一类具对数非线性项的伪p-拉普拉斯方程的整体解和爆破的注记
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摘要
该文研究了具对数非线性项的伪p-拉普拉斯方程的初边值问题.在不同的初始条件下,得到有限时间爆破和解的渐近行为的结果.这些结果改进了Nhan和Truong~([12])中的相应结果.
We consider the initial-boundary value problem for a pseudo p-Laplacian equation with logarithmic nonlinearity. Under different initial conditions, we get the results on blow up in finite time and asymptotic behavior of solutions. These results extend some recent results by Nhan and Truong~[12].
We consider the initial-boundary value problem for a pseudo p-Laplacian equation with logarithmic nonlinearity. Under different initial conditions, we get the results on blow up in finite time and asymptotic behavior of solutions. These results extend some recent results by Nhan and Truong~[12].
引文
[1] Al'shin A B, Korpusov M O, Sveshnikov A G. Blow-up in Nonlinear Sobolev Type Equations. Berlin:Walter de Gruyter&Co, 2011
[2] Ball J M. Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Quart J Math Oxford, 1977, 28(112):473-486
[3] Cao Y, Yin J, Wang C. Cauchy problems of semilinear pseudo-parabolic equations. J Differential Equations, 2009, 246(12):4568-4590
[4] Chen H, Luo P, Liu G. Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity. J Math Anal Appl, 2015, 422(1):84-98
[5] Chen H, Tian S. Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. J Differential Equations, 2015, 258(12):4424-4442
[6] Del Pino M, Dolbeault J. The optimal Euclidean L~p-Sobolev logarithmic inequality. J Funct Anal, 2003,197(1):151-161
[7] DiBenedetto E. Degenerate Parabolic Equations. New York:Springer-Verlag, 1993
[8] Drábek P, Pohozaev S I. Positive solutions for the p-Laplacian:Application of the fibrering method.Proceedings of the Royal Society of Edinburgh:Section A Mathematics, 1997, 127(4):703-726
[9] Gopala Rao V R, Ting T W. Solutions of pseudo-heat equations in the whole space. Arch Rational Mech Anal, 1972/73, 49:57-78
[10] Ladyzenskaja O A, Solonnikov V A, Ural'ceva N N. Linear and Quasilinear Equations of Parabolic Type.Providence, RI:American Mathematical Society, 1968
[11] Martinez P. A new method to obtain decay rate estimates for dissipative systems. ESAIM Control Optim Calc Var, 1999, 4:419-444
[12] Nhan L C, Truong L X. Global solution and blow-up for a class of pseudo p-Laplacian evolution equations with logarithmic nonlinearity. Comput Math Appl, 2017, 73(9):2076-2091
[13] Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. Israel J Math,1975, 22(3-4):273-303
[14] Pino M D, Dolbeault J. Nonlinear diffusions and optimal constants in sobolev type inequalities:asymptotic behaviour of equations involving the p-Laplacian. Comptes Rendus Mathematique, 2002, 334(5):365-370
[15] Showalter R E, Ting T W. Pseudoparabolic partial differential equations. SIAM Journal on Mathematical Analysis, 1970, 1(1):1-26
[16] Xu R, Su J. Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations.J Funct Anal, 2013, 264(12):2732-2763
[2] Ball J M. Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Quart J Math Oxford, 1977, 28(112):473-486
[3] Cao Y, Yin J, Wang C. Cauchy problems of semilinear pseudo-parabolic equations. J Differential Equations, 2009, 246(12):4568-4590
[4] Chen H, Luo P, Liu G. Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity. J Math Anal Appl, 2015, 422(1):84-98
[5] Chen H, Tian S. Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. J Differential Equations, 2015, 258(12):4424-4442
[6] Del Pino M, Dolbeault J. The optimal Euclidean L~p-Sobolev logarithmic inequality. J Funct Anal, 2003,197(1):151-161
[7] DiBenedetto E. Degenerate Parabolic Equations. New York:Springer-Verlag, 1993
[8] Drábek P, Pohozaev S I. Positive solutions for the p-Laplacian:Application of the fibrering method.Proceedings of the Royal Society of Edinburgh:Section A Mathematics, 1997, 127(4):703-726
[9] Gopala Rao V R, Ting T W. Solutions of pseudo-heat equations in the whole space. Arch Rational Mech Anal, 1972/73, 49:57-78
[10] Ladyzenskaja O A, Solonnikov V A, Ural'ceva N N. Linear and Quasilinear Equations of Parabolic Type.Providence, RI:American Mathematical Society, 1968
[11] Martinez P. A new method to obtain decay rate estimates for dissipative systems. ESAIM Control Optim Calc Var, 1999, 4:419-444
[12] Nhan L C, Truong L X. Global solution and blow-up for a class of pseudo p-Laplacian evolution equations with logarithmic nonlinearity. Comput Math Appl, 2017, 73(9):2076-2091
[13] Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. Israel J Math,1975, 22(3-4):273-303
[14] Pino M D, Dolbeault J. Nonlinear diffusions and optimal constants in sobolev type inequalities:asymptotic behaviour of equations involving the p-Laplacian. Comptes Rendus Mathematique, 2002, 334(5):365-370
[15] Showalter R E, Ting T W. Pseudoparabolic partial differential equations. SIAM Journal on Mathematical Analysis, 1970, 1(1):1-26
[16] Xu R, Su J. Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations.J Funct Anal, 2013, 264(12):2732-2763