, Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball,, J. Reine Angew. Math., 472 (1996), 17. doi: doi:10.1515/crll.1996.472.17. Google Scholar

[5]

L. Du and Z. Yao, Localization of blow-up points for a nonlinear nonlocal porous medium equation,, Commun. Pure Appl. Anal., 6 (2007), 183. Google Scholar

[6]

S. Filippas and A. Tertikas, On similarity solutions of a heat equation with a nonhomogeneous nonlinearity,, J. Differential Equations, 165 (2000), 468. doi: doi:10.1006/jdeq.2000.3789. Google Scholar

[7]

A. Friedman and J. B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425. doi: doi:10.1512/iumj.1985.34.34025. Google Scholar

[8]

H. Fujita, On the nonlinear equations $\Delta u+exp u=0$ and $u_t=\Delta u+exp u$,, Bull. Amer. Math. Soc., 75 (1969), 132. doi: doi:10.1090/S0002-9904-1969-12175-0. Google Scholar

[9]

V. A. Galaktionov and J. L. Vázquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions,, Comm. Pure Applied Math., 50 (1997), 1. doi: doi:10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.0.CO;2-H. Google Scholar

[10]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: dynamic case,, Nonlinear Diff. Eqns. Appl., 15 (2008), 115. doi: doi:10.1007/s00030-007-6004-1. Google Scholar

[11]

Y. Giga and R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations,, Comm. Pure Appl. Math., 38 (1985), 297. doi: doi:10.1002/cpa.3160380304. Google Scholar

[12]

Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables,, Indiana Univ. Math. J., 36 (1987), 1. doi: doi:10.1512/iumj.1987.36.36001. Google Scholar

[13]

Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations,, Comm. Pure Appl. Math., 42 (1989), 845. doi: doi:10.1002/cpa.3160420607. Google Scholar

[14]

J. S. Guo, C. S. Lin and M. Shimojo, Blow-up behavior for a parabolic equation with spatially dependent coefficient,, Dynamic Systems Appl. (to appear)., (). Google Scholar

[15]

T. Hamada, On the existence and nonexistence of global solutions of semilinear parabolic equations with slowly decaying initial data,, Tsukuba J. Math., 21 (1997), 505. Google Scholar

[16]

S. Kaplan, On the growth of solutions of quasi-linear parabolic equations,, Comm. Pure Appl. Math., 16 (1963), 305. doi: doi:10.1002/cpa.3160160307. Google Scholar

[17]

A. A. Lacey and D. Tzanetis, Complete blow-up for a semilinear diffusion equation with a sufficiently large initial condition,, IMA. J. Appl. Math., 41 (1988), 207. doi: doi:10.1093/imamat/41.3.207. Google Scholar

[18]

L. A. Lepin, Countable spectrum of the eigenfunctions of the nonlinear heat equation with distributed parameters, (Russian), Differentsial'nye Uravneniya, 24 (1988), 1226. Google Scholar

[19]

L. A. Lepin, Self-similar solutions of a semilinear heat equation, (Russian), Mat. Model., 2 (1990), 63. Google Scholar

[20]

H. Matano and F. Merle, On nonexistence of type II blowup for a supercritical nonlinear heat equation,, Comm. Pure Appl. Math., 57 (2004), 1494. doi: doi:10.1002/cpa.20044. Google Scholar

[21]

H. Matano and F. Merle, Classification of type I and type II blowup for a supercritical nonlinear heat equation,, J. Funct. Anal., 256 (2009), 992. doi: doi:10.1016/j.jfa.2008.05.021. Google Scholar

[22]

J. Matos, Unfocused blow up solutions of semilinear parabolic equations,, Discrete Contin. Dynam. Systems, 5 (1999), 905. doi: doi:10.3934/dcds.1999.5.905. Google Scholar

[23]

J. Matos, Self-similar blow up patterns in supercritical semilinear heat equations,, Comm. Appl. Anal., 5 (2001), 455. Google Scholar

[24]

