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

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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

Jong-Shenq Guo ^1, and Masahiko Shimojo ^2,

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

Abstract
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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.

Keywords: potential, semilinear heat equation, Blow up, threshold solution..

Mathematics Subject Classification: Primary: 35K55; Secondary: 35K2.

Citation: Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure and 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), 225-254. doi: doi:10.1007/BF02774019.
[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), 142-174. doi: doi:10.1016/0022-1236(87)90020-6.
[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), 160-190. doi: doi:10.1016/0022-0396(89)90081-8.
[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), 17-51. doi: doi:10.1515/crll.1996.472.17.
[5]	L. Du and Z. Yao, Localization of blow-up points for a nonlinear nonlocal porous medium equation, Commun. Pure Appl. Anal., 6 (2007), 183-190.
[6]	S. Filippas and A. Tertikas, On similarity solutions of a heat equation with a nonhomogeneous nonlinearity, J. Differential Equations, 165 (2000), 468-492. doi: doi:10.1006/jdeq.2000.3789.
[7]	A. Friedman and J. B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: doi:10.1512/iumj.1985.34.34025.
[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), 132-135. doi: doi:10.1090/S0002-9904-1969-12175-0.
[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), 1-67. doi: doi:10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.0.CO;2-H.
[10]	N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: dynamic case, Nonlinear Diff. Eqns. Appl., 15 (2008), 115-145. doi: doi:10.1007/s00030-007-6004-1.
[11]	Y. Giga and R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319. doi: doi:10.1002/cpa.3160380304.
[12]	Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40. doi: doi:10.1512/iumj.1987.36.36001.
[13]	Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42 (1989), 845-884. doi: doi:10.1002/cpa.3160420607.
[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).
[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), 505-514.
[16]	S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330. doi: doi:10.1002/cpa.3160160307.
[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), 207-215. doi: doi:10.1093/imamat/41.3.207.
[18]	L. A. Lepin, Countable spectrum of the eigenfunctions of the nonlinear heat equation with distributed parameters, (Russian) Differentsial'nye Uravneniya, 24 (1988), 1226-1234, (English Translation: Differential Equations, 24 (1988), 799-805.
[19]	L. A. Lepin, Self-similar solutions of a semilinear heat equation, (Russian) Mat. Model., 2 (1990), 63-74.
[20]	H. Matano and F. Merle, On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math., 57 (2004), 1494-1541. doi: doi:10.1002/cpa.20044.
[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), 992-1064. doi: doi:10.1016/j.jfa.2008.05.021.
[22]	J. Matos, Unfocused blow up solutions of semilinear parabolic equations, Discrete Contin. Dynam. Systems, 5 (1999), 905-928. doi: doi:10.3934/dcds.1999.5.905.
[23]	J. Matos, Self-similar blow up patterns in supercritical semilinear heat equations, Comm. Appl. Anal., 5 (2001), 455-483.
[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), 143-195. doi: doi:10.1215/S0012-7094-97-08605-1.
[25]	N. Mizoguchi, Boundedness of global solutions for a supercritical semilinear heat equation and its application, Indiana Univ. Math. J., 54 (2005), 1047-1059. doi: doi:10.1512/iumj.2005.54.2694.
[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), 591-610.
[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), 97-120. doi: doi:10.1016/0022-0396(84)90145-1.
[28]	W. M. Ni, Uniqueness, nonuniqueness and related questions of nonlinear elliptic and parabolic equations, Nonlinear Functional Analysis and its Applications, Part 2 (Berkeley, Calif., 1983), 229-241, Proc. Sympos. Pure Math., 45, Part 2, Amer. Math. Soc., Providence, RI, 1986.
[29]	R. G. Pinsky, Existence and nonexistence of global solutions for $u_ t=\Delta u+a(x)u^p$ in $\mathbb{R}^{d}$, J. Differential Equations, 133 (1997), 152-177. doi: doi:10.1006/jdeq.1996.3196.
[30]	P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts, Basler Lehrbücher, 2007.
[31]	A. Ramiandrisoa, Blow-up profile for radial solutions of the nonlinear heat equation, Asymp. Anal., 21 (1999), 221-238.
[32]	S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation, J. Differential Equations, 246 (2009), 724-748. doi: doi:10.1016/j.jde.2008.09.004.
[33]	X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590. doi: doi:10.2307/2154232.

