一类多维非线性抛物方程解的存在性与Blow-up
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摘要
本文研究以下非线性抛物方程的初边值问题解的存在性。其中，Ω(?)R~n为适当光滑的有界域，f∈C。为得到问题(1)-(3)整体广义解的存在性及唯一性，对σ(s)(1≤i≤n)，f(u)做如下假设
(H_1)σ_1，f∈C，(?)k，使(?)_i(s)=σ_i，(s)-ks-σ_i(0)非减，
(H_2) (?)A，A_1，及α＞0，使A_1|s|~α≤|(?)_i(s)|≤A|s|~α
(H_3) (?)B，B_1及γ，使|f(u)|≤B|u|~γ+B_1
其中
这里，k，A，A_1，B，B_1及α，γ均为常数，而当σ′_i(s)(1≤i≤n)，f′(u)有界时，存在唯一整体强解。本文还证明了对应非负初边值解的非负性，讨论了解的渐近性质及Blow-up。对一维情形，考虑了更为广泛的边界条件，证明了只要σ′(s)，f′(u)下方有界，即可得到整体强解的存在唯一性，并详细讨论了解的光滑性。
In this paper, we consider the initial boundary value problem for a class of multidimensional nonlinear parabolic equation
Ω(?)R~n is a propriate bounded domain and f∈C. And the existence of solution is established. Whenσ_i(s)(1≤i≤n), f(u) satisfy
(H_1)σ_i, f∈C, (?)k, satisfy (?)_i(s)=σ_i(s)-ks-σ_i(0) is not decreasing function,
where
k, A, A_1, B, B_1 andα,γare constant, we obtain the existence and uniqueness of the global generalized solution.Whenσ′_i(s)(1≤i≤n),f'(u)are bonuded there exists a unique global strong solution.The nonnegativity of the solution corresponding to the nonnegative initial boundary value, the asymptotic behavior and the blow-up of the solution are also discussed.For the case of one dimension, more general boundary conditions are considered;so long asσ'(s), f'(u) is bounded from below, the unique global strong solution can be obtained;furthermore, the smoothness of solution is discussed in detail.
(H_1)σ_1，f∈C，(?)k，使(?)_i(s)=σ_i，(s)-ks-σ_i(0)非减，
(H_2) (?)A，A_1，及α＞0，使A_1|s|~α≤|(?)_i(s)|≤A|s|~α
(H_3) (?)B，B_1及γ，使|f(u)|≤B|u|~γ+B_1
其中
这里，k，A，A_1，B，B_1及α，γ均为常数，而当σ′_i(s)(1≤i≤n)，f′(u)有界时，存在唯一整体强解。本文还证明了对应非负初边值解的非负性，讨论了解的渐近性质及Blow-up。对一维情形，考虑了更为广泛的边界条件，证明了只要σ′(s)，f′(u)下方有界，即可得到整体强解的存在唯一性，并详细讨论了解的光滑性。
In this paper, we consider the initial boundary value problem for a class of multidimensional nonlinear parabolic equation
Ω(?)R~n is a propriate bounded domain and f∈C. And the existence of solution is established. Whenσ_i(s)(1≤i≤n), f(u) satisfy
(H_1)σ_i, f∈C, (?)k, satisfy (?)_i(s)=σ_i(s)-ks-σ_i(0) is not decreasing function,
where
k, A, A_1, B, B_1 andα,γare constant, we obtain the existence and uniqueness of the global generalized solution.Whenσ′_i(s)(1≤i≤n),f'(u)are bonuded there exists a unique global strong solution.The nonnegativity of the solution corresponding to the nonnegative initial boundary value, the asymptotic behavior and the blow-up of the solution are also discussed.For the case of one dimension, more general boundary conditions are considered;so long asσ'(s), f'(u) is bounded from below, the unique global strong solution can be obtained;furthermore, the smoothness of solution is discussed in detail.