F. Merle, H. Zaag, Stability of the blow-up profile for equations of the type $u_t=\Delta u+ |u|^{p-1}u,, Duke Math. J., 86 (1997), 143. doi: doi:10.1215/S0012-7094-97-08605-1. Google Scholar

[25]

N. Mizoguchi, Boundedness of global solutions for a supercritical semilinear heat equation and its application,, Indiana Univ. Math. J., 54 (2005), 1047. doi: doi:10.1512/iumj.2005.54.2694. Google Scholar

[26]

N. Mizoguchi and E. Yanagida, Life span of solutions with large initial data in a semilinear parabolic equation,, Indiana Univ. Math. J., 50 (2001), 591. Google Scholar

[27]

W. M. Ni, P. E. Sacks and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations,, J. Differential Equations, 54 (1984), 97. doi: doi:10.1016/0022-0396(84)90145-1. Google Scholar

[28]

W. M. Ni, Uniqueness, nonuniqueness and related questions of nonlinear elliptic and parabolic equations,, Nonlinear Functional Analysis and its Applications, 45 (1983), 229. Google Scholar

[29]

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

[30]

P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States,", Birkh\, (2007). Google Scholar

[31]

A. Ramiandrisoa, Blow-up profile for radial solutions of the nonlinear heat equation,, Asymp. Anal., 21 (1999), 221. Google Scholar

[32]

S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation,, J. Differential Equations, 246 (2009), 724. doi: doi:10.1016/j.jde.2008.09.004. Google Scholar

[33]

X. Wang, On the Cauchy problem for reaction-diffusion equations,, Trans. Amer. Math. Soc., 337 (1993), 549. doi: doi:10.2307/2154232. Google Scholar

show all references

References:
[1]

H. Amann, Dual semigroups and second order linear elliptic boundary value problems,, Israel J. Math., 45 (1983), 225. doi: doi:10.1007/BF02774019. Google Scholar

[2]

P. Baras and L. Cohen, Complete blow-up after $T_{m a x}$ for the solution of a semilinear heat equation,, J. Funct. Anal., 71 (1987), 142. doi: doi:10.1016/0022-1236(87)90020-6. Google Scholar

[3]

X.Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations,, J. Differential Equations, 78 (1989), 160. doi: doi:10.1016/0022-0396(89)90081-8. Google Scholar

[4]

X. Y. Chen and P. Poláčik, Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball,, J. Reine Angew. Math., 472 (1996), 17. doi: doi:10.1515/crll.1996.472.17. Google Scholar

[5]

L. Du and Z. Yao, Localization of blow-up points for a nonlinear nonlocal porous medium equation,, Commun. Pure Appl. Anal., 6 (2007), 183. Google Scholar

[6]

S. Filippas and A. Tertikas, On similarity solutions of a heat equation with a nonhomogeneous nonlinearity,, J. Differential Equations, 165 (2000), 468. doi: doi:10.1006/jdeq.2000.3789. Google Scholar

[7]

A. Friedman and J. B. McLeod, Blow-up of positive solutions of semilinear heat equations,, Indiana Univ. Math. J., 34 (1985), 425. doi: doi:10.1512/iumj.1985.34.34025. Google Scholar

[8]

H. Fujita, On the nonlinear equations $\Delta u+exp u=0$ and $u_t=\Delta u+exp u$,, Bull. Amer. Math. Soc., 75 (1969), 132. doi: doi:10.1090/S0002-9904-1969-12175-0. Google Scholar

[9]

V. A. Galaktionov and J. L. Vázquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions,, Comm. Pure Applied Math., 50 (1997), 1. doi: doi:10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.0.CO;2-H. Google Scholar

[10]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: dynamic case,, Nonlinear Diff. Eqns. Appl., 15 (2008), 115. doi: doi:10.1007/s00030-007-6004-1. Google Scholar

[11]

Y. Giga and R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations,, Comm. Pure Appl. Math., 38 (1985), 297. doi: doi:10.1002/cpa.3160380304. Google Scholar

[12]

Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables,, Indiana Univ. Math. J., 36 (1987), 1. doi: doi:10.1512/iumj.1987.36.36001. Google Scholar

