两类具异号非线性源项的发展方程
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摘要
研究非线性波动方程和非线性反映扩散方程的初边值问题，通过对非线性项加以增长性，连续性和有界性假设，讨论了其解的相关性质。
首先引入一族位势井W_δ以及W_δ相应的井外集合V_δ，并且给出W_δ和发V_δ的一系列性质．特别是得到了位势井族深度函数的具体分析性质．利用W_δ和V_δ我们证明一些集合在在问题流之下的不变性和解的真空隔离．然后，我们获得解的整体存在性和不存在性的门槛结果，最后我们讨论了带有临界初始条件I(u_0)≥0，E(0)=d(或者J(u_0)=d)的解的整体存在性的问题和能量正定的情况．
The initial boundary problems of the nonlinear wave equation and nonlinear reaction-diffusion equation are studied. By assuming the increasing condition, the continuity and the bound, the properties of the solutions are discussed.
Firstly the family of potential wells W_δand the outside sets V_δareintroduced then their properties are given. Especially the analysis properties of the depth function of the potential wells family are obtained. By using W_δand V_δwe prove invariance of some sets in flow of the problems and the isolate of solutions. Then we get the threshold result of global existence and nonexistence ofsolutions. Finally the critical initial conditions of I(u_0)≥0, E(0) = d (orJ(u_0) = d ) and postive energy problem are discussed.
首先引入一族位势井W_δ以及W_δ相应的井外集合V_δ，并且给出W_δ和发V_δ的一系列性质．特别是得到了位势井族深度函数的具体分析性质．利用W_δ和V_δ我们证明一些集合在在问题流之下的不变性和解的真空隔离．然后，我们获得解的整体存在性和不存在性的门槛结果，最后我们讨论了带有临界初始条件I(u_0)≥0，E(0)=d(或者J(u_0)=d)的解的整体存在性的问题和能量正定的情况．
The initial boundary problems of the nonlinear wave equation and nonlinear reaction-diffusion equation are studied. By assuming the increasing condition, the continuity and the bound, the properties of the solutions are discussed.
Firstly the family of potential wells W_δand the outside sets V_δareintroduced then their properties are given. Especially the analysis properties of the depth function of the potential wells family are obtained. By using W_δand V_δwe prove invariance of some sets in flow of the problems and the isolate of solutions. Then we get the threshold result of global existence and nonexistence ofsolutions. Finally the critical initial conditions of I(u_0)≥0, E(0) = d (orJ(u_0) = d ) and postive energy problem are discussed.
引文
[1] J M Ball. Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Quart. J. Math. 1977,28:473-486P
[2] V A Galaktionov, S I Pohozaev. blow-up and critical exponents for nonlinear hyperbolic equations. Nonlinear Analysis. 2003,53:453-466P
[3] V Geogev, G Todorova. Existence of a solution of the wave equation with nonlinear damping and source terms . Journal of Differential Equations. 1994,1(9): 295-308P
[4] R T Glassay . Blow-up theorems for nonlinear wave equations. Math. Z. 1973,32:183-203P
[5] M Grillakis . Regularity for the wave equation with critical nonlinearity. Ann. Math. 1990,132:485-509P
[6] Hiroshi Uesaka. Oscillation or nonosillation property for semilinear wave equations . Journal of Computational and Applied Mathematics. 2004,164-165:723-730P
[7] H A Levine. Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu = Au + F(u) . Trans . of Math. AMS. 1974,92:1-21P
[8] H A Levine. Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. 1974,51:138-146P
[9] J L Lions. Quelques methods de resolution des problemes aux limites non linears. Dunod Gquthier-Villars. Paris(1969).
[10] LIU Yacheng. On potential wells and vacuum isolating of solutions for semilinear wave equations. Journal of Differential Equations. 2003:192: 155-169P
[11] L E Payne, D H Sattinger. Sadie points and instability of nonlinear hyperbolic equations. Israel. J. Math. 1975,22:273-303P
[12] D H Sattinger . On global solution of nonlinear hyperbolic equations. Arch. Rat. Mech. Anal. 1968,30:148-172P
[13] J Shatah, M Struwe . Regulartiy results for nonlinear wave equations. Ann. Math. 1993,138:503-518P
[14] M Tsutsumi. On solutions of semilinear differential equations in a Hilbert space. Math. Japan, 1972,17:173-193P
[2] V A Galaktionov, S I Pohozaev. blow-up and critical exponents for nonlinear hyperbolic equations. Nonlinear Analysis. 2003,53:453-466P
[3] V Geogev, G Todorova. Existence of a solution of the wave equation with nonlinear damping and source terms . Journal of Differential Equations. 1994,1(9): 295-308P
[4] R T Glassay . Blow-up theorems for nonlinear wave equations. Math. Z. 1973,32:183-203P
[5] M Grillakis . Regularity for the wave equation with critical nonlinearity. Ann. Math. 1990,132:485-509P
[6] Hiroshi Uesaka. Oscillation or nonosillation property for semilinear wave equations . Journal of Computational and Applied Mathematics. 2004,164-165:723-730P
[7] H A Levine. Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu = Au + F(u) . Trans . of Math. AMS. 1974,92:1-21P
[8] H A Levine. Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. 1974,51:138-146P
[9] J L Lions. Quelques methods de resolution des problemes aux limites non linears. Dunod Gquthier-Villars. Paris(1969).
[10] LIU Yacheng. On potential wells and vacuum isolating of solutions for semilinear wave equations. Journal of Differential Equations. 2003:192: 155-169P
[11] L E Payne, D H Sattinger. Sadie points and instability of nonlinear hyperbolic equations. Israel. J. Math. 1975,22:273-303P
[12] D H Sattinger . On global solution of nonlinear hyperbolic equations. Arch. Rat. Mech. Anal. 1968,30:148-172P
[13] J Shatah, M Struwe . Regulartiy results for nonlinear wave equations. Ann. Math. 1993,138:503-518P
[14] M Tsutsumi. On solutions of semilinear differential equations in a Hilbert space. Math. Japan, 1972,17:173-193P