引文
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[30]杜心华．一类非线性波动方程混合问题整体解的存在唯一性[J]．四川师范大学学报(自然科学版)．1994，17(4)：35-42页．
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[35]查中伟．非线性发展方程混合问题解的Blow-up[J]．三峡大学学报．2002，24(3)：276-279页．
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[42]刘亚成．半线性热方程整体解的存在性与非存在性[J]．数学年刊．18A，1997，1:65-72页．
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[2] Sattinger, D H. On global solution of nonlinear hyperbolic equations. Arch Rat. Mech. Anal. 1968,30:148-172P.
[3] Levine H.A. Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Pu1 =-A(u)+F(u), Arch Rational. mech. Anal. 1973,51:371-386P.
[4] Liu Yacheng. On potential wells and vacuum isolating of solutions for semilinear wave equations, [J].Jour. Diff. Eq, 2003,192(1): 155-169P.
[5] Tsutsumi, M. Existence and non-existence of global solutions for nonlinear equations, Publ Res Inst Math Sci. 1972,8:211-229P.
[6] 刘亚成，刘萍．关于位势井及其对强阻尼非线性波动方程的应用[J]．应用数学学报．2004，27(4)：133-146页．
[7] Otani, M. On existence of strong solutions for ((du)／(dt))(t)-(?)φ~1(u(t))-(?)φ~2(u(t))(?) f(t) J. Fac. Sci. Univ. Tokyo Sect. IA, 1977,24:575-605P.
[8] Pao, C.V. Nonlinear parabolic and elliptic equations, Plenum Press, New York and London, in 1992.
[9] L.E. Payne and D.H. Sattinger. Saddle Points and Unstability of Nonlinear Hyperboic Equations[J].Israel. J. Math, 1975,22:273-M. 303P.
[10] Tsutsumi. Existence and Nonexistence of Global Solutions for Non-linearparabolic Equations[J],Publ. RIMS .Kyoto .Univ, 1972/ 73,8: 211-229P.
[11] H.A. Levine and R.A. Smith Ames. A Potential Well Theory for the Heat Equation with a Nonlinear Boundary Condition [J].Math. Meth. in the Appl. Sci. 1987,9: 127-136P.
[12] Mitsuhiro Nakao, Kosuke Ono. Existence of Global solutins to the Cauchy problem for the semi-linear dissipative wave equations[J]. Math. Z. 1993,214: 325-342P.
[13] Ryo Ikehata and Takashi Suzuki. Stable and unstable sets for evolution equations of parabolic and hyperbolic type[J]. Hiroshima Math. J. 1996,26:475-491P.
[14] Ryo Ikehata. Some Remarks on the Wave Equations with Nonlinear Damping and Source Terms[J].Nonlinear Analysis, Theory, Methods Applications, 1996,10(27): 1161-1175P.
[15] Kosuke Ono. Global Existence, Decay, and Blowing up of Solutions for some Mildy Degenerate Nonlinear Kirchhoff Strings[J].Jounal of Differential Equations, 1997,137:273-301P.
[16] Jorge Alfredo Esquivel-Avila. A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations[J].Nonlinear Analysis, 2003,52:1111-1127P.
[17] 张宏伟，呼青英．一类偶合非线性Klein—Gordon方程组的稳定集与不稳定集[J]．纯粹数学与应用数学．2002，18(3)：207-210页．
[18] 杨晗．一类非线性热传导方程混合问题整体解的存在唯一性[J]．四川师范大学学报(自然科学版)．1995，18(1)：92-96页．
[19] Fujita H. On the blowing up of solutions of the Cauchy problem for u_1,=△u+u~(1+a) [J].J. Fac. Sci. Univ. rokyo Sect. L, 1996 (13): 109-124P.
[20] Hu Maolin. Global Solution for the Quasilinear Wave Equations with Viscosity[J].Advances in Mathematics, 2001,30(2):123-132P.