show all references

References:

[1]	H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math., 45 (1983), 225-254. doi: doi:10.1007/BF02774019.
[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), 142-174. doi: doi:10.1016/0022-1236(87)90020-6.
[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), 160-190. doi: doi:10.1016/0022-0396(89)90081-8.
[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), 17-51. doi: doi:10.1515/crll.1996.472.17.
[5]	L. Du and Z. Yao, Localization of blow-up points for a nonlinear nonlocal porous medium equation, Commun. Pure Appl. Anal., 6 (2007), 183-190.
[6]	S. Filippas and A. Tertikas, On similarity solutions of a heat equation with a nonhomogeneous nonlinearity, J. Differential Equations, 165 (2000), 468-492. doi: doi:10.1006/jdeq.2000.3789.
[7]	A. Friedman and J. B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: doi:10.1512/iumj.1985.34.34025.
[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), 132-135. doi: doi:10.1090/S0002-9904-1969-12175-0.
[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), 1-67. doi: doi:10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.0.CO;2-H.
[10]	N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: dynamic case, Nonlinear Diff. Eqns. Appl., 15 (2008), 115-145. doi: doi:10.1007/s00030-007-6004-1.
[11]	Y. Giga and R.V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38 (1985), 297-319. doi: doi:10.1002/cpa.3160380304.
[12]	Y. Giga and R. V. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J., 36 (1987), 1-40. doi: doi:10.1512/iumj.1987.36.36001.
[13]	Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42 (1989), 845-884. doi: doi:10.1002/cpa.3160420607.
[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).
[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), 505-514.
[16]	S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330. doi: doi:10.1002/cpa.3160160307.
[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), 207-215. doi: doi:10.1093/imamat/41.3.207.
[18]	L. A. Lepin, Countable spectrum of the eigenfunctions of the nonlinear heat equation with distributed parameters, (Russian) Differentsial'nye Uravneniya, 24 (1988), 1226-1234, (English Translation: Differential Equations, 24 (1988), 799-805.
[19]	L. A. Lepin, Self-similar solutions of a semilinear heat equation, (Russian) Mat. Model., 2 (1990), 63-74.
[20]	H. Matano and F. Merle, On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math., 57 (2004), 1494-1541. doi: doi:10.1002/cpa.20044.
[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), 992-1064. doi: doi:10.1016/j.jfa.2008.05.021.
[22]	J. Matos, Unfocused blow up solutions of semilinear parabolic equations, Discrete Contin. Dynam. Systems, 5 (1999), 905-928. doi: doi:10.3934/dcds.1999.5.905.
[23]	J. Matos, Self-similar blow up patterns in supercritical semilinear heat equations, Comm. Appl. Anal., 5 (2001), 455-483.
[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), 143-195. doi: doi:10.1215/S0012-7094-97-08605-1.
[25]	N. Mizoguchi, Boundedness of global solutions for a supercritical semilinear heat equation and its application, Indiana Univ. Math. J., 54 (2005), 1047-1059. doi: doi:10.1512/iumj.2005.54.2694.
[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), 591-610.
[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), 97-120. doi: doi:10.1016/0022-0396(84)90145-1.
[28]	W. M. Ni, Uniqueness, nonuniqueness and related questions of nonlinear elliptic and parabolic equations, Nonlinear Functional Analysis and its Applications, Part 2 (Berkeley, Calif., 1983), 229-241, Proc. Sympos. Pure Math., 45, Part 2, Amer. Math. Soc., Providence, RI, 1986.
[29]	R. G. Pinsky, Existence and nonexistence of global solutions for $u_ t=\Delta u+a(x)u^p$ in $\mathbb{R}^{d}$, J. Differential Equations, 133 (1997), 152-177. doi: doi:10.1006/jdeq.1996.3196.
[30]	P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States," Birkhäuser Advanced Texts, Basler Lehrbücher, 2007.
[31]	A. Ramiandrisoa, Blow-up profile for radial solutions of the nonlinear heat equation, Asymp. Anal., 21 (1999), 221-238.
[32]	S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation, J. Differential Equations, 246 (2009), 724-748. doi: doi:10.1016/j.jde.2008.09.004.
[33]	X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590. doi: doi:10.2307/2154232.

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