[13]

Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations,, Comm. Pure Appl. Math., 42 (1989), 845. doi: doi:10.1002/cpa.3160420607. Google Scholar

[14]

J. S. Guo, C. S. Lin and M. Shimojo, Blow-up behavior for a parabolic equation with spatially dependent coefficient,, Dynamic Systems Appl. (to appear)., (). Google Scholar

[15]

T. Hamada, On the existence and nonexistence of global solutions of semilinear parabolic equations with slowly decaying initial data,, Tsukuba J. Math., 21 (1997), 505. Google Scholar

[16]

S. Kaplan, On the growth of solutions of quasi-linear parabolic equations,, Comm. Pure Appl. Math., 16 (1963), 305. doi: doi:10.1002/cpa.3160160307. Google Scholar

[17]

A. A. Lacey and D. Tzanetis, Complete blow-up for a semilinear diffusion equation with a sufficiently large initial condition,, IMA. J. Appl. Math., 41 (1988), 207. doi: doi:10.1093/imamat/41.3.207. Google Scholar

[18]

L. A. Lepin, Countable spectrum of the eigenfunctions of the nonlinear heat equation with distributed parameters, (Russian), Differentsial'nye Uravneniya, 24 (1988), 1226. Google Scholar

[19]

L. A. Lepin, Self-similar solutions of a semilinear heat equation, (Russian), Mat. Model., 2 (1990), 63. Google Scholar

[20]

H. Matano and F. Merle, On nonexistence of type II blowup for a supercritical nonlinear heat equation,, Comm. Pure Appl. Math., 57 (2004), 1494. doi: doi:10.1002/cpa.20044. Google Scholar

[21]

H. Matano and F. Merle, Classification of type I and type II blowup for a supercritical nonlinear heat equation,, J. Funct. Anal., 256 (2009), 992. doi: doi:10.1016/j.jfa.2008.05.021. Google Scholar

[22]

J. Matos, Unfocused blow up solutions of semilinear parabolic equations,, Discrete Contin. Dynam. Systems, 5 (1999), 905. doi: doi:10.3934/dcds.1999.5.905. Google Scholar

[23]

J. Matos, Self-similar blow up patterns in supercritical semilinear heat equations,, Comm. Appl. Anal., 5 (2001), 455. Google Scholar

[24]

F. Merle, H. Zaag, Stability of the blow-up profile for equations of the type $u_t=\Delta u+ |u|^{p-1}u,, Duke Math. J., 86 (1997), 143. doi: doi:10.1215/S0012-7094-97-08605-1. Google Scholar

[25]

N. Mizoguchi, Boundedness of global solutions for a supercritical semilinear heat equation and its application,, Indiana Univ. Math. J., 54 (2005), 1047. doi: doi:10.1512/iumj.2005.54.2694. Google Scholar

[26]

N. Mizoguchi and E. Yanagida, Life span of solutions with large initial data in a semilinear parabolic equation,, Indiana Univ. Math. J., 50 (2001), 591. Google Scholar

[27]

W. M. Ni, P. E. Sacks and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations,, J. Differential Equations, 54 (1984), 97. doi: doi:10.1016/0022-0396(84)90145-1. Google Scholar

[28]

W. M. Ni, Uniqueness, nonuniqueness and related questions of nonlinear elliptic and parabolic equations,, Nonlinear Functional Analysis and its Applications, 45 (1983), 229. Google Scholar

[29]

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

[30]

P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States,", Birkh\, (2007). Google Scholar

[31]

A. Ramiandrisoa, Blow-up profile for radial solutions of the nonlinear heat equation,, Asymp. Anal., 21 (1999), 221. Google Scholar

[32]

S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation,, J. Differential Equations, 246 (2009), 724. doi: doi:10.1016/j.jde.2008.09.004. Google Scholar

[33]

X. Wang, On the Cauchy problem for reaction-diffusion equations,, Trans. Amer. Math. Soc., 337 (1993), 549. doi: doi:10.2307/2154232. Google Scholar

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