[21] 闫春华，刘若慧，张桂霞．具阻尼的K1ein-Gordon方程组的整体解的存 在性[J]．天中学刊，2001，16(2)：7-页．
[22]杨林．一类非线性波动方程解的存在性[J]．云南大学学报(自然科学版)．2001，23(2)：95-99页．
[23]杨林，王晓兰．一类非线性波动方程解的唯一性、光滑性[J]．云南大学学报(自然科学版)．2001，23(3)：166-168页．
[24]谭忠．具有特殊扩散过程的反应扩散方程[J]．数学学刊，22A：2001，5：597-606页．
[25]李庆霞．一类非线性双曲方程的局部解存在性[J]．数学研究．2002，35(2)：175-180页．
[26]李庆霞，谭忠．具有耗散和阻尼的Kirchhoff型方程的整体解的存在性[J]．厦门大学学报(自然科学版)．2002，41(4)：418-422页．
[27]呼青英，张宏伟．一类非线性双曲方程整体弱解的存在性与非存在性[J]．数学研究．2002，35(1)：72-78页．
[28]张宏伟，呼青英．具阻尼的Klein—Gordon方程组整体解的存在性、衰减性和爆破性[J]．数学理论与应用．2002，22(2)：34-38，77页．
[29]杨宏志，呼青英．一类非线性发展方程组整体解的存在性和不存在性[J]．数学的实践与认识．2002，32(4)：658-663页．
[30]杜心华．一类非线性波动方程混合问题整体解的存在唯一性[J]．四川师范大学学报(自然科学版)．1994，17(4)：35-42页．
[31]王培林．具有Neumann边界及临界Sobolev指数的半线性抛物方程[J]．厦门大学学报(自然科学版)．2003，42(2)：144-147页．
[32]R．A．Adams著：叶其孝等译．索伯列夫空间[M]．北京：人民教育出版社，1981．
[33]王凡彬．广义神经传播型非线性拟双曲方程解的爆破和熄灭[J]．应用数学和力学．1996：17(11)：1039-1043页．
[34]宋长明，任华国，高桂芳．一类非线性波动方程解的爆破问题[J]．河南科学．2002：20(2)：11-113页．
[35]查中伟．非线性发展方程混合问题解的Blow-up[J]．三峡大学学报．2002，24(3)：276-279页．
[36]强晓凤，两类双曲型方程解的爆破性质[J].广东职业技术师范学院学报，2000(4)：16-19页．
[37]张健．一类双曲发展系统的爆破行为[J]．应用数学学报．1993，16(3)：428-429页．
[38]杨从军．一类拟线性退化抛物型方程的初边值问题[J]．数学学报．1995，38(1)：134-139页．
[39]Levine H A. The role of critical exponents in blow-up theorems[J].SIAM Rew. 1990, 32:262-288P.
[40]SoupletPh. Blow-upin non-local reaction-diffusion equations[J]. SIAM J.Math. Anal. 1998,29:1301-1334P.
[41]张宏伟，陈国旺．一类非线性四阶波动方程的位势井方法[J]．数学物理学报．2003，23A(6)：758-768页．
[42]刘亚成．半线性热方程整体解的存在性与非存在性[J]．数学年刊．18A，1997，1:65-72页．
[43]Badiale, M. and Tarantello, G., A sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch Rational Mech Anal. 2002, 163:259-293P.
[44]Ladyzenskaja, O. A. Solonnikov, V. A. and Ural' ceva, N. N. Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, 23, Amer Math. Soc. Providence. R. L. 1968.
[45]Garcia Azorero, J. P. and Peral Alonso, L. Hardy inequalities and some critical elliptic and parabolic problems, J. of Diff. Equs. 1990, 144: 441-476P.
[46]Ye, Q.X. and Li, Z.Y. An introduction to reaction diffusion equations, Science Press, Beijing(in Chinese), 1985.
[47]刘亚成，刘大成．一类三维非线性拟双曲方程的初边值问题．数学年刊．1988，9A(2)：213-223页．
[48]Fila, M. Boundedness of global solutions of nonlinear diffusion equations,J.of Diff.Equs. 1992,98:226-240P.
[49] Knops, R.J., Levine H.A.and Payne L.E., Nonexistence instability and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastrodynamics, Arch. Rational Mech. Anal. 1974， 55:52-72P.
[50] Aguilar J.A.and Peral I.On some nonlinear parabolic equations, preprint U.A.M. 1997.
[51] Arioli G. and Gazzola F., Some resultsonp-Laplace equations with a critical growth term, Research Report in Mathematics No.14, Dept. Math. Stockholm University,1996．
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